John Earman - The “Past Hypothesis” Not even false

ARTICLE IN PRESS

Studies in History and Philosophy of

Modern Physics 37 (2006) 399ndash430

1355-2198$ -

doi101016j

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wwwelseviercomlocateshpsb

The lsquolsquoPast Hypothesisrsquorsquo Not even false

John Earman

Department of History and Philosophy of Science University of Pittsburgh Pittsburgh PA 15260 USA

Abstract

It has become something of a dogma in the philosophy of science that modern cosmology has

completed Boltzmannrsquos program for explaining the statistical validity of the Second Law of

thermodynamics by providing the low entropy initial state needed to ground the asymmetry in

entropic behavior that underwrites our inference about the past This dogma is challenged on several

grounds In particular it is argued that it is likely that the Boltzmann entropy of the initial state of

the universe is an ill-defined or severely hobbled concept It is also argued that even if the entropy of

the initial state of the universe had a well-defined low value this would not suffice to explain why

thermodynamics works as well as it does for the kinds of systems we care about Because the role of

Boltzmann entropy in our inferences to the past has been vastly overrated the failure of the

Boltzmann program does not pose a serious problem for our knowledge of the past But it does call a

different explanation of why thermodynamics works as well as it does A suggestion is offered for a

different approach

r 2006 Elsevier Ltd All rights reserved

Keywords Entropy Irreversibility Cosmology Boltzmann Second Law of thermodynamics

1 Introduction

Ludwig Boltzmann bequeathed a set of concepts and techniques that lie at the core ofstatistical mechanics an essential part of modern physics He also bequeathed a set offoundational problems and puzzles that are the subjects of a debate among physicists andphilosophers that continues unabated after nearly a century following Boltzmannrsquos deathDespite the never-ending quality of the debate a remarkable consensus has developedaround one of the key foundations issues namely the grounding of the relevant temporalasymmetries in entropic behavior observed today is to be found in the fact (posit) that the

see front matter r 2006 Elsevier Ltd All rights reserved

shpsb200603002

dress jearman+pittedu

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ARTICLE IN PRESSJ Earman Studies in History and Philosophy of Modern Physics 37 (2006) 399ndash430400

early universe was in a low entropy state (the lsquolsquoPast Hypothesisrsquorsquo) To be sure this ideacomes in a variety of forms and flavors for Albert (2000) the hypothesis of a low entropystate for the early universe has something of a neo-Kantian status in that its truth is acondition needed to make our knowledge of the past possible for others (eg Penrose1979 1989 2004) the hypothesis has a (wannabe) law status and for still others thehypothesis rests on contingent facts that may or may not call for an explanation (see Price2004 vs Callender 2004)1 Leaving these and other nuances aside the agreement in thephilosophy of science community (where disagreement is the norm) is broad enough thatwe have something that can be rightly dubbed a dogmaThis dogma I contend is ill-motivated and ill-defined and its implementation consists

mainly in furious hand waving and wishful thinking In short it is (to borrow a phrasefrom Pauli) not even false2 Such heresies are apt to get one burnt at the stake of academicopinion and if that is to be my fate I want to make sure that I am burnt at the right stakeSo at the outset let me make clear that I am not denying what is a near truism namely ifsome observed temporal asymmetry is not a consequence of laws of physics then its originmust be sought in initialboundary conditions3 Nor am I denying that there is some senseof lsquolsquoentropyrsquorsquo in which it is true that the early universe was in a low entropy state What Ido deny is that for the cosmologies described by classical general relativity theory there isany well-defined sense in which the Boltzmann entropy of the early universe has a very lowvalue Furthermore I claim that even if the Boltzmann entropy for the early universe werea well-defined quantity a low value would not suffice to explain the kinds of presentlyobserved entropic asymmetries we care about And I question a long-standing tradition inphilosophy of science that accepts the idea that solving the conceptual puzzles Boltzmannbequeathed us holds the key to understanding the fundamental ontological andepistemological asymmetries between past and the futureRoughly the standard line runs that essential components of these asymmetries hinge on

the temporal asymmetry of lsquolsquorecordsrsquorsquo or lsquolsquotracesrsquorsquo that the recordtrace asymmetry isgrounded in entropic asymmetries and that these entropic asymmetries are ultimatelytraceable to the low entropy state of the early universe None of the links in this chain Icontend can bear the weight that philosophers want to put on them when the said entropyis Boltzmann entropy4 This radical sounding contention is in fact not radical at all Inparticular it does not threaten skepticism about our knowledge of the past indeed oncewe are freed of the obsessive insistence of seeing entropy as the key to ontological andepistemological aspects of temporal asymmetries it is evident that the ill-defined nature ofclaims about entropy states of the early universe in no way compromises our knowledge ofthe past However the heresy does entail the need for a new understanding of theapproximate validity of thermodynamicsThe plan of the paper is as follows In Section 2 I sketch Boltzmannrsquos explanation of the

statistical validity of the Second Law of thermodynamics and develop the apparatus thatwill be needed for the later discussion of modern cosmology Section 3 reviews some

1For other recent versions of the Past Hypothesis see Bricmont (1996) and Goldstein (2001)2While I decry Paulirsquos haughty dismissiveness his phrase seems to me to exactly fit the present case3As Boltzmann (1904 pp 170ndash171) himself puts it while affirming the antecedent lsquolsquoSince in the differential

equations of mechanics themselves there is absolutely nothing analogous to the second law of thermodynamics

the latter can be mechanically represented only by means of assumptions regarding initial conditionsrsquorsquo4Of course these links can be maintained if one is willing to work with a sense of lsquolsquoentropyrsquorsquo that is sufficiently

loose and elastic But then the relevance of Boltzmannrsquos apparatus becomes dubious

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ARTICLE IN PRESSJ Earman Studies in History and Philosophy of Modern Physics 37 (2006) 399ndash430 401

qualms about Boltzmannrsquos explanation These qualms are waived in order to concentrateon two key problems with which Boltzmann struggled the initial state problem and theasymmetry problem These problems are reviewed in Section 4 and Boltzmannrsquos resort tocosmology in order to resolve these problems is discussed in Section 5 His own version ofa cosmological solution is judged to be unsatisfying Section 6 recounts how according toreceived opinion modern general relativistic cosmology comes to Boltzmannrsquos rescue Thisalleged rescue is questioned in Sections 7 and 8 first by presenting competing intuitionpumps that yield conflicting results about the lsquolsquoimprobabilityrsquorsquo of a low entropy initial statefor the universe and then by presenting some precise model calculations indicating that inthe very cosmologies that were supposed to answer Boltzmannrsquos prayers Boltzmannentropy is an ill-defined or severely hobbled concept Section 9 adds the complaint thateven if the Boltzmann entropy of the initial state of the universe has a well-defined lowvalue this fact does not suffice to explain the temporally asymmetric behavior of the kindsof thermodynamical systems of most interest to us Section 10 argues that the dismalprospects for using cosmology to complete Boltzmannrsquos program does not threaten toundermine our inferential practices In the course of the argument I sketch an alternativeto Boltzmannrsquos program for explaining the approximate validity of thermodynamicsSection 11 reviews various interpretations of Boltzmannrsquos speculation that the direction oftime is enslaved to the entropy gradient Conclusions are presented in Section 12

2 The logic of Boltzmannrsquos explanation of the Second Law

I am not going to retell yet again the oft told story of Boltzmannrsquos failed attempt toprovide a purely mechanical grounding for the strict validity of the Second Law ofthermodynamics his (in)famous H-theorem the reversibility objection of Loschmidt therecurrence objection of Zermelo etc5 Suffice it to say that eventually Boltzmann wasforced to a conclusion which James Clerk Maxwell had reached many years earlier on thebasis of his thought experiment involving (what we now call) Maxwellrsquos Demon6 namelythat the

5Fo

receive

and va6In

degree

tumble

purely

for the

Hamil

Icari fl

human

Knott

Minimum Theorem [H-theorem] as well as the so-called Second Law ofThermodynamics are only theorems of probability The Second Law can never beproved by means of the equations of dynamics alone (Boltzmann 1895 p 414)

That the Second Law although not strictly true retains a statistical validity is easy to saybut notoriously difficult to make precise In what follows I will make no attempt to flag all

r an authoritative version of the story see Brush (1975) Foundational issues raised by the story have

d much attention in the philosophical literature for overviews see Callender (2001) Sklar (1993 1995)

n Lith (2001)

a letter to Strutt dated 6 December 1870 Maxwell wrote lsquolsquothe 2nd law of thermodynamics has the same

of truth as the statement that if you throw a tumblerful of water into the sea you cannot get the same

rful out againrsquorsquo (Strutt 1968 p 47) In a letter to Tait Maxwell mocked Boltzmannrsquos attempts to provide a

mechanical foundation for the Second Law lsquolsquoBut it is rare sport to see these learned Germans contending

priority in the discovery that the Second Law of [thermodynamics] is the Hamiltonische Princip Thetonische Princip the while soars along in a region unvexed by statistical considerations while the German

ap their waxen wings in nephelococcygia amid those cloudy forms which the ignorance and finitude of

science have invested with the incommunicable attributes of the invisible Queen of Heavenrsquorsquo Quoted in

(1911 pp 115ndash116)

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the niceties that are needed and will concentrate on the ones that bear directly on the mainissues of concern hereI will situate the discussion in a rather general and abstract mathematical setting that

will prove useful later when addressing the cosmological context A deterministic timetranslationally invariant dynamical system is a quadruple ethX mBftTHORN where X is the statespace m is a measure on X with methX THORN frac14 1 B is the set of measurable sets of X and for eacht 2 R ft X X is a onendashone map It is required that the ft are measure preserving iefor any A 2 B and for any t 2 R methftethATHORNTHORN frac14 methATHORN and that they have the group propertyft1thornt2

frac14 ft2 ft1

with ftfrac140 frac14 id and ft frac14 f1t The intended interpretation of ft is that ifat any given time ti the state of the system is x 2 X then ftethxTHORN is the state at ti thorn t7

The paradigm instantiation of these ideas is given by Hamiltonian dynamics Abstractlya Hamiltonian system is given by a 2N-dim space X a symplectic form o (ie a closednon-degenerate 2-form) and a Hamiltonian function H X R The Hamiltonian flowft is defined by the integral curves of the vector field VH on X determined by the conditionthat VHcothorn dH frac14 08and the measure m is the volume element associated with o namelyoN frac14 o ^ o ^ ^ o (N times) That ft preserves measure is the content of Liouvillersquostheorem More concretely one can consider the Hamiltonian dynamics for a system of n

particles The state space for this case is R6n Coordinates ethq pTHORN where q frac14 ethq1 q3nTHORN

records the particle positions and p frac14 ethp1 p3nTHORN records their momenta can be chosen sothat o frac14 dq1 ^ ^ dq3n ^ dp1 ^ ^ dp3n and the associated measure is oN frac14 o ^ o ^ ^ o (N times) To assure that the measure normalizes the region X R6N of the statespace available to the system has to be limited so that X has compact closure This can beguaranteed for example by confining the particles to a box preventing the box fromexchanging energy with its environment and requiring that the intra-particle interactionpotential is bounded from below For such cases the relevant measure is m lsquolsquocut downrsquorsquo to aconstant energy surfaceNext comes coarse graining in the form of the choice of a set fmag of macrostates for

describing the outcomes of measurements that can be made on the system withmacroscopic instruments It is assumed that the macrostates supervene on the microstatesie each ma corresponds to a measurable region Ma X in the sense that at any time t thesystem is in macrostate ma just in case the microstate state xt at t belongs to Ma Introducethe operations v and amp where m m v m0 and mampm0 denote respectively themacrostate that obtains if and only if m does not obtain either m or m0 obtains and m andm0 both obtain and assume that the microndashmacro correspondence satisfies the stricturesthat (i) for any m 2 fmag m corresponds to X M and (ii) for any mm0 2 fmag m v m0

and mampm0 correspond respectively to M [M 0 and M M 0 To make a probability spaceclose fmag under and under countable v-ing and amp-ings Then associated with the closurefmag

c of the coarse graining is a probability measure Pr given by PrethmTHORN frac14methMTHORN form 2 fmag

cNext comes Boltzmann entropy The Boltzmann entropy SBethmTHORN of a macrostate m is by

definition SBethmTHORN frac14k logethPrethmTHORNTHORN frac14 k logethmethMTHORNTHORN Assume that for each x 2 X there is a

7For a non-time translationally invariant dynamics the time development of a state over a time interval Dwould depend not only on D but also on the time at which the state obtains

8In canonical coordinates qa and pa o frac14 dqa ^ dpa and dH frac14 ethqH=qqaTHORNdqa thorn ethqH=qpaTHORNdpa Thus the

Hamiltonian vector field is given by VH frac14 ethqH=qpaTHORNethq=qqaTHORN ethqH=qqaTHORNethq=qpaTHORN and the flow ft is obtained by

solving Hamiltonrsquos equations _qa frac14 qH=qpa _pa frac14 qH=qqa The story is more complicated for constrained

Hamiltonian systems since o is degenerate a relevant example is encountered in Section 8

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finest macrostate mf 2 fmagc actualized by the microstate x where lsquolsquofinestrsquorsquo means that if

m0 2 fmagc is any other macrostate actualized by x then Mf M 0 Then (relative to the

chosen coarse graining) the Boltzmann entropy SBethtTHORN of the system at t is by definitionSBethtTHORN frac14SBethm

tf THORN frac14 k logethPrethmt

f THORNTHORN where mtf is the finest macrostate actualized by the

microstate at t (From here on when I speak of the macrostate at time t I will mean thefinest such state)

We are now in a position to state the Boltzmannian version of the Second Law or rathera special case that will be the focus of the discussion that is to follow

(B) Suppose that at t frac14 0 the Boltzmann entropy SBeth0THORN of the system is low then forsome appropriate t140 it is highly probable that SBetht1THORN4SBeth0THORN

The truth of (B) depends on features of the microdynamics In particular for (B) to betrue ft must be such that for the overwhelming majority of microstates x in the region M0

corresponding to the actual low entropy initial macrostate m0 at t frac14 0 the macrostate m1

at t frac14 t1 that results from the evolution x 7ft1ethxTHORN corresponds to a region M1 such that

methM1THORNbmethM0THORN The quasi-law status of (B) rests on the presumed fact that this feature ofthe microdynamics does obtain for the sorts of systems we subject to thermodynamicanalysis and for the sorts of coarse graining relevant to explaining macroscopicobservations made on these systems over time periods of length comparable to the said t1

It should be evident by now that Boltzmannrsquos downgrading of the status of the SecondLaw signaled by the phrase lsquolsquoso-calledrsquorsquo in the quotation at the beginning of this section isjustified and further justification will be supplied in the following section Or to be morecautious the downgrading is justified if the validity of the Second Lawmdashwhatever formthat validity takesmdashis supervenient on the microdynamics of the systems of interest At theend of the 19th century there were many physicists who held that the Second Law is notjust so-called but is one of the fundamental laws of nature for them the failure of statisticalmechanics to underwrite this presumed fundamental lawlike status of the Second Law wasreason to reject Boltzmannrsquos approach and to question the atomic hypothesis By the endof the first decade of the 20th century this attitude was confined to the fringes of physicsBoltzmann had won but digesting the fruits of his victory proved to be far from simple

3 Qualms about Boltzmannrsquos explanation

The difficulties in implementing Boltzmannrsquos explanation are well known and I willrehearse them only to the extent needed to set up the discussion in the following sections Asecond reason for the rehearsal is to counteract the impression given in some of thephilosophical literature that once the contraptions of Boltzmann statistical mechanics aresupplemented with some suitable Past Hypothesis they function smoothly to underwriteinferences about the past In fact the clanking sounds of these contraptions can be heardfrom afar

The special case (B) of Boltzmannrsquos formulation of the Second Law speaks of a probableincrease in the entropy of the system What is the justification for such talk In thepreceding section I explained the truth conditions of (B) by assuming that lsquolsquoprobablersquorsquo isjudged by m-measure But this definitional move leaves unexplained the connectionbetween the sense of probability so defined and the physical probabilitymdashin a propensityor a frequency sensemdashfor the entropy to increase Boltzmann realized that a connectioncould be made between m and the limiting relative frequency sense of probability if the

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system is ergodic This property can be defined in a number of equivalent ways Here aretwo

Definition A dynamical system ethX mBftTHORN is ergodic just in case (i) for any A 2 B suchthat methATHORNa0 and for almost any x 2 X there is a t40 such that ftethxTHORN Aa+ orequivalently (ii) for any A 2 B if ftethATHORN frac14 A for all t then methATHORN frac14 0 or methATHORN frac14 19

For an ergodic dynamical system m can be interpreted in terms of the limiting relativefrequency of the time the state of the system spends in a region of X

Lemma If ethX ftB mTHORN is ergodic then for any A 2 B

methATHORN frac14 limt1

R t

0 IAethftethxTHORNTHORNdt

t

for almost any x 2 X where IAethxTHORN frac14 1 if x 2 A and 0 otherwise

While some progress has been made nagging issues remain Are the physical systems ofinterest ergodic (See Earman amp Redei 1996) If not will approximate ergodicity do (SeeVranas 1998) How is limiting relative frequency of occupation time connected withexpectations about finite stretches of time I propose to waive these naggings in order topass on to two further issuesThe first is what can be termed the subjectivity challenge to Boltzmannrsquos explanation

Here is one rough attempt to get at the worry underlying this challenge The fact to beexplainedmdashthe tendency of the entropy of an adiabatically closed system to increasemdashis anobjective physical fact about the world But Boltzmannrsquos explanation (so the complaintgoes) has to do not with such objective physical tendencies but with our epistemology Thecoarse grained description in terms of macrostates is a way of codifying our ignorance ofthe exact microstate of the system and what (B) is telling us is that a certain measure ofthis ignorance tends to grow with time This may be interesting but (the complaintconcludes) it is not what was to be explained I think this is not an unfair reading ofPopperrsquos (1981) complaint in Quantum Theory and the Schism in Physics And it is suchintuitions that surely underlie Albertrsquos (1994) form of the complaint Consider twobodies one hot and one cold which are brought into contact and whose temperaturessubsequently equilibrate Then writes Albert

9Thi

Section10Se

Nothing surely about what anybody may or may not know about these two bodies can have played any role in bringing it about (that is in causing it to happen) thatthe temperatures of those bodies subsequently approached each other (Albert 1994p 670)

This subjectivity challenge can be met at least partially by a more careful specification ofthe goals of the inquiry If the goal is to derive from statistical mechanics a fully objectiveobserver independent arrow of time then one is going to be disappointed at least if theunderlying microdynamics of the system is time reversible10 But if the goal is to explainthe temporal asymmetries we observe in the entropic behavior of macroscopic systemsthen coarse graining is perfectly legitimate since what we observe corresponds to a coarse

s is the modern formulation of ergodicity For accounts of the history of this concept see Brush (1975

1010) and Gallavotti (1999 Section 19)

e Section 4 for a definition of time reversal invariance and a discussion of its implications

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Ergodinen hypoteesi (tai ergodisuushypoteesi) on fysiikassa ja termodynamiikassa oletus jonka mukaan pitkaumln ajan kuluessa systeemin tietyssauml mikrotilojen faasiavaruuden osassa viettaumlmauml aika on verrannollinen taumlmaumln osan tilavuuteen Taumlmauml tarkoittaa ettauml systeemin kaikki tilat ovat suhteellisen pitkaumln ajan kuluessa yhtauml todennaumlkoumlisiauml Taumlmauml on yhdenpitaumlvaumlauml sen kanssa ettauml systeemin ajallinen keskiarvo on yhtauml suuri kuin keskiarvo yli tilastollisen ensemblen


grained description And given an appropriate coarse graining the increase in theBoltzmann entropy is objective in the sense that it is grounded in the underlyingmicrodynamics of the system (assuming of course that the qualms raised above have beenadequately addressed) There would be a remaining gripe if the grounding were sensitive tohow the coarse graining is donemdasheg if it worked when the energy surface is partitionedinto areas of 1

10erg but not when it partitioned into areas of 1 erg But in fact the grounding

appears to be robust under such changes11

Another worry focuses on the conditions of applicability of (B) to concrete physicalsystems One such condition is that for the system of interest the microstate at t frac14 0 will belsquolsquotypicalrsquorsquo of the microstates compatible with the observed macrostate at that time Theidea can be made more precise in the following Statistical Postulate

11Bu

subjec

space r

be rep

(SP) Let m0 be the macrostate at t frac14 0 of the system of interest Then the probabilityat t frac14 0 that the microstate of this system lies in some measurable subset A M0 ofthe state space region M0 X corresponding to m0 is methATHORN=methM0THORN

It might seem that if (B) holds and if (SP) is true at t frac14 0 then (SP) cannot be true at the t1used in (B) From (B) it follows that for the overwhelming majority of microstates x in theregion M0 the macrostate m1 at t frac14 t1 that results from the evolution x 7ft1

ethxTHORN

corresponds to a region M1 such that methM1THORNbmethM0THORN By conservation of measuremethM0THORN frac14 methft1

ethM0THORNTHORN So a fortiori for the overwhelming majority of microstates in M0 theresulting macrostate m1 at t frac14 t1 is such that methM1THORNbmethft1

ethM0THORN M1THORN ie microstates inM1 that have evolved from M0 are only a small m-fraction of the microstates of M0 Thusit seems that the microstate at t1 cannot be lsquolsquotypicalrsquorsquo of the microstates compatible withthe macrostate at t1 And now the Boltzmannian explanation threatens to unravel Fort frac14 0 was not supposed to denote some special time but an arbitrary initial time And if(SP)rsquos being true at some arbitrary initial time implies that it is false at a later time howcan (SP) reasonably be assumed to apply to the time t frac14 0 referred to in the antecedentof (B)

One response is that although of course there is nothing special about the time t frac14 0itself there is something special about the initial state namely it is part of the applicabilityof (B) that we have no information about the initial microstate other than that implied bythe observed fact that the system is in the macrostate m0 And given this state ofknowledge a principle of indifference argument can be used to justify the assumption thatthe initial microstate is equally likely to lie in equal m-volumes of M0 This response whilenot unreasonable is prey to the subjectivity complaint of Popper and Albert

The best response does not appeal to an ignorance interpretation of probability butrather exploits a non-sequitur in the argument that was supposed to show that if (B) holdsand if (SP) is true at t frac14 0 then (SP) cannot be true at t1 The fact that the macrostate m1

at t1 is such that the microstates in M1 that have evolved from the phase space regioncorresponding to the initial macrostate at t frac14 0 are only a small m-fraction of themicrostates of M1 does not entail that the microstate at t1 cannot be typical of themicrostates compatible with the macrostate at t1 for the sense of typicality relevant to

t (as emphasized to me by a referee) Boltzmannrsquos approach cannot be completely absolved of the

tivity charge For example the approach depends on taking the state space to be the momentum-position

ather than the velocity-position space and the approach has to be limited to macroscopic variables that can

resented by measurable phase functions


(SP) it suffices that the microstates that have evolved from the initial macrostate aresufficiently spread over the measurable subsets of M1 And for the needed spreading onecan appeal to a cousin of the property of mixing which is stronger than the property ofergodicity

Definition A dynamical system ethX mBftTHORN is mixing just in case for any AB 2 Blim

t1methftethATHORN BTHORN frac14 methATHORNmethBTHORN

Rather than the asymptotic property of the formal definition what is needed in supportof the reapplication of (SP) at t140 is for effective mixing to take place over the relevantmacrotime scale for those phase space regions corresponding to the macrostates at issue Inparticular it would suffice if for any macrostate m 2 fmag and any measurable subsetF Ma of any state space region Ma corresponding to a macrostate ma methft1

ethMTHORN

F THORN methMTHORNmethF THORN frac14 methft1ethMTHORNTHORNmethF THORN where the approximate equality is tight enough to

accord with what is macroscopically discriminable This effective mixing guarantees thatwhatever the initial macrostate m0 at t frac14 0 and resulting macrostate m1 at t140 the phasepoints that originate in M0 appear on the macrolevel to be uniformly spread over theregion M1 at t140Doubts can be raised about whether effective mixing can be demonstrated for the

relevant physical systems But the point I want to emphasize is independent of such doubtseffective mixing succeeds in supporting (SP) only by undermining the Boltzmannianapparatus as a means of making predictions about the macroscopic behavior of the systemFor effective mixing means that insofar as our rational expectations are governed by themeasure m the information value of knowing that at t frac14 0 the system is in a certainmacrostate m0 is effectively lost over the relevant time scale since the phase points thatoriginate in M0 are effectively uniformly spread over all the phase space regionscorresponding to macrostates at issueTo allow for knowledge of macroscopic states to have predictive power while also

providing for (SP) one could hope for a Goldilocks form of effective mixing which can beformulated as follows Let mj j frac14 0 1 2 N be any sequence of macrostatescompatible with the microdynamics with m0 being a low entropy state with the timesof occurrence of successive states differing by an amount t1 and with Nt1 being somerelevant length of observation time Then Goldilocks mixing requires that for all jmethft1ethMjTHORN Mjthorn1THORN=methMjTHORN 1 and moreover for any measurable Fjthorn1 Mjthorn1

methft1ethMjTHORN F jthorn1THORN methMjTHORNmethF jthorn1THORN In words the vast majority of phase points originating

in the region Mj corresponding to the macrostate mj evolve over a time span t1 to theregion Mjthorn1 corresponding to mjthorn1 and also appear to be spread uniformly over Mjthorn1This Goldilocks form of effective mixing requires the dynamics to be balanced betweenenough sensitive dependence on initial conditions to spread the phase points initially in Mj

over Mjthorn1 vs enough stability needed to avoid non-Goldilocks mixing that wouldundermine the predictive and retrodictive power of macrostates It remains to be seenwhether the systems to which thermodynamic reasoning is applied have a microdynamicsthat performs this balancing actThese qualms about the logic of Boltzmannrsquos program will be waived because they will

seem minor when compared to the initial state and asymmetry problems to be discussed inthe following section


4 The initial state problem and the asymmetry problem

Boltzmannrsquos (B) has the form of a conditional whose antecedent asserts that theBoltzmann entropy is low at t frac14 0 Two questions immediately arise First what is theorigin of the initial low entropy state that instantiates the antecedent of (B) Second is itreasonable to assume that the low entropy initial state satisfies the Statistical Postulate Iwill concentrate on the first question

In some instances the answer may simply be that that is the way a human agent arrangedthings eg at t frac14 0 an experimenter places a cold body in thermal contact with a hot bodyand thermally isolates this two-body system or the experimenter evacuates the gas in onechamber of a box divided into half by a partition and then at t frac14 0 removes the partitionetc But obviously such an answer does not extend to events that were not engineered aspart of an experiment Before considering answers that cover events not subject to humanintervention it will be helpful to get a second issue on the table

The second issue arises from the recognition that (B) does not capture in probabilisticformat the temporal asymmetry of the Second Law That requires something like

12Se13Se

(By) Suppose that at t frac14 0 the Boltzmann entropy SBeth0THORN of the system is low then forsome appropriate t140 it is highly probable that SBetht1THORN4SBeth0THORN and it is highlyprobable that SBeth0THORNXSBetht1THORN

But (By) is not a consequence of the Boltzmann apparatus Indeed given the presumedtime reversal invariance of the microdynamics one would expect that if (B) is true itshould also be true that

(Bn) Suppose that at t frac14 0 the Boltzmann entropy SBeth0THORN of the system is low then forthe t140 of (B) it is just as probable that SBetht1THORN4SBeth0THORN as that SBetht1THORN4SBeth0THORN

But (Bn) contradicts (By) And it also contravenes our (presumably) rational expecta-tions12 If for example at t frac14 0 we observe a thermally isolated system consisting of an icecube floating in a glass of lukewarm water we expect in accord with (B) that if there is nointervention over (say) the next 5min then at the end of that time the ice cube will havepartially melted and the water will have cooled correspondingly And further in accordwith (By) and contra-(Bn) we infer that if there was no intervention over the preceding5min then 5min earlier the ice cube was less melted and the water was correspondinglywarmer And more generally to the extent that memories and records of the past dependon entropic behavior the truth of (Bn) seems to undermine our knowledge of the past sinceit seems to imply that what we take to be memories and records are imposters because theyare not lingering traces of past events but rather the results of fluctuations from higherentropy states13

Because of the importance of this issue it is worthwhile examining in some detail theassumptions that go into generating (Bn) Time reversal invariance for a dynamical systemethX mBftTHORN is explicated as follows First there is reversal operation on states givenformally by a onendashone mapping R X X denoted by Rx having the involutionalproperty that Reth

RxTHORN frac14 x For the main application Hamiltonian particle mechanicsx frac14 ethq pTHORN where q denotes the instantaneous positions of the particles and p denotes their

e however the discussion in Section 10

e Albert (2000 Chapter 4) for an engaging discussion on this point


instantaneous momenta the reversal operation is just reversal of momenta ieRethq pTHORN frac14 ethqpTHORN14 Time reversal invariance for the dynamics means that if the systemevolves from an initial state xi to a final state xf frac14 fDethxiTHORN in a time span D then in the sametime span the reversed state Rxf evolves to Rxi frac14 fDeth

Rxf THORNSuppose that the Boltzmann apparatus is used to calculate the conditional probability

that a system initially in macrostate mi at t frac14 0 will evolve to a macrostate mf at t frac14 D40as

Prethmf ethDTHORN=mieth0THORNTHORN frac14 methfDethMiTHORN Mf THORN=methMiTHORN (1)

Make the following assumptions

(A1)

14In

(200015Al

the microdynamics is time reversal invariant

(A2)

for any Y 2 B methY THORN frac14 methRY THORN where RY frac14fx 2 X Rx 2 Y g

(A3)

for any macrostate m 2 fmag in the coarse graining if M X is the region

corresponding to m then the region RM corresponds to a macrostate Rm 2 fmag

Assumption (A3) allows us to say that a macrostate m is reversal invariant just in caseRm frac14 m Then if the initial macrostate mi is reversal invariant assumptions (A1)ndash(A2)yield

Pr ethRmf ethDTHORN=mieth0THORNTHORN frac14 methfDethMiTHORN RMf THORN=methMiTHORN

frac14 Prethmf ethDTHORN=meth0THORNTHORN eth2THORN

The proof of the second equality in (2) simply involves mechanical manipulations and isleft to the reader Since by (A2) methRMf THORN frac14 methMf THORN the macrostates mf and Rmf have thesame Boltzmann entropies Thus it follows from (2) that if the initial macrostate mi of thesystem has low Boltzmann entropy the system is as equally likely to have come from ahigher entropy state as it is to be headed towards a higher entropy stateAssumption (A2) can be motivated by the idea that the measure m should be adapted to

the symmetries of the dynamics In any case (A2) does hold for Hamiltonian mechanics(A2) would not need to be invoked in the proof of (2) if Rm frac14 m holds for all macrostatesand not just for the initial macrostate15 Whether or not this condition holds depends inpart on what is packed into the notion of a macrostate If instantaneous rates of change ofmacroparameters are admitted as part of the macrostate description then the verdict isnegative Consider for example the macrostate m of a gas in which the instantaneousrate of change of volume is positive here Rmam since the microstates in RM are those ofa gas whose volume is contracting rather than expanding On the other hand if themacrostate is defined solely by instantaneous values of macroparameters like volumeand pressure but not their instantaneous rates of change then the verdict is probablypositive It seems that nothing essential is lost by taking the latter route since the change inthe macroparameters can be captured by a succession of macrostates meth0THORNm0ethTHORNm00eth2THORN for some appropriate small 40 The result (2) can be generalized to show for examplethat if Rm frac14 m and Rm0 frac14 m0 then Prethm00eth2THORN=meth0THORNampm0ethTHORNTHORN frac14 Pr ethRm00eth2THORN=meth0THORNampm0ethTHORNTHORNBut note that it does not follow from time reversal invariance alone that

other applications the definition of the reversal operation can be a matter of contention compare Albert

Chapter 1) to Earman (2002) and Malament (2004)

bert (2000 Chapter 4) assumes that (A2) does hold for all macrostates


Prethm00eth2THORN=meth0THORNampm0ethTHORNTHORN frac14 Pr ethRm00eth2THORN=meth0THORNampm0ethTHORNTHORN Even though the apparatus is timesymmetric if time asymmetric information is put in time asymmetric conclusions cancome out

5 Boltzmannrsquos cosmological solutions to the initial state and asymmetry problems

Here is one possible solution to the initial state problem (For future reference call it(S1)) Suppose that the dynamical system of interest is ergodic Then the eventualoccurrence of low entropy states is virtually assured for ergodicity guarantees that foralmost every dynamical trajectory and for any macrostate even one with a very lowentropy there will be a time at which the microstate actuates the said macrostateFurthermore it follows from Poincarersquos recurrence theorem that the low entropymacrostate will in the fullness of time occur again and again16 The fact that we happen tobe living during an epoch that began with a low entropy state requires explanation butperhaps it suffices to use a form of anthropic reasoning viz the fact in question should notbe regarded as surprising since a sufficiently low entropy is required for the existence ofcritters like ourselves17 To my knowledge Boltzmann nowhere explicitly considers (S1)but it is implicit in the entropyndashtime graphs he draws

His own earliest attempted solution to the initial state problem (call it (S2)) appealed notto ergodicity and the fullness of time but to cosmology and the vastness of the universeThe first published version is found in a letter to Nature The account begins with theansatz that lsquolsquothe whole universe is and rests forever in thermal equilibriumrsquorsquo

16Po

system

for som

Boltzm

and B17I w18Bo

The probability that one (only one) part of the universe is in a certain state is thesmaller the further this state is from thermal equilibrium but this probability isgreater the greater is the universe itself If we assume the universe great enough wecan make the probability of one relatively small part being in any given state(however far from the state of thermal equilibrium) as great as we please We can alsomake the probability great that though the whole universe is in thermal equilibriumour world is in its present state (Boltzmann 1895 p 415)18

This ideamdashthat the vast universe which is in thermal equilibrium is composed of manylsquolsquoworldsrsquorsquo the size of our galaxy and that as a result of fluctuations some of them findthemselves in an initially lsquolsquoimprobablersquorsquo (ie low Boltzmann entropy) statemdashis repeatedseveral times (see Boltzmann 1896ndash1898 Section 90 1897a Section 7 1897b Sections4ndash5 1904 pp 171ndash172) That our own world is one of these can perhaps be explained onanthropic grounds

In at least two places Boltzmann offers (S2) as one of two possibilities the other (call it(S3)) being that the entire universe began in a low entropy state that ever since the increasein entropy has been monotonic and that the increase has been sufficiently slow as to leavethe present value of entropy well below its maximum (see Boltzmann 1897a Section 7

incarersquos recurrence theorem applies to any dynamical system ethX mBftTHORN regardless of whether or not the

is ergodic The theorem shows that for almost any x 2 X and any A 2 B such that x 2 A and methATHORN40 if

e t it is the case that ftethxTHORN A frac14+ then there is a t04t such that ftethxTHORN Aa+ Zermelorsquos objection to

annrsquos H-theorem was based on Poincarersquos recurrence theorem English translations of Zermelorsquos papers

oltzmannrsquos responses can be found in Brush (1966)

ill not broach the issue of whether anthropic explanations are genuine explanations or mere nostrums

ltzmann attributes this idea to his lsquolsquoold assistant Dr Schuetzrsquorsquo


1897b Section 4ndash5) Boltzmann obviously did not find (S3) attractive and it is not hard toimagine why Classical spacetime theories offer no provision for a beginning of time itselfsave for the artificial one of deleting all spacetime points on or below some t frac14 const timeslice In a spacetime where time extends back to t frac14 1 a beginning for the materialuniverse can be contemplated as the creation ex nihilo at some finite time in the past of thematter that makes up the constituents of Boltzmannrsquos lsquolsquoworldsrsquorsquo Neither alternative is veryprepossessing (Had he lived another two decades Boltzmann would have seen how generalrelativistic cosmological models escape between the horns of these unattractivealternativesmdashsee Section 6) But leaving aside the issue of what a beginning of theuniverse is supposed to mean in a classical spacetime setting (S3) is obviously unattractivein comparison with (S1) and (S2) since the Boltzmann apparatus suggests no mechanismfor grounding (S3) adopting (S3) has the appearance of a deus ex machinaAs far as I know Boltzmann never considered a fourth possible solution (call it (S4))

which amounts to a denial of the presupposition of the initial state problem namely thereis no lsquolsquoinitialrsquorsquo low entropy state for our lsquolsquoworldrsquorsquo or for the universe it is rather that in thepast direction there is monotonic decrease in entropy from its present value towards aminimum at t frac14 1Let us now examine Boltzmannrsquos favored alternative (S2) There is no reason to doubt

that in a large universe in thermal equilibrium relatively small parts will have low entropyat least in an intuitive sense that is some parts of the universe will display properties suchas macroscopic temperature gradients andor matter density gradients that are indicativeof thermodynamical non-equilibrium in which case one can say that these parts have lowentropy in some intuitive sense of that term But assumptions are needed to connect thisintuitive sense of entropy with the Boltzmann sense For Boltzmann entropy is a non-localquantitymdashit does not attach directly to spatial or spatiotemporal regions but rather tomacrostates of a dynamical system Presumably in suitable circumstances the Boltzmannentropy of a dynamical system will lend itself to a description in terms of a local entropydensity and an entropy flux19 Grant that these circumstances obtain for the fluctuationthat creates the initially low entropy for our lsquolsquoworldrsquorsquo Even so there is an apparently fatalobjection to Boltzmannrsquos (S2)Feynman (1994) dismisses Boltzmannrsquos fluctuation story as lsquolsquo ridiculousrsquorsquo Suppose for

sake of reductio that a fluctuation out of an equilibrium state for the universe produces anorderly patch in our neighborhood Then (according to Feynman) the most likelycondition of the rest of the universe is that it is lsquolsquomixed uprsquorsquo Thus he concludes lsquolsquotheprediction would be that if we looked at a place where we had not looked before it wouldbe disordered and a messrsquorsquo (p 115) But this prediction is falsified by what we actuallyobserve In response Boltzmann could appeal to a fluctuation large enough to encompassthe portion of the universe visible to us Such a fluctuation however is much less probablethan one which encompasses only our immediate neighborhood The logical extension ofthis line of reasoning leads to what has become known as Boltzmannrsquos brain paradox viz afluctuation that encompasses our neighborhood and produces the orderly patterns weperceive here is much less likely than a fluctuation that simply creates Boltzmannrsquos brain

19Alternatively the fluctuation can be coupled with the creation of a lsquolsquobranch systemrsquorsquo which is sufficiently

isolated that to good approximation it can be treated as a closed dynamical system in its own right Then the low

entropy initial state refers to the Boltzmann entropy of this branch system at the time of branching off But

obviously such a scenario requires additional assumptions


(or your brain or mine) complete with the faux lsquolsquomemoriesrsquorsquo of macroscopic objectsbehaving in accord with the laws of thermodynamics20 The upshot is that the position bestsupported by Boltzmannrsquos fluctuation story is solipsism of the present moment21

Boltzmannrsquos solution to the initial state problem cannot be discussed independently ofthe asymmetry problem because his solution to the former was supposed to do double dutyby also providing a solution to the latter This is made clear in Boltzmannrsquos Lectures on

Gas Theory

20Th

Tipler21It

likelyrsquorsquo

Sorbo

That in nature the transition from a probable [high Boltzmann entropy] to animprobable [low Boltzmann entropy] state does not take place as often as theconverse can be explained by assuming a very improbable initial state of the entireuniverse surrounding us in consequence of which an arbitrary system of interactingbodies will in general find itself initially in an improbable state (Boltzmann1896ndash1898 p 447)

But the solution to the initial state problem provides a solution to the asymmetry problemonly if additional assumptions are satisfied Suppose for sake of discussion that solipsismof the present moment is rejected out of hand and grant that our current perceptions andmemories of relevant temporal asymmetriesmdasheg that the temperatures of bodies inthermal contact tend to equalize rather than to become more disparatemdashare not illusionsThen the lsquolsquoinitial conditionsrsquorsquo at some time in the past must be favorable and if theBoltzmann apparatus provides the correct explanatory machinery then being lsquolsquofavorablersquorsquomeans that the initial state had low Boltzmann entropy That is not an hypothesis but adeduction What is an hypothesis by Boltzmannrsquos lights is that the time of occurrence ofthe lsquolsquoinitialrsquorsquo state is sufficiently far in the past that the inferences we make about the pastdo not overreach it We surely want to make inferences about events that predate ourearliest memories (which are stipulated to be veridical) and so the lsquolsquoinitialrsquorsquo state mustpredate them There was nothing in the cosmology of Boltzmannrsquos day to give anytheoretical backing to this hypothesis and so for him it was simply a posit One can try todignify this posit by calling it a presupposition of our knowledge of a past more distantthan the reach of our memories but that begs the question of whether this allegedknowledge deserves the name Nor will anthropic reasoning fill the gap for it is no skin offthe nose of any plausible anthropic principle that a low Boltzmann entropy state thatpredates the reach of our memories is the product of a downward fluctuation from a higherentropy state rather than the result of evolution from a state of even lower entropy

Furthermore there is an obvious tension in Boltzmannrsquos solution to the initial state andasymmetry problems To make sure that our inferences to the past do not overreach thetime of the lsquolsquoinitialrsquorsquo low entropy state the tendency is to push this state into the far pastBut the further into the past it is pushed the less it is able to guarantee that the entropy ofthe present state though higher than that of the lsquolsquoinitialrsquorsquo state is nevertheless itself lowThat guarantee requires another posit In his second reply to Zermelo Boltzmannadmitted that the solution to the asymmetry problem requires additional posits over and

is lsquolsquoBoltzmannrsquos brain paradoxrsquorsquo is discussed by a number of authors It is often attributed to Barrow and

(1986) but the basic idea may have been due to Bronstein and Landau (1933 p 72)

is interesting to note that there is still a lively controversy in cosmology over how to calculate the lsquolsquomost

outcome of a fluctuation compare for example Dyson Kleban and Susskind (2002) with Albrecht and

(2004)


above those involved in his fluctuation hypothesis His own formulation of the neededposit is labeled as Assumption A lsquolsquothat the universe considered as a mechanical systemmdashorat least a very large part of it which surrounds usmdashstarted from a very improbable stateand still is in an improbable statersquorsquo (Boltzmann 1897b pp 238ndash239) In effect Boltzmannhas been forced to move to something close to (S3) He characterizes Assumption A as alsquolsquocomprehensible physical explanation of the peculiarity of the initial state consistent withthe laws of mechanics or better it is a unified viewpoint corresponding to these lawswhich allows one to predict the type of peculiarity of the initial state in any special case forone can never expect that an explanatory principle must itself be explainedrsquorsquo (Boltzmann1897b p 239) This rhetorical flourish cannot disguise the flavor of a solution gained bytoo many posits and not enough honest toil22

This embarrassing predicament is presumably what led Boltzmann to entertain aproposal that would obviate the need for so many posits This proposal will be discussed inSection 11 But first I want to take up the contention that modern cosmology rescuesBoltzmann from the predicament by turning the posits into facts

6 Modern cosmology to the rescue ()

The standard big bang models of modern general relativistic cosmology imply abeginning for time in the sense that there is an upper bound on the proper length of anypast-directed timelike curve as measured from any point of the spacetime For the actualuniverse the bound is thought to be about 14 billion years for curves starting at the presentmoment Moreover these models cannot be extended through the big bang singularity inany physically meaningful way (eg Einsteinrsquos gravitational field equations are notmeaningful even in a distributional sense at this singularity) Of course in these modelsthere is no beginning for time in the sense of a lsquolsquofirst instantrsquorsquo But this does not matter inthe present context since for purposes of solving the initial state and asymmetry problemsthe lsquolsquoinitial statersquorsquo can be taken as the state at some time or other in the very early universeThe choice of the exact time does not matter as long as the chosen state has the requisitelow entropy and the Statistical Postulate (SP) applies Then the measure on which to basestatistical reasoning about the thermodynamics of the universe would seem to bemI ethTHORNfrac14meth=ITHORN where m is the appropriate measure on the state space for the universe and I isthe region of the state space compatible with the low entropy initial macrostate state of therelevant coarse graining The hope is that this conditionalized measure will vindicate thetemporally asymmetric inferences we routinely make about thermodynamic systemsBefore going further it is well to note that applying of the Second Law of

thermodynamics to the universe is a problematic exercise Standard cosmology describesthe large scale structure of the universe using the FriedmannndashRobertsonndashWalker (FRW)models23 Current observational data are consistent with all three types of FRW modelsk frac14 0 1 and thorn1 whose space sections have respectively zero curvature constantnegative curvature and constant positive curvature In the first two cases space is infinite24

22Note also that in order to get the Boltzmann apparatus into gear the initial state must satisfy the Statistical

Hypothesis (SP) as discussed in Section 3 requiring yet another posit23The line element can be written as ds2 frac14 aethtTHORNfrac12dr2=eth1 kr2THORN thorn r2dy2 thorn r2 sin2 ydj2 dt2 where aethtTHORN is the

scale factor and k frac14 01 or thorn124Unless topological identifications are made to make the space sections compact


and as a result such thermodynamic notions as the energy of the system the work done bythe system etc are problematic if the system is the entire universe25 To overcome thisdifficulty one could try to apply the Second Law to a finite lsquolsquocomoving patchrsquorsquo lying withinthe portion of the universe observable by us and argue that although strictly speaking thispatch is not an adiabatically closed system the observed homogeneity of the universeindicates that there is little net entropy flux across the boundary of the patch (see Carroll ampChen 2004) But the universe is homogeneous only at large scalesmdashabove 10 megaparsecsmdashand at the smaller scales at which we typically apply thermodynamic reasoningthe inhomogeneities can be expected to be associated with entropy gradients Additionallyif the expansion of the universe is acceleratingmdashas current evidence strongly indicates thatit is (see Carroll 2004)mdashthen the application of thermodynamic reasoning to a volumecomoving with the expansion is complicated by the fact that the energy of the matter fieldsin the comoving volume increases with time26 But as usual in these discussions let us agreeto suspend doubts about such niceties

Nevertheless it would seem at first blush that very early states of the standard hot bigbang models do not answer the call for a low entropy state on the contrary a thermalizedstate (needed to explain the abundances of the light elements) with a very uniformdistribution of mass-energy would seem to have high entropy instead of the requisite lowentropy The usual response is that our intuitions are misled by the failure to take intoaccount the entropy associated with the gravitational degrees of freedom Since gravitationtends to make matter clump the finer the mesh of the coarse graining the more lsquolsquospecialrsquorsquoor lsquolsquounlikelyrsquorsquo a macrostate in which matter-energy is uniformly distributed on the scale setby the coarse graining At this juncture it is standard in the literature to quote Penrosersquos(1989 Chapter 7) estimate that for a k frac14 thorn1 (spatially closed) FRW model theunlikeliness of an appropriate smooth macrostate is 1 part in 1010

123

It has to be emphasized that such numbers are the result of purely heuristic

considerations and that no connectionmdashother than a purely verbal onemdashto the Boltzmannapparatus has been made In terms of Boltzmann entropy the quoted number would haveto be taken to mean that the ratio of the state space volume corresponding to the initialsmooth macrostate to the total volume is 1010

123

27 But the basis for establishing thismeaningmdashthe definitions of the state space the measure m the dynamics and the coarsegrainingmdashare not given This might seem like a quibble and an unproductive one at thatTheoretical physicists routinely give heuristic order of magnitude estimates and it oftenturns out after a long and hard technical analysis that the estimates of the first-ratephysicists are roughly correct Thus it might be urged we should not worry about whetherPenrosersquos estimate is off by a factor of 1010 or so but rather should get on with digestingthe consequences of the extraordinary improbability of the initial state My qualm is notbased on a quibble over a factor of 1010 but rather on a worry about whether any value forthe Boltzmann entropy of the initial state is meaningful This worry will be given concreteform in Section 8 For now I want to enter into the spirit of the discussions which takePenrosersquos estimates at face value

25If gravitational energy is taken into account then in cases where space is compact one meets the difficulty that

classical general relativity supplies no well-defined notion of the energy of the entire universe26In a FRWmodel the source matter takes the form of a perfect fluid characterized by an energy density m and a

pressure p The energy of a volume element comoving with the expansion scales as a3wethtTHORN where w frac14m=p

Accelerating expansion requires that wo 13

27See for example Lebowitz (1993 p 13 1999 p S350) who puts this gloss on Penrosersquos estimate


That spirit is captured by the many approving references to Penrosersquos estimatemdashfromexperts in statistical mechanics (eg Lebowitz 1993 1999) and cosmology (eg Albrecht2004) as well as philosophers of science (eg Price 1996 p 83) A somewhat over-the-topexample from philosophy of science is Pricersquos (2004 p 228) remark that lsquolsquoIn my view thisdiscovery about the cosmological origins of low entropy is the most importantachievement of late-twentieth-century physicsrsquorsquo 28 Although other authors may not sharePricersquos importance ranking they typically agree that a low entropy initial state is adiscovery of modern cosmology and then turn to the issue of whether the specialness ofthe initial state needs explaining and if so what form that explanation should take Thefollowing section briefly reviews the controversy this issue has generated and argues that aresolution of the controversy is not to be obtained by means of intuition pumps but ratherthrough precise model calculations Such calculations threaten to undermine the notionthat modern cosmology completes Boltzmannrsquos program

7 Competing intuition pumps

The program of explaining initial conditions of the universe does not fit with themethodology of other branches of physics outside of cosmology indeed the notion thatvarious branches of physics are incomplete because they fail to offer lsquolsquotheories of initialconditionsrsquorsquo is apt to strike the practitioners in these branches as bizarre Perhaps there issomething special about cosmology that calls for a theory of initial conditions but if so itshould be based on something more substantial than the notions that the big bangrepresents a lsquolsquocreationrsquorsquo of the universe and that it is legitimate if not obligatory to inquirewhy the creation went this way rather than another The conceit of a blindfolded Creatorwho actualizes a universe by randomly throwing a dart at the cosmic dartboard of initialconditions for the universe might be useful for some purposes if for example it is takenseriously it provides an explanation of why the Statistical Postulate (SP) is satisfied by theinitial state But arguably this advantage is not worth the price of the metaphysicalbaggage used to secure it In sum there is much to be said for the position that noexplanation of the initial conditions of the universe is called for29

Another possible positionmdashwhich does resort to a theory of initial conditionsmdashis thatthe seeming specialness of the initial state of the universe is an illusion This state is specialin the comparison class of logically possible states But for physics the relevant comparisonclass is the class of physically possible states and this class (so the theory of initialconditions goes) does not include the seemingly generic lumpy initial states which areruled out by a lawlike constraint on initial conditions Penrose (1979 1989 Chapter 72004 Chapter 28) has hypothesized that there is a time asymmetric (local) physical lawthat only becomes important near spacetime singularities and that forces a smoothspacetime geometry at the big bang but not at the big crunch (should there be one as in thek frac14 thorn1 FRW model without cosmological constant) His tentative proposal for trying tomake this hypothesis precise is to take the time asymmetric law to require that the Weylcurvature tends to zero as the initial singularity is approached The details of Penrosersquos

28See also Price (2002)29For a forceful argument for this position see Callender (2004)


proposal do not matter for present purposes30 What does matter is that if his proposal ison the right track it seemingly undermines the lsquolsquodiscoveryrsquorsquo that modern cosmologyprovides the answers to Boltzmannrsquos prayers by providing an improbable or low entropyinitial state31

Inflationary cosmology offers yet a different perspective on the issue of the initial stateof the universe by claiming to provide a mechanism that obviates the need for a specialbeginning state It allows that the universe began in a generic lumpy state and postulatesthat subsequently a period of very rapid accelerated expansion (aka inflation) whichinitiated around 1035 s after the big bang effectively smoothed out the lumpiness32 Sincethe inflationary era is hypothesized to have lasted only a fraction of a second the time ofthe lsquolsquoinitial statersquorsquo can without any awkwardness for the Boltzmann program be taken tocoincide with the end of the inflationary era33 Work in inflationary cosmology wasinitially stimulated by the promise of tying cosmology to particle physics by linking thehypothesized lsquolsquoinflation fieldrsquorsquo that is supposed to drive inflation to a specific mechanismin elementary particle physics To date this promise has not been fulfilled and thelsquolsquoinflation fieldrsquorsquo is just a name for a scalar field that does what it has to do to produce theright kind of inflation at the required time But the program has been successful in offeringa natural explanation of the nearly scale free spectrum of density perturbations revealed bymeasurements of the cosmic microwave background radiation Critics of inflationarycosmology respond that the fundamental mechanism used by inflationary cosmologists intheir explanation of the spectrum of density perturbations may operate whether or notinflation occurs (see for example Hollands amp Wald 2002a) Such issues are outside thescope of this paper But what is relevant is that as with Penrosersquos solution to the cosmicdartboard problem the inflationary solution threatens Boltzmannrsquos program

If inflation is indeed an effective smoothing mechanism then it might seem to be an anti-thermodynamic machine that reduces Boltzmann entropy by taking a pre-inflation lumpymacrostate corresponding to a sizable volume of state space and evolving it into a post-inflation era smooth macrostate corresponding to a much smaller volume of phase spaceThis in itself need cause no alarm for the Boltzmann program since as noted above thelsquolsquoinitial statersquorsquo can be identified with a post-inflation state But what should cause alarm isthe recent discovery that the universe has entered a new era of inflation ( frac14 acceleratingexpansion) albeit of a less intense form than in the very early universe This is a genuinediscovery for which there are multiple independent lines of evidence based on differentobservational techniques and background assumptions

30How fast does the Weyl curvature go to zero as the initial singularity is approached does the approach to zero

take place in a parallely propagated frame or some other frame etc31How then can the entropy of the universe be expected to increase with time Penrosersquos answer is that since the

hypothesized lawlike constraint on the initial conditions is not macroscopically discernible at a time t sufficiently

later than the time of the initial state the entropy at t should be calculated not by using the phase volumes within

the manifold of states compatible with all the laws but rather by using the larger manifold of states for which the

initial constraint is not required to hold see Penrose (1979 pp 632ndash633) But any entropy increase obtained by

reference to physically impossible states is vulnerable to the objection that the increase is merely epistemic32There are precise theorems showing that for a homogeneous but non-isotropic universe inflation is an

effective mechanism in smoothing out anisotropies (see for example Wald 1983) There are also results that

attempt to show that inflation also serves to smooth out inhomogeneities but these results are based on much

more dubious and restrictive premises than the former results33See for example Albrecht (2004 p 364)


The discussion of these issues is typically carried out in terms of dueling heuristicarguments or in philosophersrsquo jargon intuition pumps Intuition pumps are notoriouslyfickle and can often be reprimed to run in the opposite direction As a case in pointconsider again a hot relatively smooth state for the very early universe Naively such astate seems to have a high entropy But this impression it was argued is due to the neglectof the gravitational degrees of freedom Now consider a lumpy state presumed by theinflationary scenario to be present at the beginning of the universe The newly tutoredintuitions suggest that such a state has relatively high entropy But it could be argued thissecond impression is also mistaken this time due to the neglect of the degrees of freedomassociated with the inflation field that drives inflation in the early universe And (theargument continues) when these additional degrees of freedom are taken into account thepre-inflation state will be seen to have low entropy Thus the threat that inflation acts asan anti-thermodynamic machine is undercut But the price paid is that by its ownstandards inflationary cosmology fails to provide a satisfactory explanation of the post-inflation smooth state because it has to appeal to special pre-inflation conditions for theinflation fieldAnd in fact a number of critics of inflationary cosmology have argued that the problem

of special initial conditions is not escaped because special conditions are needed for theuniverse to enter an inflationary era Here is one version of the criticism that applies to auniverse that begins with a big bang and ends with a big crunch

34Fo35I i

As the lsquolsquobig crunchrsquorsquo is approached it seems overwhelmingly improbablemdashandindeed in apparent blatant contradiction with the second law of thermodynamicsthat a collapsing universe would undergo an era of lsquolsquodeflationrsquorsquo just before thelsquolsquobig crunchrsquorsquo Thus the region of lsquolsquofinal data spacersquorsquo that corresponds to a universethat did not deflate should have a much larger measure than a region correspondingto a universe that did deflate But the time reverse of a collapsing universe that failsto deflate is of course an expanding universe that fails to inflate Thus [since therelevant laws are time reversal invariant] this argument strongly suggests that theregion of initial data space that fails to give rise to an era of inflation has far largermeasure than the region that does not give rise to an inflationary era ie it isoverwhelmingly unlikely that inflation did occur (Hollands amp Wald 2002a pp2045ndash2046)

And one could add if inflation did take place it was because of special pre-inflationconditions34 Needless to say inflationary cosmologists resist this conclusion but in somecases there is grudging recognition that work needs to be done to avoid it35

I do not pretend to be able to adjudicate such conflicting claims What I want toemphasize at this juncture is that despite the many confident pronouncements thatmodern cosmology has delivered Boltzmannrsquos sought-after low entropy state for the earlyuniverse the only extant grounds for this dictum come from intuition pumps and as wehave seen there are competing pumps that produce different verdicts Since there are no apriori reliable principles for identifying the lsquolsquocorrectrsquorsquo pump the way forward does not liein furiously cranking the handle of onersquos favorite pump or in producing ever more cleverpumps The only responsible way to proceed is to study models where all the relevant terms

r a similar argument see Penrose (1986)

nterpret some of the remarks in Albrecht (2004) in this way


can be precisely defined and calculations of the relevant quantities can be made In the fewcases where this has been done the results have a plague-on-all-houses character in thatwhat were taken to be clear questions about the likelihood of various states turn out tohave ill-defined answersmdashwhich may help to explain why the intuition pumps do not yieldconsistent results

8 Model calculations

In order to study the issue of how probable inflation is Hawking and Page (1988)investigated a FRW-F model in which the matter content is given by a minimally coupledmassive scalar field F Such a model is less than ideal since the FRW metric ishomogeneous and isotropic and thus does not provide a good test of inflationrsquos ability tosmooth out lumpy states but this fault is balanced by tractability The state space X forthis system is four-dimensional and can be coordinatized using the scale factor a of theFRW model the field F and their respective conjugate momenta pa and pF The equationsof motion form a constrained Hamiltonian system with the one and only constraint beingthe vanishing of the Hamiltonian H frac14 036 The three-dimensional subspace C X wherethe constraint is satisfied is called the constraint surface A reduced phase space which isfree of gauge redundancy can be formed by choosing a two-dimensional surface S that istransverse to the dynamical trajectories on C Then as noted by Hawking and Page (1988)the pullback of the (degenerate) symplectic form o frac14 dpa ^ dathorn dpF ^ dF defines avolume measure eth2THORNm on S that is invariant under dynamical evolution Alternatively asnoted by Hollands and Wald (2002b) an invariant volume measure eth3THORNm can be defined onC by the volume element eth3THORN given by the condition dH^eth3THORN frac14 eth4THORN where eth4THORN frac14dpa ^

da ^ dpF ^ dF is the Liouville volume element for XThe upshot is a deterministic dynamical system However unlike the systems

encountered in classical mechanics it cannot be deemed to be time translationallyinvariant Nor is there any reasonable sense in which the dynamics of this system can bedeemed lsquolsquoergodicrsquorsquo which undercuts one of the justifications for taking the measure to beinterpreted as probability in the relative frequency sense But these are minor worriescompared to the main one eth2THORNmethSTHORN frac14 eth3THORNmethCTHORN frac14 1 ie neither of the natural invariantmeasures normalizes Since this result holds for the k frac14 thorn1 FRW model the non-normalizability cannot be blamed on the infinity of space implied by the k frac14 0 and 1models This non-normalizability obviously poses problems for assigning probabilities tovarious macrostates

But all is not lost For any measurable M S one of three possibilities holds withrespect to eth2THORNm (see Hollands amp Wald 2002b) (i) eth2THORNmethMTHORNo1 (ii) eth2THORNmethMTHORN frac14 1 andeth2THORNmethSMTHORNo1 or (iii) eth2THORNmethMTHORN frac14 1 and eth2THORNmethSMTHORN frac14 1 Now suppose that somerelevant coarse graining has been defined for S With M being the region of Scorresponding to a macrostate m belonging to the coarse graining set eth2THORN PrethmTHORN frac14 0 in case(i) set eth2THORN PrethmTHORN frac14 1 in case (ii) and declare eth2THORN PrethmTHORN undefined in case (iii) Then it is easy toshow that eth2THORN Pr is a (partial) probability measure on the coarse graining in the followingsense 0p eth2THORN PrethmTHORNp 1 if eth2THORN PrethmTHORN is defined and eth2THORN Prethm v m0THORN frac14 eth2THORN PrethmTHORN thorn eth2THORN Prethm0THORN if

36The existence of this constraint indicates the presence of gauge freedom see Earman (2003) for an

introductory account of these matters


the region M M 0 corresponding to mampm0 is null and if eth2THORN PrethmTHORN and eth2THORN Prethm0THORN aredefined37 An exactly analogoussituation holds for eth3THORNm and its associated eth3THORN Pr defined on acoarse graining of CIn this setting the only thing that could be meant by the claim that a relevant initial

macrostate mi has low probability and correspondingly low Boltzmann entropy is thateth2THORN PrethmiTHORN frac14 0 (or eth3THORN PrethmiTHORN frac14 0) Although I do not know of any proof that this is not thecase I conjecture that it is typically false because typically the third case obtains ieeth2THORN PrethmiTHORN and

eth3THORN PrethmiTHORN are not defined Such ill-definedness is known to hold for theprobability of inflation in the FRW-F model In the best case scenario where the initialmacrostate turns out to have zero probability the conditional probability assertions of theform eth2THORN Preth=miTHORN frac14 x or eth3THORN Preth=miTHORN frac14 y are meaningless if conditional probability is givenits usual interpretation (viz PrethA=BTHORN frac14 PrethAampBTHORN=PrethBTHORN) Moreover the only changes thatcan take place in the Boltzmann entropy are maximal flip-flops corresponding to a changefrom a zero-probability macrostate to a probability-one macrostate or vice versaOf course it could be that the HawkingndashPage model is misleading and that the results

inimical to the Boltzmann program are artifacts of the idealizations of the model But Iwould expect that removing the idealizations will not remove the inimical resultsThere is an historical irony here worth remarking The opening salvo of Zermelorsquos

attack on Boltzmann used Poincarersquos recurrence theorem Part of Boltzmannrsquos (1896 p218) response was to question the applicability of Poincarersquos recurrence theorem to theuniverse in particular he deemed lsquolsquoquestionablersquorsquo the assumption that the lsquolsquospace availablefor motion and the total energy [of the particles] are finitersquorsquo If this assumption fails thenso does the normalizability of the natural invariant measure which is a necessary conditionof Poincarersquos theorem By the same token normalizability is also a necessary condition forthe applicability of a non-hamstrung version of Boltzmann entropy As illustrated by themodel considered above when the relevant dynamics is the Hamiltonian dynamics of fieldquantities normalizability may fail even when the physical space available for motion isfiniteIn sum those who advocate that the Past Hypothesis provides the answer to

Boltzmannrsquos prayers proceed on the presumption that there is a well-defined thermo-statistical mechanics for general relativity that incorporates the lsquolsquogravitational entropy ofthe universersquorsquo In fact no such theory exists and the above discussion strongly suggeststhat no such theory can exist at least not for classical general relativity It may be that aquantum theory of gravity will come to the rescue but at present that is only a pious hope

9 Worse and worse

For the sake of discussion I want to assume for the moment that there is a well-definedBoltzmann entropy of the universe which includes the gravitational contribution toentropy and that the value of this quantity for the lsquolsquoinitialrsquorsquo state of the universe is verylow Does conditionalizing on this state solve Boltzmannrsquos initial state and asymmetryproblems as some proponents of the Past Hypothesis claim The answer is negative forseveral connected reasonsFirst there is no guarantee that there will be a monotonic increase in entropy up to the

present and that nevertheless the present value of the entropy is lowmdashsuch a result

37However eth2THORN Pr need not be countably additive


depends on the details of the initial state and the dynamics of the particular cosmologicalmodel The importance of the details of the initial state is emphasized in a second relatedpoint

If the just-so story told by the advocates of the Past Hypothesis were correct then itwould seem that the story would be even better if the entropy of the lsquolsquoinitialrsquorsquo state of theuniverse were even lower The way to make it lower is to make the state more uniform Butwith a completely uniform initial state the structure formation would not have takenplace and there would be no subsystems in which statistical-thermodynamics of thefamiliar form would apply Keeping onersquos eye on what needs to be explainedmdashie whystatistical-thermodynamics of the familiar form works as well as it does for the systems ofinterest to usmdashleads to a third point

To repeat once more Boltzmann entropy is a global quantity that characterizes themacrostate of the systemmdashin this case the system of the entire universe That the value ofthis quantity is low places very little constraint on the state of a small subsystem of the typewhose behavior we are interested in explaining38 In the present instance this is doubly sosince the Boltzmann entropy of the initial state of the universe is low primarily because ofthe gravitational contribution whereas that contribution is irrelevant for the kinds ofsubsystems of interest to us The length scale of typical thermodynamic systems of interestto us is below the lsquolsquoJeans lengthrsquorsquo and as a result self-gravitation is unimportant39 Forexample for any normal box of gas considered in laboratory experiments the pressure ofthe gas is sufficient to prevent gravitational self-collapse of the gas molecules Nor is itplausible that the entropy increases we perceive in typical thermodynamic systems is theresult of an increase in gravitational entropy that is transferred to the non-gravitationaldegrees of freedom for on the time scales of typical thermodynamic observations andinferences the values of the variables that characterize the gravitational degrees of freedomare effectively constant

I am not denying the correct (if virtually tautologous) component of the PastHypothesis namely the explanation of the observed thermo-statistical asymmetries mustrely on initialboundary conditions which presumably can be traced back to the earlyuniverse In broad strokes the explanation would go as follows Immediately after the bigbang (or if inflationary cosmology is correct immediately after the inflationary era) theuniverse was in a thermalized state that was very homogeneous and isotropic But notcompletely homogeneous and isotropic The small density perturbations were amplified bythe action of gravitation and grew into the structures that populate the present universemdashstars galaxies galactic clusters as well as more exotic objects such as black holes In asubset of these structures the conditions are such that (for the reasons given immediatelyabove) one does not have to worry about any gravitational contribution to the entropyand the ordinary concept of Boltzmann entropy applies And a subsubset of thesestructures will be in thermodynamic disequilibrium with low Boltzmann entropy That ourworld belongs to this subsubset can be justified on anthropic grounds Nor does the needto resort to anthropic considerations vitiate the explanation on offer for the nature of the

38Winsberg (2004) has argued that the hypothesized low entropy state of the early universe places no effective

constraint on the microstate of a relatively small and isolated thermodynamic system because the past

interactions of the system with the rest of the universe effectively randomizes this microstate39For a box of gas the Jeans length is defined by LJ frac14

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipv2s=Gr

p where vs is the speed of sound r is the energy

density and G is the gravitational constant


initial state and the dynamics together guarantee that the subsubset in question will bewell-populated and that the thermo-statistical asymmetries of the member systems will beconcordantThe contours of this explanation sketch partially trace the just-so story that is part of the

Past Hypothesis But the departures from the story told by the advocates of the PastHypothesis should be noted in particular the new story does not invoke the just-so aspectsof the Past Hypothesis that appeal to the entropy of the universe and it recognizes theneed for an anthropic element to the storyAlthough the new story is an improvement on the Past Hypothesis story it still contains

a serious lacuna Most of the subsystems we actually encounter are never completelyisolated eg a box of gas may be thermally isolated but the thermal isolation does notshield the gas molecules from the gravitational influence of external bodies This couplingto outside systems is sometimes thought to be helpful to the Boltzmann program byproviding a kind of lsquolsquostirringrsquorsquo that intuitively at least promotes effective mixing for thesystem of interest (recall the discussion of Section 2) The trouble is however that theBoltzmann apparatus developed above and all the resultsmdashactual and desiredmdashdepend onthe assumption of a closed dynamical system Nevertheless for a subsystem of a dynamicalsystem ethX mBftTHORN one could define a Boltzmann entropy as follows Marginalize themeasure m to the degrees of freedom characterizing the subsystem to produce a measure mchoose an appropriate coarse graining of macrostates fmbg for the subsystem and takeK logethmethMTHORNTHORN as the Boltzmann entropy of the macrostate m 2 fmbg where M is the regionof the state space X of the subsystem corresponding to m Unless the degrees of freedom ofthe subsystem do not interact with those of the rest of the system ft projected down to X

will generally not define a deterministic flow on X and m will not be preserved underdynamical evolution and if not results about the behavior of the Boltzmann entropy ofthe subsystem cannot be derived by applying the Boltzmann apparatus to the subsystemNor does an increase in the Boltzmann entropy for the total dynamical system ethX mBftTHORN

provide any guarantee of an increase in the entropy of the above defined Boltzmannentropy of the subsystem For one thing the increase in the entropy for the total systemmay be largely due to the behavior of a class of degrees of freedom that is irrelevant to theentropy of the subsystem and for another the increase in the Boltzmann entropy for thetotal system can be due to the establishment of correlations among the subsystems ratherthan increases in the Boltzmann entropies of the subsystemsIn sum the Past Hypothesis even if true does not explain why ordinary

thermodynamics works as well as it does for the types of systems of interest to us andthe widespread celebration of this hypothesis has been counter-productive in obscuring thehard work that still needs to be done to secure a satisfactory explanation

10 But not to worry but get to work

The combination of the worry of Section 8mdashthat the Boltzmann entropy of the earlyuniverse is ill-definedmdashand the worry of Section 9mdashthat even if the Boltzmann entropy ofthe early universe is well-defined and has a low value it does not suffice to ground thetemporally asymmetric behavior for the (sub)systems we care aboutmdashshould create aserious concern for those who believe that our knowledge of the past depends in essentialways on thermodynamic reasoning and that Boltzmannrsquos apparatus is needed to explainthe extent to which this reasoning is sound I want to suggest that the importance of


statistical mechanics in grounding our knowledge of the past should be downgradedespecially when Boltzmann entropy is being relied upon to do the grounding but moregenerally as well The suggestion can be fleshed out by a series of remarks the last of whichwill offer an alternative to Boltzmannrsquos program for explaining the approximate validity ofthermodynamics

First as a preliminary I want to urge that philosophers have been too credulous inbuying into one crucial aspect of Boltzmannrsquos attempt to resolve the asymmetry problemBegin by acknowledging the obvious namely that if time symmetric informationabout the present state of affairs is plugged into time reversal invariant laws then onlytime symmetric predictions can emerge But do not swallow the claim that the onlyway to get out time asymmetric predictions is with the help of additional information orposits about the distant past state of the system whether in the form of the entropy valueof this state or some other For we can and do get temporally asymmetric predictionsand retrodictions by plugging in time asymmetric information about the present andnear past states that comes for example from a short temporal sequence of observa-tions (see Section 4) This procedure does presume that our short-term memoriesare veridical But without this presumption both common-sensical and scientific knowl-edge would be impossible and issues about inferences to the more remote past would bemoot

Second note that if the Boltzmann apparatus is essential in generating knowledge of thepart of the past that lies beyond the reach of our short-term memories and if the only wayto solve the asymmetry problem is with the help of a low entropy lsquolsquoinitialrsquorsquo state then onpain of circularity the knowledge of this state can only be presuppositional But theknowledge of the early universe generated by modern cosmology is not presuppositional inthis sense though it does presuppose a number of theories More generally while much ofour knowledge of the past is based on statistical reasoning broadly construed much of it isnot in need of grounding or justification from the Boltzmann version of statisticalmechanics For instance we know with more precision and certainty the state of theuniverse at the time of lsquolsquolast scatteringrsquorsquomdashover 13 billion years agomdashthan we know thestate of the earth 100000 years ago This knowledge of the early universe comes fromlsquolsquoobservingrsquorsquo it with optical and radio telescopes Ellis Nel Maartens Stoeger andWhitman (1985) have shown that the field equations of general relativity theory plus datathat is in principle observable on a past null cone are together sufficient to determine thespacetime geometry and the matter content on and in the interior of this cone To be sureactual data is noisy and needs to be massaged by various statistical procedures and filteredthrough a number of scientific theories both low level and high level in order to produceuseful conclusions and of course the reliability of these inferences presupposes that weare not being deceived by Cartesian demons or conspiratorial conditions that producesystematically misleading impressions The point is not that empirical knowledge does notrely on substantial presuppositionsmdashit most certainly doesmdashbut that the validity ofBoltzmannian statistical mechanics and the Past Hypothesis do not figure prominentlyamong the presuppositions

The third remark calls attention to the fact that many of our inferences to the past donot concern closed systems to which Boltzmannian statistical mechanics might applyrather the object is to make probable inferences about interventions on the system Thehackneyed example of footprint-like indentations on a beach serves as a useful illustrationThe outline of the common sense inference that the indentations were made by a creature


walking across the beach can be reconstructed as follows First we ask supposing that thesand were an isolated system how long would we have to wait on average for aspontaneous fluctuation to produce a pattern of indentations similar to the ones observedBy analogy with simpler cases where the calculations have been carried out we are justifiedin concluding that the waiting time is longer than the age of the universe Thus it would bean extraordinary coincidence if we happened to be observing the beach shortly after such afluctuation occurred Next we reflect on the fact that the beach is not a closed system but isacted on by the tide and the winds But again we estimate that the likelihood that theseforces would by chance conspire to produce the observed pattern is so small as to beignorable Finally we use plausibility considerations some of a statistical variety andothers of a kind that subjective Bayesians would incorporate in lsquolsquoprior probabilitiesrsquorsquo toexclude various possibilities such as a visitation by an alien spacecraft with foot-shapedlanding gear And in Sherlock Holmes fashion after eliminating all of the othercontenders that come to mind we arrive at the Robinson Crusoe solution This crudeoutline begs for more detail but it seems wholly implausible that any useful precisificationwill result from attempts to bring to bear Boltzmann entropy or from the investigation ofthe initial conditions of the universeIn sum the notion that our knowledge of the past would crumble unless the Boltzmann

program can be made to work with the help of the Past Hypothesis is a kind of hysteriathat can only be generated by the febrile minds of philosophers who have become tooenamored with the Boltzmann apparatus Nevertheless it is certainly the case that some ofour reasoning about the past does need to be grounded on thermo-statistical physics If theexplanation of why this physics works as well as it does is not to be based on Boltzmannentropy and the Past Hypothesis it is a fair challenge to ask how an alternativeexplanation might go Here I can offer only a sketchy suggestionBegin by observing that the temporally asymmetric behavior we want to explain can be

characterized independently of the concept of Boltzmann entropy In particular what wewant is an explanation of the approximate validity of the Poisson heat equationmdashwhichdescribes the phenomenology of heat diffusionmdashand of the transport equationsmdashwhichdescribe the phenomenology of mass diffusion viscosity and thermal conductivity Andwe want the explanation to apply to relatively small subsystems such as a box of gasor an ice cube in a glass of lukewarm water The fact that such subsystems are typicallynot dynamically isolated from the environment may play an important role in the soughtafter explanation40 The coupling to the environment may help to drive a large basinof states in the subsystem state space to follow trajectories that fulfill to goodapproximation the transport equations Needless to say some sort of lsquolsquoPast Hypothesisrsquorsquoin the form of a restriction on initial conditions will have to be used But it is to behoped the restriction can be justified without having to appeal to dubious and perhapsill-defined notions such as the Boltzmann entropy of the universe In any case thisprogram is best pursued in the Gibbsian rather than the Boltzmannian approach tostatistical mechanics41

40For some attempts to implement this idea see Bergmann and Lebowitz (1955) and Blatt (1959)41In fairness to Boltzmann it should be noted that he was one of the first writers to emphasize the impact of the

environment on a thermodynamical system for example Boltzmann (1871) bases the ergodic hypothesis on the

multiplicity of outside forces But Boltzmann did not seem to realize that this insight is not easily exploited within

his version of statistical mechanics


11 From the ill-defined to the ill-considered and the ridiculous

In his second reply to Zermelo Boltzmann considered a proposal for avoiding the posit-your-way solution to the asymmetry problem In reading the passage quoted below recallthat Boltzmann is hypothesizing that the universe is in thermal equilibrium but that withinit are many lsquolsquoworldsrsquorsquo in non-equilibrium

42In

physic43Sin

precise

of an o

order

For the universe as a whole the two directions of time are indistinguishable just as inspace there is no up or down However just as at a particular place on the earthrsquossurface we can call lsquolsquodownrsquorsquo the direction toward the center of the earth so a livingbeing that finds itself in such a world at a certain period of time can define the timedirection as going from the less probable [lower entropy] to the more probable[higher entropy] states (the former will be the lsquolsquopastrsquorsquo and the latter will be thelsquolsquofuturersquorsquo) and by virtue of this definition he will find that this small region isolatedfrom the rest of the universe is lsquolsquoinitiallyrsquorsquo always in an improbable [low entropy]state (Boltzmann 1897b p 242)

It is not clear how seriously Boltzmann took this proposal for a define-your-waysolution to the asymmetry problem For example in the paragraph immediately followingthe quoted remarks he says that lsquolsquoWhether one chooses to indulge such speculations is amatter of tastersquorsquo (Boltzmann 1897b p 243) Except for the fact that the proposal isrepeated in nearly identical language in the Lectures on Gas Theory (Boltzmann1896ndash1898 p 447) one could ignore it as an expedient for escaping the embarrassmentrevealed in his exchanges with Zermelo of having to posit what is needed for a solution tothe initial state and asymmetry problems42

Commentators tend to take Boltzmannrsquos proposal seriously and those who do aredeeply divided about its viability Reichenbach (1971 p 128) saw it as lsquolsquoone of the keenestinsights into the problem of timersquorsquo while Popper (1974 p 128) found it lsquolsquostaggering in itsboldness and beautyrsquorsquo but ultimately lsquolsquountenablersquorsquo My assessment lies more with Popperthan with Reichenbach but instead of boldness and beauty I find muddles I willdistinguish five interpretations of Boltzmannrsquos proposal and will argue that all of themsuccumb to one of two ills they are either tenable but uninteresting or else interesting butuntenable

The first interpretation (I1) would have it that there is an objective directionality for timethat is independent both of observers and of the entropic behavior of physical systems butthat the subjective perception of time order for critters such as us is enslaved to the entropygradient of their environment43 In more detail consider a time translationally invariantdeterministic dynamical system ethX mBftTHORN that describes not only the physics of gasesand liquids but also the physics of critters like us and stipulate that the mental states ofthese critters supervene on their physical states Suppose that in some time intervaltiptptf of a dynamically possible history generated by ftethxTHORN x 2 X as t ranges from 1to thorn1 the Boltzmann entropy is monotonically increasing Then in the time interval

Curdrsquos (1982 p 282) opinion Boltzmannrsquos proposal was introduced lsquolsquoto appease Zermelorsquos demand for a

al explanation of Assumption Arsquorsquo

ce lsquolsquothe directionality of timersquorsquo is used in a number of different ways in the literature it is necessary to say

ly what one means by this phrase I will do that below But for the moment the reader can take the existence

bjective directionality for time to be equivalent to the existence of an objective observer independent time

of (relatively timelike) events Nuances will be added at the appropriate place below


tf ptp ti of the time reversed history generated by fTt ethxTHORN the entropy is monotonically

decreasing And using time translation invariance we can shift events in fTt ethxTHORN by an

amount ethtf thorn tiTHORN so that the interval of monotonic entropy decrease coincides withtiptptf Boltzmannrsquos proposal (on its present interpretation) implies that the critters inthe time reversed (and time translated) history have the subjective impression that entropyis increasing as time increases from ti to tf and hence that they perceive the same series ofmacrostates as the critters in the original history but in reverse temporal order44 So (thestory goes) critters such as us will not be able to distinguish between these two historiessince both will think that they are inhabiting the one in which entropy is increasing I thinkthat Maudlinrsquos (2004) reaction to such stories is exactly right we do not have to wait forneuro-psychology to become a mature science to know that this just-so story is just as falseas it can bemdashthe critters in the time reversed history will not have the subjective impressionof increasing entropy rather they will not have any coherent temporal perceptions at allTo arrive at the second and third interpretations focus not on subjective temporal

perceptions but on objective temporal structures and take Boltzmannrsquos proposal to be thatobjective temporal directionality is enslaved to the entropy gradient Here two versions canbe distinguished depending upon whether the directionality at issue is the directionality oftime itself or of physical processes within time Consider first the second reading yieldinginterpretation (I2) If the temporal directionality of physical processes is identified with themonotonic increase in some genuine physical magnitude45 assigned to these processes thenin Boltzmannrsquos day it would have been plausible to think that such behavior has to beidentical with or parasitic upon entropy increase Modern cosmology provides analternative that is meaningful even when the concept of Boltzmann entropy of the universeis not namely the expansion of the universe Up to the present this expansion has beenmonotonic and if the equation of state for the lsquolsquodark energyrsquorsquo that is driving the currentacceleration of expansion does not change the universe will continue to expandmdashadinfinitum in the cases where the dark energy is supplied by a positive cosmological constantor by quintessence and for a finite time in the case where the dark energy is supplied byphantom matter until the density of this matter becomes infinite and the universe ends in alsquolsquobig splatrsquorsquo46 In addition to the expansion of the universe there are other non-thermodynamic lsquolsquoarrows of timersquorsquo such as the electrodynamic arrow making Boltzmannrsquosconcentration on the entropic arrow seem parochialNow consider the alternative reading that would have it that the directionality of time

itself is enslaved to the entropy gradient yielding (I3) I will offer a dilemma for (I3)depending on two different ways to understand time reversal invariance taking the timereversal transformation as either an active or a passive transformation In preparation it isnecessary to be more precise about the presuppositions of the treatment of time reversalinvariance given in Section 4 Suppose that the state of a dynamical system is given by thevalues of particle and field variables and possibly the values of their time derivatives onsome time slice of spacetime Then consider what properties the spacetime must have tomake the definition of time reversal invariance offered in Section 4 meaningful First the

44This is assuming that if M is the region of state space corresponding to a macrostate m then RM frac14M see the

discussion in Section 445Some weasel words here are needed to rule out monotonic lsquolsquoCambridge changersquorsquo46See Caldwell Kamionkowski and Winberg (2003) Before the lsquolsquobig splatrsquorsquo there will be a lsquolsquobig riprsquorsquo in which

gravitationally bound matter will be torn apart


spacetime must be time orientable47 Any such spacetime admits two possible timeorientations48 with one of these orientations singled out a relation of temporal precedence5 is defined on spacetime points by the condition that p15p2 iff there is a future directedtimelike curve from p1 to p2 Requiring that there are no closed timelike curves assures that5 is not only transitive but asymmetric and thus is an order relation49 But still more isrequired to guarantee that there is a global time function in the guise of a smooth functiont from the spacetime to R such that tethp1THORNotethp2THORN iff p1op2

50 With all of these conditions onspacetime structure satisfied the definition of time reversal invariance offered in Section 4can be applied

That definition implicitly assumes the active reading of time reversal ie for any x 2 X

the dynamical trajectories generated by ftethxTHORN and fTt ethxTHORN as t ranges from 1 to thorn1

correspond to physically distinct histories51 and what time reversal invariance of the lawsof motion means is that one of the histories satisfies the laws if and only if the other doesTake a case where in ftethxTHORN the Boltzmann entropy SBethtTHORN increases monotonically with timeThen in fT

t ethxTHORN the Boltzmann entropy SBethtTHORN decreases monotonically with time Butaccording to (I3) this latter history is not a live possibility which is just to say that (I3)must reject the active reading of time reversal invariance

On the passive reading of time reversal invariance the dynamical trajectories generatedby ftethxTHORN and fT

t ethxTHORN as t ranges from 1 to thorn1 are merely different descriptions of thesame history This lsquolsquogaugersquorsquo interpretation of the time reversal operation might seem to bein blatant contradiction with the presuppositions of the definition of time reversalinvariance However if all of the fundamental laws are time reversal invariant some of thestructures enumerated above can be regarded as ladders that can be kicked away once theyhave served their purpose of getting us to the distinction between time reversal invariantand non-time reversal invariant laws But notice that this Wittgensteinian ladder trickamounts to regarding the relation of temporal precedence5 as a mere auxiliary device andto denying that events stripped of their gauge-dependent dressing are ordered as to earlierand later52 But this removes all literal meaning from (I3)

47A spacetime classical or relativistic is time orientable just in case it admits a continuous non-vanishing

timelike vector field For a classical spacetime a timelike vector is one that is oblique to the planes of absolute

simultaneity while for a relativistic spacetime it is a vector that lies within the local null cone Any spacetime that

has not been subjected to nasty Mobius type identifications admits such a vector field for relativistic spacetimes

the relevant result is that any simply connected spacetime admits the required vector field48For sake of definiteness concentrate on relativistic spacetimes In a time orientable spacetime choose some

continuous non-vanishing timelike vector field Call a timelike (tangent) vector at a spacetime point p future

pointing (respectively past pointing) just in case it points into the same lobe of the local null cone at p as the vector

from the reference field It is easy to see that the freedom in the choice of the reference fields leads to exactly two

possible choices for the division of timelike vectors into past and future pointing The choice of one of the two

constitutes a time orientation49If there are closed timelike curves then although there is a globally consistent directionality for time there is

no globally consistent time order If there are no closed timelike curves then the existence of a time direction or

orientation is equivalent to the existence of a time order I ignore this distinction since for present purposes it

makes no difference50For a relativistic spacetime to admit a global time function it must be stably causalmdashintuitively there is a

widening the null cones by some finite amount that does not result in closed timelike curves51Except in degenerate cases eg a single particle which is always at rest52Although assuming that the spacetime admits a global time function they are ordered by the three-place

relation of temporal betweenness Betethp1 p2 p3THORN which when the relation5 of temporal precedence is added to the

spacetime structure is interpreted to mean that either p15p25p3 or p35p25p1


A fourth interpretation (I4) emerges from the passive reading of the time reversalsymmetry Take Boltzmannrsquos proposal to mean that we should view the time reversalsymmetry as a gauge symmetry and that we should work in the lsquolsquoBoltzmann gaugersquorsquo inwhich the relation of temporal precedence 5 is assigned in such a way that entropyincreases with increasing5 order As such (I4) is harmless but insignificant in just the waythe proposal to work in the Lorentz gauge in electromagnetism is harmless butinsignificant (I4) can be made more significant by combining it with the idea that thelsquolsquoBoltzmann gaugersquorsquo is to be preferred since it corresponds to our subjective impressions oftime order Here I have an objection similar to the one made against (I2) namely theaugmented (I4) entails empirically false predictions According to the augmented (I4) sincethe dynamical trajectories generated by ftethxTHORN and fT

t ethxTHORN as t ranges from 1 to thorn1 aremerely different descriptions of the same physics the critters described by ftethxTHORN and fT

t ethxTHORN

must have the same temporal experiences if they have any at all My conjecture is that thisprediction is false and thus that in order to obtain an explanation of the differences intemporal experiences of critters in what (I4) regards as the physically the same historyunder different gauge dressings it is necessary to use the active interpretation of timereversal invarianceIt is also worth noting that there is now strong evidence that the presupposition

of (I3) and (I4)mdashnamely that all of the fundamental laws of physics are time reversalinvariantmdashis false This fact is often brushed aside on the grounds that the non-timereversal invariant laws concern the weak interactions of elementary particles53 and that theexotic processes created in particle accelerators to demonstrate the failure of time reversalinvariance in these interactions have no connection with the temporal asymmetries ofordinary thermodynamic processes or with our temporal perceptions Even if this iscorrectmdashand I think the rush to judgment could prove prematuremdashthe existence ofnon-time reversal invariant laws does have an obvious and important consequence forthe issues at hand namely the time reversal operation cannot be interpreted as agauge symmetry And in order to distinguish between which of two processes that aretime reverse images of each other is physically possible and which is not there must beat least locally in spacetime a time orientation and if the physical laws governingthese process are the same in all regions of spacetime then that orientation must beglobally defined54

A fifth interpretation (I5) of Boltzmannrsquos proposal has been offered by Curd (1982)Without going into the details the basic idea is that entropy increase in lsquolsquobranch systemsrsquorsquosatisfying appropriate boundary conditions serves as a criterion for the relation oftemporal precedence If lsquolsquocriterionrsquorsquo simply means that entropy increase serves as a reliableindicator of temporal precedence then (I5) is surely acceptable to the extent that the(temporally asymmetric) Second Law has statistical validity indeed (I5) is just a statementof a consequence of this statistical validity But there are many other lsquolsquocriteriarsquorsquo in the senseof reliable indicators of temporal precedence and no reason has been given for whyentropy increase should take pride of place among all these criteria especially since someof them (such as the expansion of the universe) are more reliable indicators On the otherhand if lsquolsquocriterionrsquorsquo means something stronger eg that entropy increase is part of the

53See Sachs (1987 Chapter 9) for a summary of the evidence for the violation of time reversal invariance54For the argument see Earman (2002) One possible escape is to redefine the time reversal operation by the

combination CPT where T is the usual time reversal operation and to appeal to CPT invariance


meaning of temporal precedence or that it is physically necessary for temporal precedenceor that it holds the key to our perceptions of temporal order then for all of the reasonsgiven above I would reject (I5)

Once philosophers get hold of a muddle they are reluctant to let it go But thephilosophy of time would be better off if the muddle started by Boltzmannrsquos definitionalploy were bid adieu

12 Conclusion

The following line of reasoning although a caricature helps to explain the tenor ofrecent discussions in the philosophical literature many of the temporal asymmetries thatunderlie the direction of time and undergird our knowledge of the past invoke entropy inessential ways this entropy is to be understood in Boltzmannrsquos sense hence Boltzmannrsquosprogram for explaining the statistical validity of the Second Law must be made to workand to make it work requires a low entropy initial state for the universe thus it will befound that cosmology fits the bill if not by honest toil then by hand waving intuitionpumps or by ignoring lsquolsquomerely technicalrsquorsquo considerations

On the contrary I claim there is no lsquolsquomustrsquorsquo about it Even before the initial state andasymmetry problems are encountered there are serious difficulties with the logic ofBoltzmannrsquos explanation of the Second Law Moreover there are good reasons forthinking that the alleged solutions to these problems that invoke the Past Hypothesis isbadly flawed The initial state of the universe can have a low Boltzmann entropy onlybecause of the gravitational contribution to the entropy budget But there does not exist atpresent and probably cannot exist a Boltzmannian statistical mechanics for gravitatingsystems as described by classical general relativity theory and as a result theBoltzmannian entropy of the universe is an ill-defined concept Moreover even if theentropy of the initial state of the universe had a well-defined low value this would notsuffice to explain why thermodynamics works as well as it does for the kinds of systems wecare about

These negative verdicts on Boltzmannrsquos program do not I contend threaten a disasterfor the inferential practices we use to generate conclusions about the past But if correctthey do require an alternative to Boltzmannrsquos program for understanding the statisticalvalidity of thermodynamics I have given only the sketchiest indications of how thisalternative program might work The difficulties for orthodoxy seem to me sufficientlygreat that more investigation of heterodox approaches is in order Extant heterodoxiesinclude the idea that thermodynamic asymmetries have to be grounded on quantumconsiderationsmdashperhaps using the state vector reduction scheme of Girardi Rimini andWeber (as favored by Albert 2000 Chapter 7) or by using the phenomenon of quantumdecoherence (as favored by Hemmo amp Shenker 2001) Since the world is at base quantummechanical the explanation of thermodynamical asymmetries must of course be statablein quantum mechanical terms But I would be surprised if the asymmetries at issue wereinexplicable without the essential use of quantum concepts55

55I also have qualms about the mentioned mechanisms I doubt that there is a satisfactory relativistically

invariant account of state vector reduction and while I have no doubt that quantum decoherence is an important

part of the emergence of the classical world from the quantum mechanics I do doubt that decoherence by itself

can explain the emergence


Acknowledgments

I am very grateful to Jos Uffink for number of helpful suggestions on a earlier draft ofthis paper needless to say this does not imply that he shares the conclusions I advanceThe comments of two anonymous referees are also gratefully acknowledged

References

Albert D (1994) The foundations of quantum mechanics and the approach to thermodynamic equilibrium

British Journal for the Philosophy of Science 45 669ndash677

Albert D (2000) Time and chance Cambridge MA Harvard University Press

Albrecht A (2004) Cosmic inflation and the arrow of time In J D Barrow P C W Davies amp C L Harper

(Eds) Science and ultimate reality Quantum theory cosmology and complexity (pp 363ndash401) Cambridge

MA Cambridge University Press

Albrecht A amp Sorbo L (2004) Can the universe afford inflation Physical Review D 70 063528-1-10 hep-th

0405270

Barrow J amp Tipler F (1986) The anthropic cosmological principle Oxford Oxford University Press

Bergmann P G amp Lebowitz J L (1955) New approach to nonequilibrium processes Physical Review 99

578ndash587

Blatt J M (1959) An alternative approach to the ergodic problem Progress of Theoretical Physics 22

745ndash756

Boltzmann L (1871) Einige allgemeine Satze uber Warmegleichgewicht Sitzungsberichte der Kaiserlichen

Akademie der Wissenschaften Mathematisch-Naturwissenschaftliche Klasse Wien 63 679ndash711 Reprinted in

F Hasenohrl (Ed) Ludwig Boltzmann Wissenschaftliche Abhandlungen (Vol 1) (pp 259ndash287) New York

Chelsea Publishing Co 1968

Boltzmann L (1895) On certain questions of the theory of gases Nature 51 413ndash415

Boltzmann L (1896) Entgegung auf die warmetheoretischen Betrachtungungen des Hrn E Zermelo Annalen

der Physik 57 773ndash784 (S G Brush English Transl) Kinetic theory (Vol 2) (pp 218ndash228) New York

Pergamon Press 1966

Boltzmann L (1896ndash1898) Lectures on gas theory (S G Brush English Transl) Berkeley CA University of

California Press 1964

Boltzmann L (1897a) Uber einige meiner weniger bekannten Abhandlungen uber Gastheorie und deren

Verhaltnis zu derselben Verhandlungen der Gesellschaft Deutscher Naturforscher und Arzte 69 19ndash26

Reprinted in F Hasenohrl (Ed) Ludwig Boltzmann Wissenschaftliche Abhandlungen (Vol 3) (pp 598ndash698)

New York Chelsea Publishing Co 1968

Boltzmann L (1897b) Zu Hr Zermelos Abhandlung lsquoUber die mechanische Erklarung irreversibler Vorgangersquo

Annalen der Physik 60 392ndash398 (S G Brush English Transl) Kinetic theory (Vol 2) (pp 238ndash245) New

York Pergamon Press 1966

Boltzmann L (1904) On statistical mechanics Populare Schriften Essay 19 Reprinted in B McGinness

(Ed) Ludwig Boltzmann Theoretical physics and philosophical problems (pp 158ndash172) Dordrecht D Reidel

1974

Bricmont J (1996) Science of chaos or chaos of science In P Gross N Levitt amp M Lewis (Eds) The flight

from science and reason (pp 131ndash175) New York New York Academy of Sciences

Bronstein M amp Landau L (1933) On the second law of thermodynamics and the universe Reprinted in English

translation in D Ter Harr (Ed) Collected papers of L D Landau (pp 69ndash72) New York Gordon and

Breach 1967

Brush S G (1966) Kinetic theory (Vol 2) New York Pergamon Press

Brush S G (1975) The kind of motion we call heat (Vols 1 2) Amsterdam North-Holland

Caldwell R R Kamionkowski M amp Winberg N N (2003) Phantom energy and cosmic doomsday Physical

Review Letters 91 071301 astro-ph0302505

Callender C (2001) Thermodynamic asymmetry in time Stanford encyclopedia of philosophy hhttp

wwwplatostanfordeduentriestime-thermoi

Callender C (2004) There is no puzzle about the low-entropy past In C Hitchcock (Ed) Contemporary debates

in philosophy of science (pp 240ndash255) London Blackwell


Carroll S (2004) Why is the universe accelerating In W L Friedman (Ed) Measuring and modeling the

universe Cambridge MA Cambridge University Press astro-ph0310342

Carroll S amp Chen J (2004) Spontaneous inflation and the origin of the arrow of time hep-th0410270

Curd M (1982) Popper on the direction of time In R Sexl amp J Blackmore (Eds) Ludwig Boltzmann

internationale Tagung anlasslich des 75 Jahrestages seines Todes 5-8 September 1981 ausgewahlte

Abhandlungen (pp 263ndash303) Graz Akademische Druck-u Verlagsanstalt

Dyson L Kleban M amp Susskind L (2002) Disturbing implications of a cosmological constant Journal of

High Energy Physics 0210 011

Earman J (2002) What time reversal invariance is and why it matters International Studies in Philosophy of

Science 16 245ndash264

Earman J (2003) Tracking down gauge An ode to the constrained Hamiltonian formalism In K Brading amp E

Castellani (Eds) Symmetries in physics Philosophical reflections (pp 140ndash162) Cambridge MA Cambridge

University Press

Earman J amp Redei M (1996) Why ergodic theory does not explain the success of equilibrium statistical

mechanics British Journal for the Philosophy of Science 47 63ndash78

Ellis G F R Nel S D Maartens R Stoeger W R amp Whitman A P (1985) Ideal observational

cosmology Physics Reports 124 315ndash417

Feynman R P (1994) The character of physical law Cambridge MA MIT Press

Gallavotti G (1999) Statistical mechanics A short treatise New York Springer

Goldstein S (2001) Boltzmannrsquos approach to statistical mechanics In J Bricmont et al (Eds) Chance in

physics Foundations and perspectives Lecture notes in physics (Vol 574) (pp 39ndash54) New York Springer

Hawking S W amp Page D N (1988) How probable is inflation Nuclear Physics B 298 789ndash809

Hemmo M amp Shenker O (2001) Can we explain thermodynamics by quantum decoherence Studies in History

and Philosophy of Modern Physics 32 555ndash568

Hollands S amp Wald R M (2002a) An alternative to inflation General Relativity and Gravitation 34 2043ndash2055

gr-qc0205058

Hollands S amp Wald R M (2002b) Comment on inflation and alternative cosmology hep-th0210001

Knott C G (1911) Life and scientific work of Peter Guthrie Tait Cambridge MA Cambridge University Press

Lebowitz J L (1993) Macroscopic laws microscopic dynamics timersquos arrow and Boltzmannrsquos entropy Physica

A 194 1ndash27

Lebowitz J L (1999) Statistical mechanics A selective review of two central issues Reviews of Modern Physics

71 S346ndashS357

Maudlin T (2004) On the passing of time preprint

Malament D (2004) On the time reversal invariance of classical electromagnetic theory Studies in History and

Philosophy of Modern Physics 35 295ndash315

Penrose R (1979) Singularities and time asymmetry In S W Hawking amp W Israel (Eds) General relativity An

Einstein centenary (pp 581ndash638) Cambridge MA Cambridge University Press

Penrose R (1986) Review of G W Gibbons S W Hawking amp S T C Siklos (Eds) The very early universe

Cambridge MA Cambridge University Press Observatory 106 20ndash21

Penrose R (1989) The emperorrsquos new mind Concerning computers minds and the laws of physics Oxford Oxford

University Press

Penrose R (2004) The road to reality A complete guide to the laws of the universe London Jonathan Cape

Popper K (1974) Boltzmann and the arrow of time In P A Schilpp (Ed) The philosophy of Karl Popper

(pp 124ndash129) La Salle IL Open Court

Popper K (1981) Quantum theory and the schism in physics Totowa NJ Roman and Littlefield

Price H (1996) Timersquos arrow and Archimedesrsquo point New York Oxford University Press

Price H (2002) Boltzmannrsquos time bomb British Journal for the Philosophy of Science 53 83ndash119

Price H (2004) On the origins of the arrow of time Why there is still a puzzle about the low-entropy past In C

Hitchcock (Ed) Contemporary debates in philosophy of science (pp 219ndash239) London Blackwell

Reichenbach H (1971) The direction of time Los Angeles University of California Press

Sachs R G (1987) The physics of time reversal Chicago University of Chicago Press

Sklar L (1993) Physics and chance Philosophical issues in the foundations of statistical mechanics Cambridge


Sklar L (1995) The elusive object of desire In pursuit of the kinetic equations and the Second Law In S F

Savitt (Ed) Timersquos arrows today (pp 191ndash216) Cambridge MA Cambridge University Press

Strutt R J (1968) The life of John William Strutt Madison WI University of Wisconsin Press


van Lith J (2001) Stir in stillness A study of the foundations of equilibrium statistical mechanics PhD

dissertation University of Utrecht

Vranas P B M (1998) Epsilon-ergodicity and the success of equilibrium statistical mechanics Philosophy of


Wald R M (1983) Asymptotic behavior of homogeneous cosmological models in the presence of a positive

cosmological constant Physical Review D 28 2118ndash2120

Winsberg E (2004) Can conditioning on the lsquoPast Hypothesisrsquo militate against the reversibility objection

Philosophy of Science 71 489ndash504

The rsquorsquoPast Hypothesisrsquorsquo Not even false

Introduction

The logic of Boltzmannaposs explanation of the Second Law

Qualms about Boltzmannaposs explanation

The initial state problem and the asymmetry problem

Boltzmannaposs cosmological solutions to the initial state and asymmetry problems

Modern cosmology to the rescue ()

Competing intuition pumps

Model calculations

Worse and worse

But not to worry but get to work

From the ill-defined to the ill-considered and the ridiculous

Conclusion

Acknowledgments

References


early universe was in a low entropy state (the lsquolsquoPast Hypothesisrsquorsquo) To be sure this ideacomes in a variety of forms and flavors for Albert (2000) the hypothesis of a low entropystate for the early universe has something of a neo-Kantian status in that its truth is acondition needed to make our knowledge of the past possible for others (eg Penrose1979 1989 2004) the hypothesis has a (wannabe) law status and for still others thehypothesis rests on contingent facts that may or may not call for an explanation (see Price2004 vs Callender 2004)1 Leaving these and other nuances aside the agreement in thephilosophy of science community (where disagreement is the norm) is broad enough thatwe have something that can be rightly dubbed a dogmaThis dogma I contend is ill-motivated and ill-defined and its implementation consists

mainly in furious hand waving and wishful thinking In short it is (to borrow a phrasefrom Pauli) not even false2 Such heresies are apt to get one burnt at the stake of academicopinion and if that is to be my fate I want to make sure that I am burnt at the right stakeSo at the outset let me make clear that I am not denying what is a near truism namely ifsome observed temporal asymmetry is not a consequence of laws of physics then its originmust be sought in initialboundary conditions3 Nor am I denying that there is some senseof lsquolsquoentropyrsquorsquo in which it is true that the early universe was in a low entropy state What Ido deny is that for the cosmologies described by classical general relativity theory there isany well-defined sense in which the Boltzmann entropy of the early universe has a very lowvalue Furthermore I claim that even if the Boltzmann entropy for the early universe werea well-defined quantity a low value would not suffice to explain the kinds of presentlyobserved entropic asymmetries we care about And I question a long-standing tradition inphilosophy of science that accepts the idea that solving the conceptual puzzles Boltzmannbequeathed us holds the key to understanding the fundamental ontological andepistemological asymmetries between past and the futureRoughly the standard line runs that essential components of these asymmetries hinge on

the temporal asymmetry of lsquolsquorecordsrsquorsquo or lsquolsquotracesrsquorsquo that the recordtrace asymmetry isgrounded in entropic asymmetries and that these entropic asymmetries are ultimatelytraceable to the low entropy state of the early universe None of the links in this chain Icontend can bear the weight that philosophers want to put on them when the said entropyis Boltzmann entropy4 This radical sounding contention is in fact not radical at all Inparticular it does not threaten skepticism about our knowledge of the past indeed oncewe are freed of the obsessive insistence of seeing entropy as the key to ontological andepistemological aspects of temporal asymmetries it is evident that the ill-defined nature ofclaims about entropy states of the early universe in no way compromises our knowledge ofthe past However the heresy does entail the need for a new understanding of theapproximate validity of thermodynamicsThe plan of the paper is as follows In Section 2 I sketch Boltzmannrsquos explanation of the

statistical validity of the Second Law of thermodynamics and develop the apparatus thatwill be needed for the later discussion of modern cosmology Section 3 reviews some

1For other recent versions of the Past Hypothesis see Bricmont (1996) and Goldstein (2001)2While I decry Paulirsquos haughty dismissiveness his phrase seems to me to exactly fit the present case3As Boltzmann (1904 pp 170ndash171) himself puts it while affirming the antecedent lsquolsquoSince in the differential

equations of mechanics themselves there is absolutely nothing analogous to the second law of thermodynamics

the latter can be mechanically represented only by means of assumptions regarding initial conditionsrsquorsquo4Of course these links can be maintained if one is willing to work with a sense of lsquolsquoentropyrsquorsquo that is sufficiently

loose and elastic But then the relevance of Boltzmannrsquos apparatus becomes dubious

Ville
Korostus
Ville
Highlight
Ville
Highlight
Ville
Highlight
Ville
Highlight
Ville
Highlight
Ville
Highlight





5Fo

receive

and va6In

degree

tumble

purely

for the

Hamil

Icari fl

human

Knott





n Lith (2001)








(1911 pp 115ndash116)

Ville
Highlight




frac14 ft2 ft1















Ville
Highlight


















Ville
Highlight
Ville
Highlight
Ville
Highlight
Ville
Highlight







R t


t




9Thi

Section10Se






Ville
Highlight






11Bu

subjec

space r

be rep



ethxTHORN
















ethMTHORN














12Se13Se















(A1)

14In

(200015Al


(A2)


(A3)
















16Po

system

for som

Boltzm

and B17I w18Bo























Gas Theory

20Th

Tipler21It

likelyrsquorsquo

Sorbo








(2004)













123



123






























34Fo35I i






















9 Worse and worse








































42In

physic43Sin

precise

of an o

order
















































t ethxTHORN









12 Conclusion









Acknowledgments


References








0405270



578ndash587


745ndash756








Pergamon Press 1966












1974





Breach 1967






















University Press













gr-qc0205058




A 194 1ndash27


71 S346ndashS357









University Press


























Introduction







Model calculations

Worse and worse



Conclusion

Acknowledgments

References





5Fo

receive

and va6In

degree

tumble

purely

for the

Hamil

Icari fl

human

Knott





n Lith (2001)








(1911 pp 115ndash116)

Ville
Highlight




frac14 ft2 ft1















Ville
Highlight


















Ville
Highlight
Ville
Highlight
Ville
Highlight
Ville
Highlight







R t


t




9Thi

Section10Se






Ville
Highlight






11Bu

subjec

space r

be rep



ethxTHORN
















ethMTHORN














12Se13Se















(A1)

14In

(200015Al


(A2)


(A3)
















16Po

system

for som

Boltzm

and B17I w18Bo























Gas Theory

20Th

Tipler21It

likelyrsquorsquo

Sorbo








(2004)













123



123






























34Fo35I i






















9 Worse and worse








































42In

physic43Sin

precise

of an o

order
















































t ethxTHORN









12 Conclusion









Acknowledgments


References








0405270



578ndash587


745ndash756








Pergamon Press 1966












1974





Breach 1967






















University Press













gr-qc0205058




A 194 1ndash27


71 S346ndashS357









University Press


























Introduction







Model calculations

Worse and worse



Conclusion

Acknowledgments

References




frac14 ft2 ft1















Ville
Highlight


















Ville
Highlight
Ville
Highlight
Ville
Highlight
Ville
Highlight







R t


t




9Thi

Section10Se






Ville
Highlight






11Bu

subjec

space r

be rep



ethxTHORN
















ethMTHORN














12Se13Se















(A1)

14In

(200015Al


(A2)


(A3)
















16Po

system

for som

Boltzm

and B17I w18Bo























Gas Theory

20Th

Tipler21It

likelyrsquorsquo

Sorbo








(2004)













123



123






























34Fo35I i






















9 Worse and worse








































42In

physic43Sin

precise

of an o

order
















































t ethxTHORN









12 Conclusion









Acknowledgments


References








0405270



578ndash587


745ndash756








Pergamon Press 1966












1974





Breach 1967






















University Press













gr-qc0205058




A 194 1ndash27


71 S346ndashS357









University Press


























Introduction







Model calculations

Worse and worse



Conclusion

Acknowledgments

References


















Ville
Highlight
Ville
Highlight
Ville
Highlight
Ville
Highlight







R t


t




9Thi

Section10Se






Ville
Highlight






11Bu

subjec

space r

be rep



ethxTHORN
















ethMTHORN














12Se13Se















(A1)

14In

(200015Al


(A2)


(A3)
















16Po

system

for som

Boltzm

and B17I w18Bo























Gas Theory

20Th

Tipler21It

likelyrsquorsquo

Sorbo








(2004)













123



123






























34Fo35I i






















9 Worse and worse








































42In

physic43Sin

precise

of an o

order
















































t ethxTHORN









12 Conclusion









Acknowledgments


References








0405270



578ndash587


745ndash756








Pergamon Press 1966












1974





Breach 1967






















University Press













gr-qc0205058




A 194 1ndash27


71 S346ndashS357









University Press


























Introduction







Model calculations

Worse and worse



Conclusion

Acknowledgments

References







R t


t




9Thi

Section10Se






Ville
Highlight






11Bu

subjec

space r

be rep



ethxTHORN
















ethMTHORN














12Se13Se















(A1)

14In

(200015Al


(A2)


(A3)
















16Po

system

for som

Boltzm

and B17I w18Bo























Gas Theory

20Th

Tipler21It

likelyrsquorsquo

Sorbo








(2004)













123



123






























34Fo35I i






















9 Worse and worse








































42In

physic43Sin

precise

of an o

order
















































t ethxTHORN









12 Conclusion









Acknowledgments


References








0405270



578ndash587


745ndash756








Pergamon Press 1966












1974





Breach 1967






















University Press













gr-qc0205058




A 194 1ndash27


71 S346ndashS357









University Press


























Introduction







Model calculations

Worse and worse



Conclusion

Acknowledgments

References






11Bu

subjec

space r

be rep



ethxTHORN
















ethMTHORN














12Se13Se















(A1)

14In

(200015Al


(A2)


(A3)
















16Po

system

for som

Boltzm

and B17I w18Bo























Gas Theory

20Th

Tipler21It

likelyrsquorsquo

Sorbo








(2004)













123



123






























34Fo35I i






















9 Worse and worse








































42In

physic43Sin

precise

of an o

order
















































t ethxTHORN









12 Conclusion









Acknowledgments


References








0405270



578ndash587


745ndash756








Pergamon Press 1966












1974





Breach 1967






















University Press













gr-qc0205058




A 194 1ndash27


71 S346ndashS357









University Press


























Introduction







Model calculations

Worse and worse



Conclusion

Acknowledgments

References






ethMTHORN














12Se13Se















(A1)

14In

(200015Al


(A2)


(A3)
















16Po

system

for som

Boltzm

and B17I w18Bo























Gas Theory

20Th

Tipler21It

likelyrsquorsquo

Sorbo








(2004)













123



123






























34Fo35I i






















9 Worse and worse








































42In

physic43Sin

precise

of an o

order
















































t ethxTHORN









12 Conclusion









Acknowledgments


References








0405270



578ndash587


745ndash756








Pergamon Press 1966












1974





Breach 1967






















University Press













gr-qc0205058




A 194 1ndash27


71 S346ndashS357









University Press


























Introduction







Model calculations

Worse and worse



Conclusion

Acknowledgments

References






12Se13Se















(A1)

14In

(200015Al


(A2)


(A3)
















16Po

system

for som

Boltzm

and B17I w18Bo























Gas Theory

20Th

Tipler21It

likelyrsquorsquo

Sorbo








(2004)













123



123






























34Fo35I i






















9 Worse and worse








































42In

physic43Sin

precise

of an o

order
















































t ethxTHORN









12 Conclusion









Acknowledgments


References








0405270



578ndash587


745ndash756








Pergamon Press 1966












1974





Breach 1967






















University Press













gr-qc0205058




A 194 1ndash27


71 S346ndashS357









University Press


























Introduction







Model calculations

Worse and worse



Conclusion

Acknowledgments

References







(A1)

14In

(200015Al


(A2)


(A3)
















16Po

system

for som

Boltzm

and B17I w18Bo























Gas Theory

20Th

Tipler21It

likelyrsquorsquo

Sorbo








(2004)













123



123






























34Fo35I i






















9 Worse and worse








































42In

physic43Sin

precise

of an o

order
















































t ethxTHORN









12 Conclusion









Acknowledgments


References








0405270



578ndash587


745ndash756








Pergamon Press 1966












1974





Breach 1967






















University Press













gr-qc0205058




A 194 1ndash27


71 S346ndashS357









University Press


























Introduction







Model calculations

Worse and worse



Conclusion

Acknowledgments

References






16Po

system

for som

Boltzm

and B17I w18Bo























Gas Theory

20Th

Tipler21It

likelyrsquorsquo

Sorbo








(2004)













123



123






























34Fo35I i






















9 Worse and worse








































42In

physic43Sin

precise

of an o

order
















































t ethxTHORN









12 Conclusion









Acknowledgments


References








0405270



578ndash587


745ndash756








Pergamon Press 1966












1974





Breach 1967






















University Press













gr-qc0205058




A 194 1ndash27


71 S346ndashS357









University Press


























Introduction







Model calculations

Worse and worse



Conclusion

Acknowledgments

References













Gas Theory

20Th

Tipler21It

likelyrsquorsquo

Sorbo








(2004)













123



123






























34Fo35I i






















9 Worse and worse








































42In

physic43Sin

precise

of an o

order
















































t ethxTHORN









12 Conclusion









Acknowledgments


References








0405270



578ndash587


745ndash756








Pergamon Press 1966












1974





Breach 1967






















University Press













gr-qc0205058




A 194 1ndash27


71 S346ndashS357









University Press


























Introduction







Model calculations

Worse and worse



Conclusion

Acknowledgments

References













123



123






























34Fo35I i






















9 Worse and worse








































42In

physic43Sin

precise

of an o

order
















































t ethxTHORN









12 Conclusion









Acknowledgments


References








0405270



578ndash587


745ndash756








Pergamon Press 1966












1974





Breach 1967






















University Press













gr-qc0205058




A 194 1ndash27


71 S346ndashS357









University Press


























Introduction







Model calculations

Worse and worse



Conclusion

Acknowledgments

References
























34Fo35I i






















9 Worse and worse








































42In

physic43Sin

precise

of an o

order
















































t ethxTHORN









12 Conclusion









Acknowledgments


References








0405270



578ndash587


745ndash756








Pergamon Press 1966












1974





Breach 1967






















University Press













gr-qc0205058




A 194 1ndash27


71 S346ndashS357









University Press


























Introduction







Model calculations

Worse and worse



Conclusion

Acknowledgments

References

















9 Worse and worse








































42In

physic43Sin

precise

of an o

order
















































t ethxTHORN









12 Conclusion









Acknowledgments


References








0405270



578ndash587


745ndash756








Pergamon Press 1966












1974





Breach 1967






















University Press













gr-qc0205058




A 194 1ndash27


71 S346ndashS357









University Press


























Introduction







Model calculations

Worse and worse



Conclusion

Acknowledgments

References





































42In

physic43Sin

precise

of an o

order
















































t ethxTHORN









12 Conclusion









Acknowledgments


References








0405270



578ndash587


745ndash756








Pergamon Press 1966












1974





Breach 1967






















University Press













gr-qc0205058




A 194 1ndash27


71 S346ndashS357









University Press


























Introduction







Model calculations

Worse and worse



Conclusion

Acknowledgments

References






































t ethxTHORN









12 Conclusion









Acknowledgments


References








0405270



578ndash587


745ndash756








Pergamon Press 1966












1974





Breach 1967






















University Press













gr-qc0205058




A 194 1ndash27


71 S346ndashS357









University Press


























Introduction







Model calculations

Worse and worse



Conclusion

Acknowledgments

References




12 Conclusion









Acknowledgments


References








0405270



578ndash587


745ndash756








Pergamon Press 1966












1974





Breach 1967






















University Press













gr-qc0205058




A 194 1ndash27


71 S346ndashS357









University Press


























Introduction







Model calculations

Worse and worse



Conclusion

Acknowledgments

References


Acknowledgments


References








0405270



578ndash587


745ndash756








Pergamon Press 1966












1974





Breach 1967






















University Press













gr-qc0205058




A 194 1ndash27


71 S346ndashS357









University Press


























Introduction







Model calculations

Worse and worse



Conclusion

Acknowledgments

References














University Press













gr-qc0205058




A 194 1ndash27


71 S346ndashS357









University Press


























Introduction







Model calculations

Worse and worse



Conclusion

Acknowledgments

References











Introduction







Model calculations

Worse and worse



Conclusion

Acknowledgments

References

John Earman - The “Past Hypothesis” Not even false

Documents

Transcript of John Earman - The “Past Hypothesis” Not even false