Entropy production and fluctuation phenomena in nonequilibrium systems
Haye HinrichsenFaculty for Physics and AstronomyUniversity of Würzburg, Germany
Workshop on Large Fluctuations in NonEquilibrium SystemsMPIPKS Dresden, July 2011
In collaboration with:
Andre Barato, ICTP, Trieste, ItalyUrna Basu, SAHA Institute, Kolkata, IndiaRaphael Chetrite, Lyon and CNRSChristian Gogolin, PotsdamPeter Janotta, WürzburgDavid Mukamel, Weizmann Insititute, Israel
Non-equilibrium Dynamics, Thermalization and Entropy ProductionH. Hinrichsen, C. Gogolin, and P. JanottaJ. Phys.: Conf. Ser. 297 012011 (2011)
Entropy production and fluctuation relations for a KPZ interfaceA. C. Barato, R. Chetrite, H. Hinrichsen, and D. MukamelJ. Stat. Mech.: Theor. Exp. P10008 (2010)
Outline
1) Introduction to entropy production
2) Fluctuation theorem revisited
3) Entropy production and renormalization
Nonequilibrium systemsNonequilibrium systems
T1 T2
μ1 μ2
Flow of heat
… typically driven systems
Flow of particles
Environment
Nonequilibrium systemsNonequilibrium systems
System
Environment
Nonequilibrium systemsNonequilibrium systems
System
drive
entropy
Models of classical nonequilibrium systemsModels of classical nonequilibrium systems
Systementropy
Model
Models of classical nonequilibrium systemsModels of classical nonequilibrium systems
Systementropy
Set of configurations Ωsys
(state space)
configurations c∈ sys
Model
Models of classical nonequilibrium systemsModels of classical nonequilibrium systems
Systementropy
Model Irreversible dynamics byspontaneous transitions
at ratecc ' wcc '
Ωsys
Configurational entropy
Environmental entropy
Total entropy
S sys t = −ln P c , t
S env t
S tot (t)=Ssys(t )+Senv(t )
EnvironmentSystem
drive
entropy
Actual time evolution:
Sequence of transitions(stochastic path)
at times
Our partial knowledge:
Probability distribution P(c,t)evolving deterministicallyby the master equation.
c1c2c3 ...cN
t1 , t 2 , t3 , ... , t N
⟨Ssys(t)⟩ = −∑c∈Ωsys
P(c , t) ln P(c , t)
Ssys(t) = −ln P (c(t), t)
ddt
P (c , t) = ∑c '∈Ω
P(c ' , t)wc ' c−P (c ,t )w cc '
Configurational entropy
Mean entropy
Entropy of the systemEntropy of the system
⟨Ssys(t)⟩ = −∑c∈Ωsys
P(c , t) ln P(c , t)
Ssys(t) = −ln P (c(t), t)Configurational entropy
Mean entropy
Entropy of the systemEntropy of the system
Change of conf. entropy S sys(t) = −P (c (t), t)P (c (t), t)
−∑j
δ(t−t j) lnP (c j , t )
P (c j−1 , t )
⟨ Ssys(t)⟩ = − ∑c , c '∈Ωsys
P (c ,t )wc→c ' lnP (c ,t )P(c ' , t)
Change of mean entropy
Configurational entropy
Environmental entropy
Total entropy
S sys t = −ln P c , t
S env t
S tot (t)=Ssys(t )+Senv(t )
EnvironmentSystem
drive
entropy
??
Senv(t) = ∑j
δ(t−t j) lnωc j−1→ c j
ωc j→c j−1
Andrieux and Gaspard, J. Chem. Phys. 2004U. Seifert, PRL 2005
Commonly accepted formula for theCommonly accepted formula for theenvironmental entropyenvironmental entropy
Where does it come from?
1976
X 1 X 2 X N
P(c , t)
[X i](t )
probability
concentration
Schnakenberg: The master equation
is mapped to a fictitious chemical system evolving according to the law of mass action (= mean field equation)
Fictitious chemical systemFictitious chemical system
ddt
P(c , t) = ∑c '∈Ω
(P(c ' , t)wc '→c−P(c , t)wc→c ')
Isothermal / isochroric → minimize F.
Extent of reactionExtent of reaction = average number of forward reactions c→c' minus backward reactions c'→c.
Brief summary of Schnakenbergs argument (1)Brief summary of Schnakenbergs argument (1)
Thermodynamic fluxThermodynamic flux Conjugate thermodynamic forceConjugate thermodynamic force
Extent of reaction Extent of reaction ξξcc′cc′
Chemical a nityffiChemical a nityffi
Compare
with
→ Chemical affinity is chemical potential difference
With and
we arrive at:
Brief summary of Schnakenbergs argument (2)Brief summary of Schnakenbergs argument (2)
F = ∑cc '
Acc '˙ξcc ' = −∑
cc '
A cc '˙N cc '
Brief summary of Schnakenbergs argument (3)Brief summary of Schnakenbergs argument (3)
In the stationary state we have
With . Hence turns intoF=∑cc '
Acc '˙ξcc '
E,T constant
Brief summary of Schnakenbergs argument (4)Brief summary of Schnakenbergs argument (4)
S=−kB∑c , c '
ξc , c ' ln[X c ' ]wc '→c
[X c ]wc→c '
S = −kB∑c , c '
ξc , c ' ln[X c ' ]
[X c]− kB∑
c , c '
ξc , c ' lnwc '→c
wc→c '
Brief summary of Schnakenbergs argument (4)Brief summary of Schnakenbergs argument (4)
⟨ Ssys(t)⟩ = − ∑c , c '∈Ωsys
P (c ,t )wc→c ' lnP(c ' , t)P (c ,t )
⟨ Stot ⟩ = ⟨ S sys⟩ + ⟨ Senv⟩
S=−kB∑c , c '
ξc , c ' ln[X c ' ]wc '→c
[X c ]wc→c '
S = −kB∑c , c '
ξc , c ' ln[X c ' ]
[X c]− kB∑
c , c '
ξc , c ' lnwc '→c
wc→c '
Brief summary of Schnakenbergs argument (4)Brief summary of Schnakenbergs argument (4)
⟨ Stot ⟩ = ⟨ S sys⟩ + ⟨ Senv⟩
⟨ Senv(t)⟩ = ∑c , c '∈Ωsys
P(c , t)wc→c ' lnwc→c '
wc '→c
S=−kB∑c , c '
ξc , c ' ln[X c ' ]wc '→c
[X c ]wc→c '
S = −kB∑c , c '
ξc , c ' ln[X c ' ]
[X c]− kB∑
c , c '
ξc , c ' lnwc '→c
wc→c '
⟨ Ssys(t)⟩ = − ∑c , c '∈Ωsys
P (c ,t )wc→c ' lnP(c ' , t)P (c ,t )
Environmental entropy productionEnvironmental entropy production
⟨ Senv(t) ⟩ = ∑c ,c '∈Ωsys
P(c , t )wc ,→c ' lnwc→c '
wc '→ c
Senv(t) = ∑j
δ(t−t j) lnwc j−1→ c j
wc j→c j−1
Important consequence:
Irreversible transitions do not exist.In Nature, there are no „absorbing states“.
totTotal state space
sysSystem state space
EnvironmentSystem
drive
Explaining entropy productionExplaining entropy productionin terms of microstatesin terms of microstates
Simplest example:Simplest example:
Stochastic clock in a stationary state
Counting the number of cycles, we may think of a linear chain of transitions
Each configuration corresponds to a certain number of configurations of the environment.
Assume equal rates among all transitions
Subsystem is driven by an entropic force.
wcc '
wc 'c
=N c ' N c
Environmental entropy production
wcc '
wc 'c
=N env c '
N env c S env = −ln N env c '
ln N envc
S env = lnwcc '
wc 'c
Senv(t) = ∑j
δ(t−t j) lnωc j−1→ c j
ωc j→ c j−1
Question
Under which conditions is this formula correct?
Answer:Answer:
● The formula is correct if the environment The formula is correct if the environment equilibrates instantaneouslyequilibrates instantaneously after each transition. after each transition.
● In realistic systems this is not necessarily true.In realistic systems this is not necessarily true.
● The formula could provide an upper bound in the The formula could provide an upper bound in the longtime limit (ongoing research)longtime limit (ongoing research)
Senv(t) = ∑j
δ(t−t j) lnωc j−1→ c j
ωc j→ c j−1
Question:Question: Under which conditions is Under which conditions is this formula correct? this formula correct?
// Example: biased random walkconst double p=0.3;int x=0; double S_env=0;...if (rnd()<p)
{x++;S_env += ln(p)/ln(1p)}
else{x;S_env = ln(p)/ln(1p);}
Environmental entropy production is easily accessible in numerical simulations.
Whenever the configuration changes, simply add lnwc→c '
wc '→c
p1-p
Ssys(t ) = −P (c (t ), t)P (c (t ), t)
−∑j
δ(t−t j) lnP(c j , t)
P(c j−1 , t)
Senv(t) = ∑j
δ(t−t j) lnωc j→ c j+1
ωc j+1→ c j
Stot (t ) = −P (c (t ), t)P (c (t ), t)
−∑j
δ(t−t j) lnP (c j , t)wc j−1→c j
P (c j−1 , t)wc j→c j−1
No entropy production in the stationary state Detailed balance↔
Two equivalent definitions of detailed balance:Two equivalent definitions of detailed balance:
Probability currents in the stationary state cancel
pairwise:
∀ c ,c '∈ :
P cwcc ' = P c ' wc 'c
For each closed stochastic path
the product of all rates along this path is equal to the product of
the rates in reverse direction
wc1c2wc2c3
...wcN−1cNwcNc1
=
wcNcN−1wcN−1cN−2
...wc2c1wc1cN
c1 c2 ...cN c1
● does not rely on P(c)● difficult to prove● easy to disprove
● requires knowledge of P(c)● easy to prove
2. Fluctuation theorem revisited
t
Stot(t)
t
ΔStot(t)
Second law:
but it fluctuates sometimes even in opposite direction
⟨ S tot ⟩ ≥0
t
Stot(t)
t
ΔStot(t)
P(ΔStot)
ΔStot
t
Stot(t)
t
ΔStot(t)
P(ΔStot)
ΔStot
P S tot
P − S tot= e S tot
Fluctuation theorem:
To prove the fluctuation theorem,
1) prove it for a single transition c↔c'
2) show that it will hold for any sequence of transitions
YAP yet another proofYAP yet another proofof the fluctuation relationof the fluctuation relation
P (Δ Stot)
P(−ΔS tot)= eΔ S tot
c c'
First step:Consider a single transition c↔c'
P(ΔStot)
ΔStot
First step:Consider a single transition c↔c'
c c'
P(ΔStot)
ΔStot
S tot = lnP cwcc '
P c ' wc 'c
P S tot ∝ P c wcc '
P − S tot ∝ P c ' wc 'c
P S tot
P − S tot = exp S tot
c c'
Fluctuation theorem holds trivially !
Second step: Show that the FR holds for any sequence.
Prove invariance under → convolution:
f x = f −x e x
g x =g −x e x
f∗g x = ∫ f ( y)g(x− y)dy
= ∫ f (− y)e y g(−x+ y)ex− y
= ex∫ f (−y)g( y−x)dy
= ex∫ f ( y)g(− y−x)dy = ex(f∗g)(−x)
Fluctuation relation
holds exactly for the total entropy
holds approximately for the environmental entropyproduction in a nonequilibrium steady state in
the long time limit
P(Δ Senv)
P(−Δ Senv)≈ exp(Δ Senv)
P S tot
P − S tot= exp S tot
Distribution itself is systemdependent
3. Entropy production and renormalization
Arrow can be interpreted as ' time'
Contact process:
A → 2A2A → AA → 0
Example: Directed percolation (DP)Example: Directed percolation (DP)
Bonds openwith probability p
Toy model for epidemic spreading
Absorbing states Infinite entropy production↔
Renormalization scheme for DP by logical ORRenormalization scheme for DP by logical OR
Let be the probability to find adjacent blocks of size mat time t in the bit pattern p.
Example:
P101(5)
000101001010000001011010110
1 1 0 1 1
Pp(m)(t )
Let be the probability to find adjacent blocks of size mat time t in the bit pattern p.
Example:
In a critical DP process increases with timewhile all other decrease with time.
saturates as
P101(5)
000101001010000001011010110
1 1 0 1 1
Pp(m)(t )
P000(m) (t )
S p(m)(t) :=
P p(m)(t)
1−P000(m)(t)
t→∞
Perform two limits:Perform two limits:
1. Take time
2. Take block size
Observation: These quantities are universal.
S p(m) := lim
t→∞S p(m)(t) =
Pp(m)(t)
1−P000(m)(t )
t→∞
m→∞
S p* := lim
m→∞S p(m)
t→∞
m→∞
Useful for:
● Verification whether a given model belongs to DP● Definition of a „clean“ contact process
Number of bits Number of univ. quantities
2 2
3 5
4 9
5 17
space
time
space
time
space
time
space
time
space
time
0 1 →
1 0 →
space
time
0 1 →
1 0 →
space
time
0 1 →
1 0 →
space
time
0 1 →
1 0 →
space
time
0 1 →
1 0 →
w100
w111
Effective transition rates wp in the coarsegrained dynamics
There are
● Reversible transitions 110 111↔
● Irreversible transitions 010 000↔
● Impossible transitions 000 010→
The allowed transitions are expected to decreasewith increasing block size.
Example: Effective 3bit ratesExample: Effective 3bit rates
Example: Effective 3bit ratesExample: Effective 3bit rates
m=4
m=32
t→∞
Example: Effective 3bit ratesExample: Effective 3bit rates
Example: Effective 3bit ratesExample: Effective 3bit rates
Observation:
1. The irreversible rates decrease faster than the reversible rates with increasing block size.
w prev∼m−2 , w p
irr∼m−2.6
Example: Effective 3bit Example: Effective 3bit currentscurrents
Example: Effective 3bit Example: Effective 3bit currentscurrents
m→∞
Observation:Observation:
1. The irreversible currents decrease faster than the reversible currents with increasing block size.
2. The reversible currents approach each other as if they would satisfy detailed balance in the limit
J prev∼m−2 , J p
irr∼m−2.6
m→∞
Summary
● The commonly accepted formula for environmental entropy production holds only if the environment equilibrates instantaneously.
● The fluctuation theorem is a property that it is invariant under convolution.
● It is very difficult (although not impossible) to find other physical quantities which obey the fluctuation theorem.
● Directed percolation has infinite entropy production. Under block renormalization, however, the currents of irreversible transitions vanish faster while reversible transitions seem to approach detailed balance.
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