Unique additive information measures – Boltzmann-Gibbs-Shannon, Fisher and beyond

Peter VánBME, Department of Chemical Physics

Thermodynamic Research GroupHungary

1. Introduction – the observation of Jaynes2. Weakly nonlocal additive information measures3. Dynamics – quantum mechanics4. Power law tails

Microcanonical and canonical equilibrium distributions

5. Conclusions

1 Entropy in local statistical physics(information theoretical, predictive, bayesian)

(Jaynes, 1957):

The measure of information is unique under general physical conditions.

(Shannon, 1948; Rényi, 1963)

– Extensivity (density)– Additivity

)()()( 2121 fsfsffs

fkfs ln)( (unique solution)

The importance of being unique:

Robustness and stability: link to thermodynamics.

Why thermodynamics?

• thermodynamics is not the theory of temperature• thermodynamics is not a generalized energetics• thermodynamics is a theory of STABILITY

(in contemporary nonequilibrium thermodynamics)

•The most general well-posedness theorems use explicite thermodynamic methods (generalizations of Lax idea)

•The best numerical FEM programs offer an explicite thermodynamic structure (balances, etc…)

Link to statistical physics: universality (e.g. fluid dynamics)

)()()( 2121 fsfsffs

)(')(')(

)(')(')(

2211212

1212211

fsffsfffsf

fsffsfffsf

1

2

2

121

)(')(')('

f

fs

f

fsffs

.)(' constfsf

Cfdff

fs ln)(

ffs ln)(

2 Entropy in weakly nonlocal statistical physics (estimation theoretical, Fisher based)

(Fisher, …., Frieden, Plastino, Hall, Reginatto, Garbaczewski, …):

– Isotropy

))(,(),( 2DffsDffs

– Extensivity

– Additivity

),(~),(~))(,( 22112121 DffsDffsffDffs

2

2

12 )(

ln))(,(f

DfkfkDffs

Fisher

Boltzmann-Gibbs-Shannon

(unique solution)

There is no estimation theory behind!

2

2

12 )(

ln))(,(f

DfkfkDffs

– What about higher order derivatives?

– What is the physics behind the second term? What could be the value of k1?

– What is the physics behind the combination of the two terms?

Equilibrium distributions with power law tails

Dynamics - quantum systems

Dimension dependence

Second order weakly nonlocal information measures

– Isotropy in 3D

))(,)(),(

,)(,,)(,(),,(32222

22223

23

fDTrfDTrfDTr

DffDDfDffDDfDffsfDDffs

– Extensivity

– Additivity

),,(~),,(~))(),(,(

22

222

11

212

2121

fDDffsfDDffs

ffDffDffs

(unique solution)

))(1

)(1

)(1

)(1

)()32(

1

)()(

)()(

)(ln

),,(

3236

2225

24

2243

25

2

632

32

6

6

634

4

522

2

1

23

fDTrf

k

fDTrf

kfDTrf

kDffDDff

k

DffDDff

DfkkDffDDf

fk

f

Dfkk

f

Dfkk

f

Dfkfk

fDDffs

Depends on the dimension of the phase space.

ln

)())(,(

2

2

12 kks

FisherBoltzmann-Gibbs-Shannon

– Mass-scale invariance (particle interpretation)

–

),(),( ss

f probability distribution

3 Quantum systems – the meaning of k1

21 k

Why quantum?

048

ln2

''1

2

11

Rk

kRp

mkRR

k

kR

02

''2

22

ExD

m

Time independent Schrödinger equation

Probability conservationMomentum conservation+ Second Law (entropy inequality)

Equations of Korteweg fluidswith a potential

0 v0Pv )C(

2

2

1 mk

0)C()C(s s j

Schrödinger-Madelung fluid

222),,(

22v

v

SchM

SchMs

2

8

1 2rSchM IP

(Fisher entropy)

Bernoulli equation

Schrödinger equation

v m

ie

21 k

2

)(

22),,(

22v

vm

s SchMSchM

Logarithmic and Fisher together?

Nonlinear Schrödinger eqaution of Bialynicki-Birula and Mycielski (1976)(additivity is preserved):

Existence of non dispersive free solutions – solutions with finite support - Gaussons

Ubmt

i2

2

ln2

Stationary equation for the wave function:

02

ln''2

22

ExD

km

12

ln)(

.

2

2

2

1

fdxEdxm

pf

dpfkf

Dfkfextr

4 MaxEnt calculations – the interaction of the two terms(1D ideal gas)

fR :

048

ln2

''1

2

11

Rk

kRp

mkRR

k

kR

boundarys RRJ |'

048

ln2

''1

2

11

Rk

kRp

mkRR

k

kR

CR

R

)0(

0)0('

There is no analitic solution in general

There is no partition function formalism

symmetry

Numerical normalization

One free parameter: Js

-2 -1 1 2x

0.2

0.4

0.6

0.8

1

1.2

R

18

,12

),5,2,5.1,3.1,2.1,1.1,1(4 111

mkk

k

k

k

Conclusions

– Corrections to equilibrium (?) distributions

– Corrections to equations of motion

– Unique = universal• Independence of micro details• General principles in background

Thermodynamics Statistical physics

Information theory?

Concavity properties and stability:

ff

Dff

Df

f

Df

fk

k

kfsff

4

1

4

44

)(1

)(ˆ

2

23

2

11

2

Positive definite

If k is zero (pure Fisher): positive semidefinite!

One component weakly nonlocal fluid

),,,(C vv ),,,,(Cwnl vv

)C(),C(),C(s Pjs

Liu procedure (Farkas’s lemma):

constitutive state

constitutive functions

0 v

0)C()C(s s j0Pv )C(

... Pvjs2

)(s),(s2

e

vv

2),(s),,(s

2

e

vv

),( v basic state

0:s2

ss2

1 22

s

vIP

rv PPP

reversible pressurerP

Potential form: Qr U P

)()( eeQ ssU Euler-Lagrange form

Variational origin

Schrödinger-Madelung fluid

222),,(

22v

v

SchM

SchMs

2

8

1 2rSchM IP

(Fisher entropy)

Bernoulli equation

Schrödinger equation

v ie

Alternate fluid

2)(),(

AltAlts

)(

42IP Altr

Alt

2Alt

AltU

Korteweg fluids:

22)( IP prKor