Mathematical Structures of Statistical Mechanics: from equilibrium to nonequilibrium and beyond

Hao Ge

Beijing International Center for Mathematical Researchand Biodynamic Optical Imaging Center

Peking University, China

Equilibrium statistical mechanics and maximum entropy principle

Microcanonical ensemble and equal probability priori

In statistical physics, the microcanonical ensemble refers to an isolated system, where all the possible macrostates of the system have the same energy; The probability for the system to be in any given microstate is the same; Khinchin was the first mathematician to make this statement rigorous; Equal probability priori could also be viewed as maximum entropy principle for microcanonical ensemble.

∑−=i

ii ppS log

Canonical ensemble: Boltzmann/Gibbs/Darwin-Fowler

Suppose one has a large microcanonical ensemble consisting of N closed, identical small canonical ensembles; Let Xi represent the microstate of the i-th canonical ensemble, say (p, q). So the high dimensional microstate vector X=(X1,X2,...,XN);Then assume the function g(Xi) is the energy of the i-th canonical ensemble.

totn Exgxgxg =+++ )(...)()( 21

Canonical ensemble: Boltzmann/Gibbs/Darwin-Fowler

Boltzmann's approach: Suppose the probability of X is only confined in the subspace {Etot=H} and equally distributed (zero probability outside {Etot=H}), then the marginal distribution of each Xi is exponentially dependent on g(Xi) when N tends to infinity; Boltzmann’s most probable state method, Gibbs’ partition function method and Darwin-Fowler’s steepest descent method are equivalent; In mathematical language, they all concern about the convergence of marginal distribution or empirical distribution.

Canonical ensemble: Maximum entropy principle approach

E. T. Jaynes proposed an alternative approach to statistical mechanics based on maximum entropy principle. He called this framework as “subjective thermodynamics”; Canonical distribution (Boltzmann’s law) is exactly the posterior distribution from the maximum entropy principle up to the given averaged energy.

Can Maximum Entropy really replace Boltzmann/Gibbs's theory as the foundation of statistical mechanics?

What is maximum entropy principle

Given that some quantity averaged over a large number of individual random variables shows highly unlikely behavior, what is the conditional distribution of an individual sample?Maximum entropy principle is a special case of the minimum relative entropy principle:

{Xi | i=1,2,…,N} i.i.d. with probability density fpriori

Under the conditiong(X1) + g(X2) +…+ g(XN) = α = NaWhat is the asymptotic posterior distribution?

Minimum relative entropy principle

Xi

Xj

Now, if N is very large, then a is essentially the expected value of the posterior distribution for each Xi!

Minimum relative entropy:

dxxfxf

xfpriori

posteriorposterior )(

)(ln)(∫

subjected to: .)()( adxxfxg posterior =∫

Several remarks

The mathematical result is a consequence of the theory of large deviations in stochastic analysis, and is called Gibbs conditioning (Level II); It is essentially the content of E.T. Jaynes’ Principle of Maximum Entropy (MaxEnt), when choosing the prior distribution as uniformly random; Shannon theory’s application to communication engineering aside, the “information theory” as a new paradigm essentially based on this mathematical theorem. It really has nothing to do with real “physics”.

Markov processes follow from MaxEnt approach in the time domain

Markov processes are a natural consequence of the dynamical principle of Maximum Caliber;

(1) Assuming general stochastic processes a la A. Kolmogorov;

(2) Conditioned on observing frequencies of singlet or pair-wise statistics.

H. Ge, S. Pressé, K. Ghosh and K. Dill: (preprint) (2011)

Equivalence of Boltzmann’s and MaxEnt arguments for canonical ensemble

Boltzmann assumed energy conservation following classical mechanics , and equal probability a priori in microcanonical ensemble; MaxEnt assumes uniform prior distribution in canonical ensemble due to ignorance and constraint on observed energy. They are mathematically equivalent! But for MaxEnt approach, we should assume there is a prior distribution for canonical ensemble when not up to any energy constrain. It is not a big deal if we accept Jaynes’ subjective point of view.

Grand canonical ensemble and Gibbs paradox

In classic textbooks, people just imitate Boltzmann’s most probable state approach to express the microstate probability distribution of grand canonical ensemble with energy and particle number; However, this approach could not give the correct distribution of particle numbers (missing the n factorial); And we would not be able to catch this n factorial term either applying the MaxEnt principle; We must appeal to quantum mechanics?

=

Tkn

n1

TZ1p

Bn

µµ

exp!),(

Grand canonical ensemble and Gibbs paradox

If we correctly apply Botzmann’s equal probability priori of the large microcanonical ensemble, we would definitely see the n factorial term of the particle number statistics comes out, no matter the particles are distinguishable or not; For undistinguishable particles, the n factorial merges due to the calculation of phase space volume; for distinguishable particles, it is due to the partition of particles into grand canonical ensemble; This is known as Poisson statistics for a point process. Minimum relative entropy principle is a purely mathematical theorem, which could really substitute equilibrium statistical mechanics.

Superstructure and possible new ingredients for nonequilibrium statistical mechanics

Oono, Y. and Paniconi, M., Prog. Theor. Phys. Suppl. (1998)

M. Esposito, U. Harbola and S. Mukamel, Phys. Rev. E. (2007)

H. Ge, Phys. Rev. E. (2009)

H. Ge and H. Qian, Phys. Rev. E (2010)

M. Esposito and C. van den Broeck, Phys. Rev. Lett. (2010)

Master equation model

No matter starting from any initial distribution, it will finally approach its stationary distribution satisfying

( ) 01

=−∑=

N

jij

ssiji

ssj kckc

∑ −=j

ijijiji kckcdt

tdc )()(

Consider a motor protein with N different conformations R1,R2,…,RN. kij is the first-order or pseudo-first-order rate constants for the reaction Ri→Rj.

Self-assembly or self-organization

ijssiji

ssj kckc =

Detailed balance

Superstructure of thermodynamics

;dfdtdF

−= fd: free energy dissipation rate

;dhk hQdtdE

−=hd: heat dissipation/work out Qhk: house keeping heat/work in

;dp hedtdS

−= ep: entropy production rate

Relative entropy

( ) ( ) ;log)()()( log)()()(ji

ij

jijijijid

jij

iji

jijijijip k

kktcktcth

kckc

ktcktcte ∑∑>>

−=−= ；

( ) . log)()( )( ∑>

−=ji ji

ssj

ijssi

jijijihk kckc

ktcktctQ

.,,0

00≥+=

≥≥

hkdp

hkd

QfeQf

ep characterizes total time irreversibility in aMarkov process.

When system reaches stationary, fd = 0.When system is closed (i.e., no active energy

drive, detailed balaned) Qhk = 0. Boltzmann: fd = ep >0 but Qhk=0;

Prigogine (Brussel school, NESS): Qhk=ep > 0 but fd=0.fd ≥ 0 in driven systems is “self-organization”.

Two origins of irreversibility

Recover the fundamental equation of classical thermodynamics

0)()(≥=−= dfdt

tdFdt

tdS

Entropy increases for systems with stationary equal probability, such as microcanonical ensemble

Isothermal system, not entropy increasing, but free energy decreases (Helmholtz); However, the rate of free energy decrease is precisely entropy production. In equilibrium steady state, entropy production = 0, heat dissipation = 0, time is reversible. If the equilibrium distribution is constant, i.e. equal probability a priori, we have

Time-inhomogeneous systems

)()()( thtedt

tdSdp −=

)()()( tQtWdt

tdEex

ext −=

)()()( tftWdt

tdFd

ext −=

∑ −=j

ijijiji tkctkcdt

tdc ))()(()(

)()()()()( tQthtftetQ exddphk −=−=

Dissipative work in Jarzynski equality

Entropy in Hatano-Sasa equality. We would like to call it intrinsic entropy, which could be defined at individual level.

Two faces of the Second Law

dhdtdS

−≥

In non-detailed balance case, the new one is stronger than the traditional one.

In detailed-balance case, they are equivalent.

extWdtdF

≤

0 , ≡= hkdp Qfe

exexext Q

dtdSQ

dtdUW

dtdF

−≥⇒+=≤

dhdtdS

−≥

Phase transition and emergent landscape for singular-perturbed diffusion

Ge, H., Qian, H.: Phys. Rev. Lett. (2009)

Qian, H.: J. Stat. Phys. (2010)

Ge, H., Qian, H.: J. Roy. Soc. Interface (2011)

Ge, H., Qian, H.: (in preparation) (2011)

( ) . ,0)()()( Nssss RxxFxpxp ∈=−∇⋅∇ εεε

Emergent landscape

.0)()(loglim0

≤−=→

xxpss φε εε

)(xFdtdx

=

( ) .0)()()())(( 2 ≤∇−=⋅∇= xxFxdt

txd φφφ

The corresponding deterministic nonlinear chemical dynamics follow the downhill of the function φ(x) (Lyapunov property).

Related to Hamiltonian-Jacobi equation as well as Freidlin-Wentzel large deviation theory (action along a path).

Large deviation rate function

Emergent dynamic landscape

)(xφ

Global minimum

Local minimum

Maximum: the barrier

Stable fixed points of deterministic dynamics

Unstable fixed point of deterministic dynamics

)(xFdtdx

=

Relative stability of stable steady states

Many nonlinear dynamical systems have multiple, locally stable steady states.Is one attractor more “important” than another?

( )( ) .0,/exp)( →−≈ εεφε xxpss

The most important steady state when εis small would be the global minimum of dynamic landscape.

Maxwell construction

),( θxFdtdx

=θ *

φ (x,θ )

θ

Steady States x*

x

Global minimum(LLN) abruptly transferred (discontinuous, phase transition, symmetry breaking).

Related to “non-convexity”of the landscape and “broken ergodicity” of the system

Kramers’ theory

., /12

/21

1221 εε →→ −→

−→ ∝∝ HH eTeT

21→H

The switching time between attractors:

12→H

The barrier H here may not be the same as the barrier in the previous landscape φ(x) for multistable cases.

Local-global conflictionBA ⇔ CB ⇔ AC ⇔(a)

(b)

(c)

AB

C

Completely nonlinear, nonequilibrium phenomenon.

Just cut and glue on the local landscapes (non-derivative point).

The emergent Markovian jumping process being nonequilibrium is equivalent to the discontinuity of the local landscapes (time symmetry breaking).

Dynamic on a ring as an example.

Local landscape and Kramers’theory in multistable case

According to the large deviation theory of Freidlin and Wentzel, the local landscape in each attractor equals logarithm of the probability of the trajectory with the highest probability starting from the stable fixed point or limit cycle in this attractor.

In this case, Kramers’ theory says that the barrier crossing time from the i-th attractor to the j-th attractor is just exponentially dependent on the lowest barrier of the local landscape along the boundary between the two attractors.

Within these transition rates in hand, one could build a emergent Markovian jumping process in the time scale of inter-attractorial dynamics, whose state space is just the attractors of the diffusion process.The steady distribution of this emergent process would help to construct the global landscape from the local ones.

Possible future generalization

The relation of Boltzmann’s approach and maximum entropy approach in continuous case, i.e. in the real high-dimensional phase space; Thermodynamic superstructure for reaction-diffusion process, or even for fluid mechanics; Maxwell construction and local-global landscape confliction of phase transition occur for high-dimensional chemical master equation (when V tends to infinity).

Summary

Minimum relative entropy principle is a purely mathematical result, which then could be applied beyond statistical mechanics. Thermodynamic superstructure would explicitly distinguish Boltzmann and Prigogine’s thesis, which are actually two faces of the Second Law; Maxwell construction of phase transition occurs in singular-perturbed diffusion as well as chemical master equation, and local-global landscape confliction is the origin of nonequilibrium emergent Markovian jumping process.

Acknowledgement

Prof. Hong Qian

University of WashingtonDepartment of Applied Mathematics

Thanks for your attention!