Dynamical Fluctuations & Entropy Production Principles

Karel Netočný Institute of Physics AS CR

Seminar, 6 March 2007

To be discussed

Min- and Max-entropy production principles: various examples From variational principles to fluctuation laws: equilibrium case Static versus dynamical fluctuations Onsager-Machlup macroscopic fluctuation theory Stochastic models of nonequilibrium Conclusions, open problems, outlook,...

(In collaboration with C. Maes, K. U. Leuven)

Motivation: Modeling Earth climate[Ozawa et al, Rev. Geoph. 41(4), 1018 (2003)]

Linear electrical networks explaining MinEP/MaxEP principles

U22

Kirchhoff’s loop law:

Entropy production rate:

MinEP principle:

Stationary values of voltages minimize the entropy production rate

Not valid under inhomogeneous temperature!

X

k

Uj k =X

k

E j k

¾(U) = ¯Q(U) = ¯X

j ;k

U2j k

R j k

Linear electrical networks explaining MinEP/MaxEP principles

U22

Kirchhoff’s current law:

Entropy production rate:

Work done by sources:

(Constrained) MaxEP principle:

Stationary values of currents maximize the entropy production under constraint

X

j

I j k =0

¾(I ) = ¯Q(I ) = ¯X

j ;k

R j k I 2j k

Q(I ) =W(I )

W(I ) =X

j k

E j kI j k

Linear electrical networks summary of MinEP/MaxEP principles

Current law+

Loop law

MaxEP principle+

Current law

Loop law+

MinEP principleGeneralized

variational principle

I

U

U, I

From principles to fluctuation laws Questions and ideas

How to go beyond approximate and ad hoc thermodynamical principles?

Inspiration from thermostatics:

Equilibrium variational principles are intimately related to structure of equilibrium fluctuations

Is there a nonequilibrium analogy of thermodynamical fluctuation theory?

From principles to fluctuation laws Equilibrium fluctuations

H(x) = Ne

M (x) = Nmeq(e)

H (x) = Ne

Typical value

P (M (x) = Nm) = eN [s(e;m)¡ seq(e)]

Probability of fluctuation

H h(x) = H(x) ¡ hM (x) = N[e¡ hm]

The fluctuation made typical!s(e;m) = sheq(e¡ hm)

add

field

From principles to fluctuation laws Equilibrium fluctuations

Fluctuationfunctional

Variational

functional

Thermodynamic

potentialEntropy (Generalized)

free energy

From principles to fluctuation laws Static versus dynamical fluctuations

Empirical ergodic average:

Ergodic theorem:

Dynamical fluctuations:

Interpolating between static and dynamical fluctuations:

H(x) = Ne

P ( ¹mT =m) = e¡ T I (m)

P¡1N

P Nk=1

m(x¿k) =m¢= e¡ N I ( ¿ ) (m)

Static: ¿ ! 1I (1 )(m) =s(e) ¡ s(e;m)

Dynamic: ¿ ! 0

¹mT ! meq(e); T ! 1

¹mT = 1T

RT0m(xt) dt

Effective model of macrofluctuationsOnsager-Machlup theory

Dynamics:

Equilibrium:

Path distribution:

R dmtdt = ¡ smt +

q2RN wt

P (m1 =m) / e¡ 12N sm2

S(m) ¡ S(0)

P (! ) = exp£¡ N

4

RT0

R2

¡dmtdt + s

Rmt¢2¤

Effective model of macrofluctuationsOnsager-Machlup theory

Dynamics: Path distribution:

Dynamical fluctuations:

(Typical immediate) entropy production rate:

P (! ) = exp£¡ N

4

RT0

R2

¡dmtdt + s

Rmt¢2¤

¾(m) = dS(mt )dt = N s2

2R m2

I (m) = 14¾(m)

P ( ¹mT =m) = P (mt =m; 0 · t · T) = exp£¡ T

Ns2

8Rm2

¤

R dmtdt = ¡ smt +

q2RN »t

Towards general theory

Equilibrium Nonequilibrium

ClosedHamiltonian dynamics

OpenStochastic dynamics

MesoscopicMacroscopic

Linear electrical networks revisitedDynamical fluctuations

Fluctuating dynamics:

Johnson-Nyquist noise:

Empirical ergodic average:

Dynamical fluctuation law:

R1 R2

E

C

E f1 E f

2

E f =q

2R¯ »

white noise

U

total dissipated heat

E = U +R2I +E f2

I = C _U +U ¡ E f

1

R1

¹UT = 1T

RT0Ut dt

¡ 1TlogP ( ¹UT =U) = 1

4¯ 1¯ 2 (R 1+R 2 )¯ 1R 1+¯ 2R 2

hU 2

R 1+ (E ¡ U )2

R 2¡ E 2

R 1+R 2

i

Stochastic models of nonequilibriumjump Markov processes

Local detailed balance:

Global detailed balance generally broken:

Markov dynamics:

log k(x;y)k(y;x) = ¢ s(x;y) = ¡ ¢ s(y;x)

entropy changein the

environment

breaking term¢ s(x;y) = s(y) ¡ s(x) + ²(x;y)

x y

k(x;y)

k(y;x)

d½t(x)dt =

X

y

£½t(y)k(y;x) ¡ ½t(x)k(x;y)

¤

Stochastic models of nonequilibriumjump Markov processes

Entropy of the system:

Entropy fluxes:

Mean entropy production rate:

S(½) = ¡P

x ½(x) log½(x)

Warning:Only for time-reversal invariant observables!

x y

k(x;y)

k(y;x)

¾(½) =dS(½t)dt

+12

X

(x;y)

j½(x;y)¢ s(x;y)

=X

x;y

½(x)k(x;y)½(x)k(x;y)½(y)k(y;x)

j½(x;y) =½(x)k(x;y) ¡ ½(y)k(y;x)| {z }

zero at detailed balance

Stochastic models of nonequilibriumjump Markov processes

(“Microscopic”) MinEP principle:

Can we again recognize entropy production as a fluctuation functional?

x y

k(x;y)

k(y;x)

In the first order in the expansionof the breaking term:

¾(½) =min , ½=½s

Stochastic models of nonequilibriumjump Markov processes

Empirical occupation times:

Ergodic theorem:

Fluctuation law for occupation times?

Note:

¹pT (x) ! ½s(x); T ! 1

P (¹pT =½) = e¡ T I (½)

¹pT (x) = 1T

RT0Â(! t = x)dt

x y

k(x;y)

k(y;x)

I (½s) = 0 Natural variational functional

Stochastic models of nonequilibriumjump Markov processes

Idea: Make the empirical distribution typical by modifying dynamics:

The “field” a(x) is such that p(x) is stationary distribution for the modified dynamics:

Comparing both processes yields the fluctuation law:

Py6=x

£p(x)ka(y;x) ¡ p(y)ka(x;y)

¤= 0

k(x;y) ¡ ! ka(x;y) = k(x;y) ea(y)¡ a(x)

I (p) =P

y6=x

£k(x;y) ¡ ka(x;y)

¤

difference of escape rates

Stochastic models of nonequilibriumfluctuations versus MinEP principle

General observation:

In the first order approximation around detailed balance

The variational functional is recognized as an approximate fluctuation functional

A consequence: A natural way how to go beyond MinEP principle is to study various fluctuation laws

I (½) = 14

£¾(½) ¡ ¾(½s)

¤+o(²2)

General conclusionsopen problems

Similarly, MaxEP principle is obtained from the fluctuation law of empirical current

Is there a natural computational scheme for the fluctuation functional far from equilibrium (e.g. for ratchets)?

Natural mathematical formalism: the large deviation and Donsker-Varadhan theory

Might the dynamical fluctuation theory provide a natural nonequilibrium thermodynamical formalism far from equilibrium?

Outlook from fluctuations to nonequilibrium thermodynamics?

Fluctuationfunctional

Variational

functional

Nonequilibriumpotential

“Corrected”entropy production

??

References C. Maes, K. N. (2006). math-ph/0612063. S. Bruers, C. Maes, K. N. (2007). cond-mat/0701035. C. Maes, K. N. (2006). cond-mat/0612525.

M. D. Donsker and S. R. Varadhan. Comm. Pure Appl. Math., 28:1–47 (1975).

M. J. Klein and P. H. E. Meijer. Phys. Rev., 96:250-255 (1954). I. Prigogine. Introduction to Non-Equilibrium Thermodynamics. Wiley-

Interscience, New York (1962). G. Eyink, J. L. Lebowitz, and H. Spohn. In: Chaos, Soviet-American Perspectives in Nonlinear Science, Ed. D. Campbell, p. 367–391 (1990). E. T. Jaynes. Ann. Rev. Phys. Chem., 31:579–601 (1980). R. Landauer. Phys. Rev. A, 12:636–638 (1975)