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 U 22  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 U j k = X k E j k ¾ ( U ) = ¯ Q ( U ) = ¯ X j ; k U 2 j k R j k

Linear electrical networks explaining MinEP/MaxEP principles U 22  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 2 j k Q ( I ) = W ( I ) W ( I ) = X j k E j k I j k

Linear electrical networks summary of MinEP/MaxEP principles Current law + Loop law MaxEP principle + Current law Loop law + MinEP principle Generalized 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 ) = N e M ( x ) = N m eq ( e ) H ( x ) = N e Typical value P ( M ( x ) = N m ) = e N [ s ( e ; m ) ¡ s eq ( e )] Probability of fluctuation H h ( x ) = H ( x ) ¡ h M ( x ) = N [ e ¡ h m ] The fluctuation made typical! s ( e ; m ) = s h eq ( e ¡ h m ) add field

From principles to fluctuation laws Equilibrium fluctuations Fluctuation functional Variational functional Thermodynamic potential Entropy (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 ) = N e P ( ¹ m T = m ) = e ¡ TI ( m ) P ¡ 1 N P N k = 1 m ( x ¿ k ) = m ¢ = e ¡ NI ( ¿ ) ( m ) S t a t i c:¿ ! 1 I ( 1 ) ( m ) = s ( e ) ¡ s ( e ; m ) D ynam i c:¿ ! 0 ¹ m T ! m eq ( e ) ; T ! 1 ¹ m T = 1 T R T 0 m ( x t ) d t

Effective model of macrofluctuations Onsager-Machlup theory  Dynamics:  Equilibrium:  Path distribution: R d m t d t = ¡ sm t + q 2 R N w t P ( m 1 = m ) / e ¡ 1 2 N sm 2 S ( m ) ¡ S ( 0 ) P ( ! ) = exp £ ¡ N 4 R T 0 R 2 ¡ d m t d t + s R m t ¢ 2 ¤

Effective model of macrofluctuations Onsager-Machlup theory  Dynamics:  Path distribution:  Dynamical fluctuations:  (Typical immediate) entropy production rate: P ( ! ) = exp £ ¡ N 4 R T 0 R 2 ¡ d m t d t + s R m t ¢ 2 ¤ ¾ ( m ) = d S ( m t ) d t = N s 2 2 R m 2 I ( m ) = 1 4 ¾ ( m ) P ( ¹ m T = m ) = P ( m t = m; 0 · t · T ) = exp £ ¡ T N s 2 8 R m 2 ¤ R d m t d t = ¡ sm t + q 2 R N » t

Towards general theory EquilibriumNonequilibrium Closed Hamiltonian dynamics Open Stochastic dynamics Mesoscopic Macroscopic

Linear electrical networks revisited Dynamical fluctuations  Fluctuating dynamics:  Johnson-Nyquist noise:  Empirical ergodic average:  Dynamical fluctuation law: R 1 R 2 E C E f 1 E f 2 E f = q 2 R ¯ » white noise U total dissipated heat E = U + R 2 I + E f 2 I = C _ U + U ¡ E f 1 R 1 ¹ U T = 1 T R T 0 U t d t ¡ 1 T l og P ( ¹ U T = U ) = 1 4 ¯ 1 ¯ 2 ( R 1 + R 2 ) ¯ 1 R 1 + ¯ 2 R 2 h U 2 R 1 + ( E ¡ U ) 2 R 2 ¡ E 2 R 1 + R 2 i

Stochastic models of nonequilibrium jump Markov processes  Local detailed balance:  Global detailed balance generally broken:  Markov dynamics: l og k ( x ; y ) k ( y ; x ) = ¢ s ( x ; y ) = ¡ ¢ s ( y ; x ) entropy change in the environment breaking term ¢ s ( x ; y ) = s ( y ) ¡ s ( x ) + ² ( x ; y ) x y k ( x ; y ) k ( y ; x ) d ½ t ( x ) d t = X y £ ½ t ( y ) k ( y ; x ) ¡ ½ t ( x ) k ( x ; y ) ¤

Stochastic models of nonequilibrium jump Markov processes  Entropy of the system:  Entropy fluxes:  Mean entropy production rate: S ( ½ ) = ¡ P x ½ ( x ) l og½ ( x ) Warning: Only for time-reversal invariant observables! x y k ( x ; y ) k ( y ; x ) ¾ ( ½ ) = d S ( ½ t ) d t X ( x ; y ) j ½ ( x ; y ) ¢ s ( x ; y ) = X x ; y ½ ( x ) k ( x ; y ) ½ ( x ) k ( x ; y ) ½ ( y ) k ( y ; x ) j ½ ( x ; y ) = ½ ( x ) k ( x ; y ) ¡ ½ ( y ) k ( y ; x ) |{z} zeroa t d e t a i l e db a l ance

Stochastic models of nonequilibrium jump 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 expansion of the breaking term : ¾ ( ½ ) = m i n, ½ = ½ s

Stochastic models of nonequilibrium jump Markov processes  Empirical occupation times:  Ergodic theorem:  Fluctuation law for occupation times?  Note: ¹ p T ( x ) ! ½ s ( x ) ; T ! 1 P ( ¹ p T = ½ ) = e ¡ TI ( ½ ) ¹ p T ( x ) = 1 T R T 0 Â ( ! t = x ) d t x y k ( x ; y ) k ( y ; x ) I ( ½ s ) = 0 Natural variational functional

Stochastic models of nonequilibrium jump 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: P y 6 = x £ p ( x ) k a ( y ; x ) ¡ p ( y ) k a ( x ; y ) ¤ = 0 k ( x ; y ) ¡ ! k a ( x ; y ) = k ( x ; y ) e a ( y ) ¡ a ( x ) I ( p ) = P y 6 = x £ k ( x ; y ) ¡ k a ( x ; y ) ¤ difference of escape rates

Stochastic models of nonequilibrium fluctuations 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 ) = 1 4 £ ¾ ( ½ ) ¡ ¾ ( ½ s ) ¤ + o ( ² 2 )

General conclusions open 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? Fluctuation functional Variational functional Nonequilibrium potential “Corrected” entropy production ? ?

References  C. Maes, K. N. (2006). math-ph/  S. Bruers, C. Maes, K. N. (2007). cond-mat/  C. Maes, K. N. (2006). cond-mat/  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: (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)