Beyond Onsager-Machlup Theory of Fluctuations Karel Netočný Institute of Physics AS CR Seminar, 13 May 2008

Why to worry about fluctuations? Structure of equilibrium fluctuations is well understood and useful Second law S = m a x Thermodynamics Fluctuations P = e x p ¡ S k ¢ S = k l n W Entropy thermodynamic potential variational functional fluctuation functional

Example: fluctuations of energy System spontaneously exchanges energy with reservoir Free energy and its variational extension determine fluctuations Mathematical structure of large deviations Reservoir System E ¯ = 1 k T P ( E ) = e ¡ 1 k T f S F g ¡ l o g P F = m i n E f ¡ T S ( ) g 1 E E e q

Search for nonequilibrium ensembles Reservoir System E ¯ = 1 k T Engine W Q McLennan’s extension of Gibbs ensembles to weak nonequilibrium: P ( x ) ' 1 Z e ¡ ¯ H + ¢ S Gibbs corrected by “excess entropy”

Systems close to equilibrium Linear response theory and fluctuation-dissipation relations (Kubo, Mori,…) MaxEnt formalism (Jaynes) and Zubarev ensembles Linear irreversible thermodynamics (Prigogine, Nicolis,…) Local equilibrium methods (Lebowitz and Bergmann,…) Dynamical fluctuation theory (Onsager and Machlup,…)

Landauer: essential role of dynamics Blowtorch theorem says that noise plays a fundamental role in determining steady state No “magic” universal principles are to be expected to describe open systems One cannot avoid studying the full kinetics Precist si poradne Landauera!

Onsager-Machlup fluctuation theory

Onsager-Machlup fluctuation theory Modelling small macroscopic fluctuations around exponential relaxation to equilibrium Action has a generic structure: _ m i = ¡ L j + \ n o s e R k P ( ! ) = e ¡ A path A ( ! ) = 1 2 R d t £ D ¡ _ S + E ¤ Time locality D = R j k _ m S = ¡ s j k m E = L j k m

Onsager-Machlup fluctuation theory Describes normal fluctuations away from criticality It is consistent with the fluctuation-dissipation relation It provides a “microscopic” basis for the least dissipation principle: It provides a dynamical extension of Einstein’s theory of macroscopic fluctuations D 2 ¡ _ S = m i n

Nonlinear systems with weak noise A method to study dynamical systems is to add a noise and to analyze fluctuations Friedlin-Wentzel theory: Method of effective potential: _ x = Á ( ) + p " : w h i t e n o s P ( ! ) = e x p ¡ 1 " A + c o r t i n s ¢ A = 1 4 R d t [ _ x ¡ Á ( ) ] 2 Field-theoretical methods to compute the effective potential ª ( x ) = m a 1 T R d t ! l o g P

Brillouin paradox _ x = Á ( ) + p " : w h i t e n o s Nonlinear stochastic model often leads to inconsistencies (e.g. to rectification of fluctuations violating second law) _ x = Á ( ) + p " : w h i t e n o s N. G. van Kampen: “Statistical mechanics leads, on the macroscopic level to a stochastic description in terms of a master equation. Subsequently deterministic equations plus fluctuations can be extracted from it by suitable limiting processes. The misconception that one should start from the known macroscopic equations and then somehow add the fluctuations on to them is responsible for much confusion in the literature.”

Nonequilibrium macrostatistics Initiated by Derrida, Jona-Lasinio, Bodineau,… Many-body stochastic system analyzed in hydrodynamic limit The scaling yields a diffusion equation + noise Here it is possible to add white noise here because of the diffusive regime @ ½ t = r ¢ [ D ( ) + n o i s e ]

Some questions Can one extend the Onsager-Machlup approach to mesoscopic systems possibly far from equilibrium? (strong noise!) What would be the natural observables there? Can one understand the mesoscopic stationary variational principles this way? How to extend to quantum open systems?

Stochastic nonequilibrium network 1 Q 2 S S Dissipation modeled as the rate asymmetry Local detailed balance condition z y y k ( x ! y ) = e ¢ S x

Some trivias Description using statistical distributions Single-time distribution for Markov system evolves as Local energy currents are Stationary distribution and stationary currents observed as typical long-time averages d ½ t ( x ) = X y £ k ; ¡ ¤ - x J ½ ( x ; y ) = k ¡ | { z } e r o a t d i l b n c Ergodic average of empirical residence times Ergodic average of empirical directed number of jumps

Result I. Natural fluctuation observables It is useful to consider jointly all empirical residence times and all empirical currents as a collection of “canonical” observables A natural expansion parameter of the fluctuation theory is the inverse observation time (remember that noise is no longer weak) P ( p T = ; j ) e ¡ I + c o r t i n s I Mesoscopic Onsager-Machlup functional stationary

Result II. Mesoscopic Onsager-Machlup functional It takes the explicit form I ( p ; j ) = 1 2 £ G + H F ¡ _ S ¤ Entropy current _ S = 1 2 X x ; y F ( ) j Conjugated dissipation functional @ H F = j G Onsager-like dissipation functional H = 2 ( T F ¡ ) Driving from reference equilibrium k F ( x ; y ) = e 2 Traffic T = X x ; y p ( ) k +

Result III. Close-to-equilibrium decoupling In the linear response regime, the time-symmetric and time-antisymmetric fluctuations mutually decouple Moreover, the traffic excess equals to the total entropy production excess Then the total entropy production fully determines the structure of fluctuations I ( p ; j ) = 1 + 2 T F ¡ = E E = _ S + d s y t

Various remarks and outlook Implies (mesoscopic) minimum and maximum entropy production principles close to equilibrium The relation between the traffic and the entropy production excesses remains true in the diffusion regime far from equilibrium Apart from the two cases, the traffic emerges as a novel relevant functional determining dynamical fluctuations More coarse-grained fluctuation laws follow by solving corresponding variational problems Extension of the Onsager-Machlup formalism to open systems with quantum coherence remains to be understood…

FCS approach to mesoscopic systems More standard FCS methods yield a direct access to the (partial) current cummulants [See Flindt, Novotný, Braggio, Sassetti, and Jauho, PRL (2008).] The FCS methods can also be adapted to study the time-symmetric fluctuations (e.g. the residence-time distribution) Extensions to time-dependent (AC) driving

References [1] C. Maes and K. Netočný, Europhys. Lett. 82 (2008) 30003. [2] C. Maes, K. Netočný, and B. Wynants, Physica A 387 (2008) 2675. [3] C. Maes, K. Netočný, and B. Wynants, arXiv:cond-mat/0709.4327.