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Two Approaches to Dynamical Fluctuations in Small Non-Equilibrium Systems M. Baiesi #, C. Maes #, K. Netočný *, and B. Wynants # * Institute of Physics AS CR Prague, Czech Republic Prague, Czech Republic & # Instituut voor Theoretische Fysica, K.U.Leuven, Belgium K.U.Leuven, Belgium MECO34 Universität Leipzig, Germany 30 March – 1 April 2009

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Outlook From the Einstein’s (static) and Onsager’s (dynamic) equilibrium fluctuation theories towards nonequilibrium macrostatistics and dynamical mesoscopic fluctuations

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Outlook From the Einstein’s (static) and Onsager’s (dynamic) equilibrium fluctuation theories towards nonequilibrium macrostatistics and dynamical mesoscopic fluctuations An exact Onsager-Machlup framework for small open systems, possibly with high noise and beyond Gaussian approximation

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Outlook From the Einstein’s (static) and Onsager’s (dynamic) equilibrium fluctuation theories towards nonequilibrium macrostatistics and dynamical mesoscopic fluctuations An exact Onsager-Machlup framework for small open systems, possibly with high noise and beyond Gaussian approximation Towards non-equilibrium variational principles; role of time-symmetric fluctuations

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Outlook From the Einstein’s (static) and Onsager’s (dynamic) equilibrium fluctuation theories towards nonequilibrium macrostatistics and dynamical mesoscopic fluctuations An exact Onsager-Machlup framework for small open systems, possibly with high noise and beyond Gaussian approximation Towards non-equilibrium variational principles; role of time-symmetric fluctuations Generalized O.-M. formalism versus a systematic perturbation approach to current cumulants

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Generic example: (A)SEP with open boundaries

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Local detailed balance principle: Breaking detailed balance µ 1 > µ 2 Not a mathematical property but a physical principle!

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Generic example: (A)SEP with open boundaries Macroscopic description: fluctuations around diffusion limit, noneq. boundaries Static fluctuation theory Time-dependent fluctuations (Einstein) (Onsager-Machlup)

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Generic example: (A)SEP with open boundaries Macroscopic description: fluctuations around diffusion limit, noneq. boundaries Static fluctuation theory Time-dependent fluctuations Small noise theory

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Generic example: (A)SEP with open boundaries Macroscopic description: fluctuations around diffusion limit, noneq. boundaries L. Bertini, A. D. Sole, D. G. G. Jona-Lasinio, C. Landim, Phys. Rev. Let 94 (2005) T. Bodineau, B. Derrida, Phys. Rev. Lett. 92 (2004)

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Generic example: (A)SEP with open boundaries Mesoscopic description: large fluctuations for small or moderate L, high noise Time span is the only large parameter Fluctuations around ergodic averages

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General: Stochastic nonequilibrium network W Q y x y z SS Dissipation modeled as the transition rate asymmetry Local detailed balance principle Non-equilibrium driving Q Q’

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How to unify? Ruelle’s thermodynamic formalism Evans-Gallavotti-Cohen fluctuation theorems Min/Max entropy production principles (Prigogine, Klein-Meijer) Donsker-Varadhan large deviation theory Onsager-Machlup framework

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How to unify? Ruelle’s thermodynamic formalism Evans-Gallavotti-Cohen fluctuation theorems Min/Max entropy production principles (Prigogine, Klein-Meijer) Donsker-Varadhan large deviation theory Onsager-Machlup framework ?

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Occupation-current formalism Consider jointly the empirical occupation times and empirical currents - x y xtxt time

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Occupation-current formalism Consider jointly the empirical occupation times and empirical currents Compute the path distribution of the stochastic process and apply standard large deviation methods (Kramer’s trick) Do the resolution of the fluctuation functional w.r.t. the time-reversal (apply the local detailed balance condition)

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Occupation-current formalism Consider jointly the empirical occupation times and empirical currents General structure of the fluctuation functional: (Compare to the Onsager-Machlup)

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Occupation-current formalism Dynamical activity (“traffic”) Entropy flux Equilibrium fluctuation functional

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Occupation-current formalism Dynamical activity (“traffic”) Entropy flux Equilibrium fluctuation functional Time-symmetric sector Evans-Gallovotti-Cohen fluctuation symmetry

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Towards coarse-grained levels of description Various other fluctuation functionals are related via variational formulas E.g. the fluctuations of a current J (again in the sense of ergodic avarage) can be computed as Rather hard to apply analytically but very useful to draw general conclusions For specific calculations better to apply a “grand canonical” scheme

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Fluctuations of empirical times alone: MinEP principle: fluctuation origin

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Fluctuations of empirical times alone: MinEP principle: fluctuation origin Expected entropy flux Expected rate of system entropy change

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Fluctuations or empirical times alone: This gives a fluctuation-based derivation of the MinEP principle as an approximatate variational principle for the stationary distribution Systematic corrections are possible, although they do not seem to reveal immediately useful improvements MaxEP principle for stationary current can be understood analogously MinEP principle: fluctuation origin Expected entropy flux Expected rate of system entropy change

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Some remarks and extensions The formalism is not restricted to jump processes or even not to Markov process, and generalizations are available (e.tg. to diffusions, semi-Markov systems,…) Transition from mesoscopic to macroscopic is easy for noninteracting or mean-field models but needs to be better understood in more general cases The status of the EP-based variational principles is by now clear: they only occur under very special conditions: close to equilibrium and for Markov systems Close to equilibrium, the time-symmetric and time-anti- symmetric sectors become decoupled and the dynamical activity is intimately related to the expected entropy production rate Explains the emergence of the EP-based linear irreversible thermodynamics

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Perturbation approach to mesoscopic systems Full counting statistics (FCS) method relies on the calculation of cumulant- generating functions like for a collection of “macroscopic’’ currents J B This can be done systematically by a perturbation expansion in λ and derivatives at λ = 0 yield current cumulants This gives a numerically exact method useful for moderately-large systems and for arbitrarily high cumulants A drawback: In contrast to the direct (O.-M.) method, it is harder to reveal general principles! Rayleigh–Schrödinger perturbation scheme generalized to non-symmetric operators

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References [1] C. Maes and K. Netočný, Europhys. Lett. 82 (2008) [2] C. Maes, K. Netočný, and B. Wynants, Physica A 387 (2008) [3] C. Maes, K. Netočný, and B. Wynants, Markov Processes Relat. Fields 14 (2008) 445. [4] M. Baiesi, C. Maes, and K. Netočný, to appear in J. Stat. Phys (2009). [5] C. Maes, K. Netočný, and B. Wynants, in preparation.

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