Thermodynamic formula for the cumulant generating function of time-averaged current

Abstract

The cumulant generating function of time-averaged current is studied from an operational viewpoint. Specifically, for interacting Brownian particles under non-equilibrium conditions, we show that the first derivative of the cumulant generating function is equal to the expectation value of the current in a modified system with an extra force added, where the modified system is characterized by a variational principle. The formula reminds us of Einstein's fluctuation theory in equilibrium statistical mechanics. Furthermore, since the formula leads to the fluctuation-dissipation relation when the linear response regime is focused on, it is regarded as an extension of the linear response theory to that valid beyond the linear response regime. The formula is also related to previously known theories such as the Donsker-Varadhan theory, the additivity principle, and the least dissipation principle, but it is not derived from them. Examples of its application are presented for a driven Brownian particle on a ring subject to a periodic potential.

21 pages, 2 figures; In ver. 2, a new appendix has been added

References in corpus (22)

• Non equilibrium steady states: fluctuations and large deviations of the density and of the current
• First-order dynamical phase transition in models of glasses: an approach based on ensembles of histories
• Fluctuations and response of nonequilibrium states
• Integral fluctuation theorem for the housekeeping heat
• The Einstein relation generalized to non-equilibrium
• Experimental verification of a modified fluctuation-dissipation relation for a micron-sized particle in a non-equilibrium steady state
• Steady state statistics of driven diffusions
• Cumulants and large deviations of the current through non-equilibrium steady states
• Steady State Thermodynamics for Heat Conduction -- Microscopic Derivation
• Symmetries in Fluctuations Far from Equilibrium
• Fluctuation relations in simple examples of non-equilibrium steady states
• An expression for stationary distribution in nonequilibrium steady state
• Work probability distribution in systems driven out of equilibrium
• Large deviation function for entropy production in driven one-dimensional systems
• Minimum entropy production principle from a dynamical fluctuation law
• Entropy and Nonlinear Nonequilibrium Thermodynamic Relation for Heat Conducting Steady States
• Work distribution and path integrals in general mean-field systems
• Vortices in the two-dimensional Simple Exclusion Process
• Variational formula for experimental determination of high-order correlations of current fluctuations in driven systems
• Entropy production in the non-equilibrium steady states of interacting many-body systems
• Extended Clausius Relation and Entropy for Nonequilibrium Steady States in Heat Conducting Quantum Systems
• On the nonequilibrium relation between potential and stationary distribution for driven diffusion

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