Fluctuations in Stationary Nonequilibrium States of Irreversible Processes

Abstract

In this paper we formulate a dynamical fluctuation theory for stationary non equilibrium states (SNS) which covers situations in a nonlinear hydrodynamic regime and is verified explicitly in stochastic models of interacting particles. In our theory a crucial role is played by the time reversed dynamics. Our results include the modification of the Onsager-Machlup theory in the SNS, a general Hamilton-Jacobi equation for the macroscopic entropy and a non equilibrium, non linear fluctuation dissipation relation valid for a wide class of systems. The Boltzmann-Einstein theory of equilibrium thermodynamic fluctuations, as described for example in Landau-Lifshitz [1], states that the probability for a fluctuation from equilibrium in a macroscopic region of volume V is proportional to exp{V ∆S/k} where ∆S is the variation of entropy density calculated along a reversible transformation creating the fluctuation and k is the Boltzmann constant. This theory is well established and has received a rigorous mathematical formulation in classical equilibrium statistical mechanics via the so called large deviation theory [2]. The rigorous study of large deviations has been extended to hydrodynamic evolutions of stochastic interacting particle systems [3]. In a dynamical setting one may asks new questions, for example what is the most probable trajectory followed by the system in the spontaneous emergence of a fluctuation or in its relaxation to equilibrium. The Onsager-Machlup approach [4] answers precisely to this question: in the situation of a linear hydrodynamic equation, that is, close to equilibrium, the most probable emergence and relaxation trajectories are one the time reversal of the other. Developing the methods of [3], this theory has been extended to nonlinear regimes [5]. Onsager-Machlup assume the reversibility of the microscopic dynamics; however microscopically non reversible models were constructed where the above results still hold, [6,7].Emergence of large fluctuations, including OnsagerMachlup symmetry, has been observed in stochastically perturbed gradient type electronic devices [8]. In their work, these authors study also non gradient (i.e. non reversible) systems and observe violation of OnsagerMachlup symmetry.In the present paper we formulate a general theory of large deviations for irreversible processes, i.e. when detailed balance condition does not hold. This question was previously addressed in [10]. Natural examples are boundary driven stationary non equilibrium states (SNS), e.g. a thermodynamic system in contact with two reservoirs, but our theory covers, as a special case, also the model systems considered in [8]. In our approach a crucial role is played by the time reversed dynamics with respect to the stationary non equilibrium ensemble.Our results are: 1. The Onsager-Machlup relationship has to be modified: the emergence of a fluctuation takes place along a trajectory which is determined by the time reversed process.2. We show that the macroscopic entropy solves a Hamilton-Jacobi equation ge...

Renormalized time scale for anticipating and lagging synchronization

“…By using this fluctuating hydrodynamic picture together with a path integral formulation, we derive a general form for the action associated to a history of the density, current and dissipation fields (that is, a path in mesoscopic phase space). Remarkably, this action takes the same form as in conservative nonequilibrium systems [7][8][9][10][11][12][13][14], simplifying the analysis in the dissipative case. This is both an important and a surprising result, which stems from the quasi-elastic character of the underlying microscopic dynamics in the large system size limit.…”

“…The study of fluctuation statistics of macroscopic observables provides an alternative path to obtain thermodynamic potentials, a complementary approach to the usual ensemble description. This observation, valid both in equilibrium [1] and nonequilibrium [7,9], is most relevant in the latter case because no general bottom-up approach, connecting microscopic dynamics to macroscopic nonequilibrium properties, has been found yet. In this way, the large deviation function (LDF) controlling the statistics of these fluctuations may play in nonequilibrium statistical mechanics a role similar to the equilibrium free energy [4,10].…”

“…These intriguing "symmetries" appear as a consequence of the invariance under time reversal of the underlying microscopic dynamics. These and other recent results [5][6][7][8][9][10][11][12][13][14] are however restricted to nonequilibrium "conservative" systems characterized by currents.…”

“…A classical example is the fluctuation-dissipation theorem, which relates the linear response of a system to an external perturbation to the fluctuation properties of the system in thermal equilibrium [2,3]. More recently, the investigation of general properties of fluctuations in nonequilibrium steady states is opening new paths for understanding physics far from equilibrium [4][5][6][7][8][9][10][11][12][13][14]. The study of fluctuation statistics of macroscopic observables provides an alternative path to obtain thermodynamic potentials, a complementary approach to the usual ensemble description.…”

“…This is done by combining a suitable generalization of macroscopic fluctuation theory (MFT) [7] to the realm of dissipative media, and extensive numerical simulations of a particular albeit broad class of microscopic models. Our starting point is a general fluctuating balance equation for the (energy) density, with a drift term proportional to the spatial derivative of the current and a sink term.…”