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Stationary non-equilibrium states describe steady flows through macroscopic systems. Although they represent the simplest generalization of equilibrium states, they exhibit a variety of new phenomena. Within a statistical mechanics approach, these states have been the subject of several theoretical investigations, both analytic and numerical. The macroscopic fluctuation theory, based on a formula for the probability of joint space-time fluctuations of thermodynamic variables and currents, provides a unified macroscopic treatment of such states for driven diffusive systems. We give a detailed review of this theory including its main predictions and most relevant applications.

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...

We study current fluctuations in lattice gases in the macroscopic limit extending the dynamic approach for density fluctuations developed in previous articles. More precisely, we establish a large deviation theory for the space-time fluctuations of the empirical current which include the previous results. We then estimate the probability of a fluctuation of the average current over a large time interval. It turns out that recent results by Bodineau and Derrida [Phys. Rev. Lett. 92, 180601 (2004)]] in certain cases underestimate this probability due to the occurrence of dynamical phase transitions.

Abstract. We propose a new definition of metastability of Markov processes on countable state spaces. We obtain sufficient conditions for a sequence of processes to be metastable. In the reversible case these conditions are expressed in terms of the capacity and of the stationary measure of the metastable states.

We study current fluctuations in lattice gases in the macroscopic limit extending the dynamic approach for density fluctuations developed in previous articles. More precisely, we establish a large deviation principle for a space-time fluctuation j of the empirical current with a rate functional I(j). We then estimate the probability of a fluctuation of the average current over a large time interval; this probability can be obtained by solving a variational problem for the functional I. We discuss several possible scenarios, interpreted as dynamical phase transitions, for this variational problem. They actually occur in specific models. We finally discuss the time reversal properties of I and derive a fluctuation relationship akin to the Gallavotti-Cohen theorem for the entropy production.

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This paper provides an introduction to some stochastic models of lattice gases out of equilibrium and a discussion of results of various kinds obtained in recent years. Although these models are different in their microscopic features, a unified picture is emerging at the macroscopic level, applicable, in our view, to real phenomena where diffusion is the dominating physical mechanism. We rely mainly on an approach developed by the authors based on the study of dynamical large fluctuations in stationary states of open systems. The outcome of this approach is a theory connecting the non equilibrium thermodynamics to the transport coefficients via a variational principle. This leads ultimately to a functional derivative equation of Hamilton-Jacobi type for the non equilibrium free energy in which local thermodynamic variables are the independent arguments. In the first part of the paper we give a detailed introduction to the microscopic dynamics considered, while the second part, devoted to the macroscopic properties, illustrates many consequences of the Hamilton-Jacobi equation. In both parts several novelties are included.