The Heat theorem reveals the second law of equilibrium Thermodynamics (i.e.existence of Entropy) as a manifestation of a general property of Hamiltonian Mechanics and of the Ergodic Hypothesis, valid for 1 as well as 10 23 degrees of freedom systems, i.e. for simple as well as very complex systems, and reflecting the Hamiltonian nature of the microscopic motion. In Nonequilibrium Thermodynamics theorems of comparable generality do not seem to be available. Yet it is possible to find general, model independent, properties valid even for simple chaotic systems (i.e. the hyperbolic ones), which acquire special interest for large systems: the Chaotic Hypothesis leads to the Fluctuation Theorem which provides general properties of certain very large fluctuations and reflects the time-reversal symmetry. Implications on Fluids and Quantum systems are briefly hinted. The physical meaning of the Chaotic Hypothesis, of SRB distributions and of the Fluctuation Theorem

This paper presents sufficient conditions for proving ergodicity of noise-driven dynamical systems. The essential conditions are weak irreducibility and closure under second randomization of the driving noise. With these conditions one can ascertain ergodicity of Langevin processes even if the diffusion and drift matrices associated to the momentums are degenerate. The paper illustrates how to check these conditions practically in the context of a simple mechanical system governed by Langevin equations (a simple stochastic rigid body system). 1

We derive the first two terms in an ε-expansion for the invariant measure of a class of semilinear parabolic SPDEs near a change of stability, when the noise strength and the linear instability are of comparable order ε 2. This result gives insight into the stochastic bifurcation and allows to rigorously approximate correlation functions. The error between the approximate and the true invariant measure is bounded in both the Wasserstein and the total variation distance.

The paper presents a unified approach to different fluctuation relations for classical nonequilibrium dynamics described by diffusion processes. Such relations compare the statistics of fluctuations of the entropy production or work in the original process to the similar statistics in the time-reversed process. The origin of a variety of fluctuation relations is traced to the use of different time reversals. It is also shown how the application of the presented approach to the tangent process describing the joint evolution of infinitesimally close trajectories of the original process leads to a multiplicative extension of the fluctuation relations. 1

We propose a thermodynamic formalism that is expected to apply to a large class of nonequilibrium steady states including a heat conducting fluid, a sheared fluid, and an electrically conducting fluid. We call our theory steady state thermodynamics (SST) after Oono and Paniconi’s original proposal. The construction of SST is based on a careful examination of how the basic notions in thermodynamics should be modified in nonequilibrium steady states. We define all thermodynamic quantities through operational procedures, which can be (in principle) realized experimentally. Based on SST thus constructed, we make some nontrivial predictions, including an extension of Einstein’s formula on density fluctuation, an extension of the minimum work principle, the existence of a new osmotic pressure of a purely nonequilibrium origin, and a shift of coexistence temperature. All these predictions may be checked experimentally to test SST for its quantitative validity. Contents

Abstract. We consider a stochastic Klein-Gordon wave equation modeling heat flow in a linear field that is coupled to thermal reservoirs at different temperatures. We discuss, in a perturbative context, the approach to a stationary, non-equilibrium state, and show that the state is supported on field configurations which are Hölder continuous, with any exponent less than 1/2. We determine the heat flux to lowest order in perturbation theory. 1.

We study the model of a strongly non-linear chain of particles coupled to two heat baths at different temperatures. Our main result is the existence and uniqueness of a stationary state at all temperatures. This result extends those of Eckmann, Pillet, Rey-Bellet [EPR99a,EPR99b] to potentials with essentially arbitrary growth at infinity. This extension is possible by introducing a stronger version of Hrmander's theorem for Kolmogorov equations to vector fields with polynomially bounded coefficients on unbounded domains. 1.

Equilibrium states are the central objects of equilibrium statistical mechanics. To a large extent the success of this theory is due to the fact that the equilibrium states of a system can be constructed and characterized without explicit reference to the dynamics of this system. This is a truly amazing fact because the concept of equilibrium state and the ultimate justification of the whole theory, as formulated by Boltzmann and its followers, do depend on the dynamics. Nonequilibrium statistical mechanics has a completely different status. It is a more difficult theory, both conceptually and technically, because a deep control of the dynamics is required to understand even the most basic nonequilibrium properties of the system. The simplest conceivable elements of the large manifold of nonequilibrium states of a system are the steady states in which it settles under the action of weak, stationary external forces. Such a forcing can be achieved either by applying an external field or by imposing a sustained gradient of intensive thermodynamic parameters (e.g., a constant temperature drop) across the system. Even though definite progresses have been made in the recent years we are still far from a coherent theory of steady states if such a theory exists at all. 1 Phenomenological theory and linear response To appreciate recent rigorous results in the field a basic knowledge of nonequilibrium thermodynamics is required.

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Isotropic hypoellipticity and trend to the equilibrium for the Fokker-Planck equation with high degree potential