We present a selective overview of the current state of our knowledge (more precisely of our ignorance) regarding the derivation of Fourier's Law, J(r) = −κ∇T (r); J the heat flux, T the temperature and κ, the heat conductivity. This law is empirically well tested for both fluids and crystals, when the temperature varies slowly on the microscopic scale, with κ an intrinsic property which depends only on the system's equilibrium parameters, such as the local temperature and density. There is however at present no rigorous mathematical derivation of Fourier's law and ipso facto of Kubo's formula for κ, involving integrals over equilibrium time correlations, for any system (or model) with a deterministic, e.g. Hamiltonian, microscopic evolution.

We consider a d-dimensional harmonic crystal in contact with a stochastic Langevin type heat bath at each site. The temperatures of the ''exterior'' left and right heat baths are at specified values T L and T R , respectively, while the temperatures of the ''interior'' baths are chosen self-consistently so that there is no average flux of energy between them and the system in the steady state. We prove that this requirement uniquely fixes the temperatures and the self consistent system has a unique steady state. For the infinite system this state is one of local thermal equilibrium. The corresponding heat current satisfies Fourier's law with a finite positive thermal conductivity which can also be computed using the Green-Kubo formula. For the harmonic chain (d=1) the conductivity agrees with the expression obtained by Bolsterli, Rich, and Visscher in 1970 who first studied this model. In the other limit, d ± 1, the stationary infinite volume heat conductivity behaves as (a d d) −1 where a d is the coupling to the intermediate reservoirs. We also analyze the effect of having a non-uniform distribution of the heat bath couplings. These results are proven rigorously by controlling the behavior of the correlations in the thermodynamic limit.

The chaotic hypothesis discussed in Gallavotti and Cohen (1995) is tested experimentally in a simple conduction model. Besides a confirmation of the hypothesis predictions the results suggest the validity of the hypothesis in the much wider context in which, as the forcing strength grows, the attractor ceases to be an Anosov system and becomes an Axiom A attractor. A first text of the new predictions is also attempted

Note that the "classical" Green-Kubo relation involves the total flux J t = J w t + J H t . For the latter one has, in the adiabatic approximation,

No abstract

The chaotic hypothesis has several implications which have generated interest in the literature because of their generality and because a few exact predictions are among them. However its application to Physics problems requires attention and can lead to apparent inconsistencies. In particular there are several cases that have been considered in the literature in which singularities are built in the models: for instance when among the forces there are Lennard-Jones potentials (which are infinite in the origin) and the constraints imposed on the system do not forbid arbitrarily close approach to the singularity even though the average kinetic energy is bounded. The situation is well understood in certain special cases in which the system is subject to Gaussian noise; here the treatment of rather general singular systems is considered and the predictions of the chaotic hypothesis for such situations are derived. The main conclusion is that the chaotic hypothesis is perfectly adequate to describe the singular physical systems we consider, i.e. deterministic systems with thermostat forces acting according to Gauss' principle for the constraint of constant total kinetic energy ("isokinetic Gaussian thermostats"), close and far from equilibrium. Near equilibrium it even predicts a fluctuation relation which, in deterministic cases with more general thermostat forces (i.e. not necessarily of Gaussian isokinetic nature), extends recent relations obtained in situations in which the thermostatting forces satisfy Gauss' principle. This relation agrees, where expected, with the fluctuation theorem for perfectly chaotic systems. The results are compared with some recent works in the literature.

We describe a way of interpreting the chaotic principle of [GC1] more extensively than it was meant in the original works. Mathematically the analysis is based on the dynamical notions of Axiom A and Axiom B and on the notion of Axiom C, that we introduce arguing that it is suggested by the results of an experiment ([BGG]) on chaotic motions. Physically we interpret a breakdown of the Anosov property of a time reversible attractor (replaced, as a control parameter changes, by an Axiom A property) as a spontaneous breakdown of the time reversal symmetry: the relation between time reversal and the symmetry that remains after the breakdown is analogous to the breakdown of T -invariance while T CP still holds.Keywords: Strange attractors, Chaoticity principle, Time reversal, TCP §1: Introduction.In reference [GC2] a general mechanical system in a non equilibrium situation was considered. Calling C the (compact or "finite") phase space of the "observed events", µ 0 the volume measure on it and S the map describing the time evolution (regarded as a discrete invertible mapping of "observed events" into the "next ones", i.e. as a Poincaré's map on some surface in the full phase space) a principle holding when motions have an empirically chaotic nature was introduced:Chaotic hypothesis: A chaotic many particle system in a stationary state can be regarded, for the purpose of computing macroscopic properties, as a smooth dynamical system with a transitive Axiom A globally attracting set. In reversible systems it can be regarded, for the same purposes, as a smooth transitive Anosov system.•Chaotic is an empirical qualitative notion that means that most points of the attracting set have a stable and an unstable manifold with positive dimension. In the applications in [GC1], [GC2], [G1] the use of the hypothesis, which is a natural extension of a principle proposed by Ruelle, was based on reversibility * mp arc@math.utexas.edu, #96-50, and chao-dyn@xyz.lanl.gov,#9512??.

In this paper we study a model of randomly colliding particles interacting with a thermal bath. Collisions between particles are modeled via the Kac master equation while the thermostat is seen as an infinite gas at thermal equilibrium at inverse temperature $\beta$. The system admits the canonical distribution at inverse temperature $\beta$ as the unique equilibrium state. We prove that any initial distribution approaches the equilibrium distribution exponentially fast both by computing the gap of the generator of the evolution, in a proper function space, as well as by proving exponential decay in relative entropy. We also show that the evolution propagates chaos and that the one-particle marginal, in the large-system limit, satisfies an effective Boltzmann-type equation.Comment: 21 pages; obtained a stronger entropy decay result and hence modified the statement and proof of Theorem 1.3 accordingly; rewrote proof of Theorem 1.2 to improve clarity; and made other changes to address referee comment