A Mechanical Model for Fourier’s Law of Heat Conduction

Abstract

Nonequilibrium statistical mechanics close to equilibrium is a physically satisfactory theory centered on the linear response formula of Green-Kubo. This formula results from a formal first order perturbation calculation without rigorous justification. A rigorous derivation of Fourier's law for heat conduction from the laws of mechanics remains thus a major unsolved problem. In this note we present a deterministic mechanical model of a heat-conducting chain with nontrivial interactions, where kinetic energy fluctuations at the nodes of the chain are removed. In this model the derivation of Fourier's law can proceed rigorously.

Energy Transfer in a Fast-Slow Hamiltonian System

“…Lately several results have appeared trying to bring new perspective to the above problem in a collective effort to attack the problem from different points of views. Let us just mention, as examples, papers considering stochastic models [3,4,5], approaches starting from kinetic equations or assuming extra hypotheses [2,26,7] or papers trying to take advantage of the point of view and results developed in the field of Dynamical Systems [16,13,14,15,8,9,29]. This paper belongs to the latter category but it is closely related to results obtained for stochastic models (e.g., [25]).…”

Statistical properties of mostly contracting fast-slow partially hyperbolic systems

“…We now show that µ ε,i is indeed a SRB measure in the sense of Remark 2.6: consider a measurable partition {I ξ } ξ∈Ξ of T × T ,i in horizontal segments 40 of length between δ/2 and δ with indices in some measure space Ξ. That is, we let I ξ = [a ξ , b ξ ] × {y ξ } for some a ξ , b ξ , y ξ ∈ T with δ/2 ≤ b ξ − a ξ ≤ δ.…”

“…It would then be of great interest in the field of Dynamical Systems, but also, e.g., for non-equilibrium Statistical Mechanics, to extend the class of systems for which statistical properties are well understood. See [38,33] for a discussion of some aspects of these issues and [40] for an interesting application to non-equilibrium Statistical Mechanics.…”

Map Lattices Coupled by Collisions: Hitting Time Statistics and Collisions Per Lattice Unit

“…Consider first the case when none of the elements of S = {(a v , a −v )} v∈V + are periodic points, that is, S per = ∅. Then, by the expression (31) it is evident that in the ε → 0 limit we get q k = 0 for all k ≥ 0. As a consequence, θ = 1.…”

“…The state of each unit, called a site, evolves according to the interplay between the local dynamics and the effect of the interaction with other sites. Such systems were introduced in the 80s (see [26] and references therein), and since then, they have seen a remarkable amount of research due to their paramount importance in applications and in studying non-equilibrium thermodynamics [5,31]. From a mathematical viewpoint, the main interest of coupled map lattices is that they provide natural examples of dynamical systems with an infinite-dimensional state space.…”