Thermalization in Harmonic Particle Chains with Velocity Flips

Abstract

We propose a new mathematical tool for the study of transport properties of models for lattice vibrations in crystalline solids. By replication of dynamical degrees of freedom, we aim at a new dynamical system where the "local" dynamics can be isolated and solved independently from the "global" evolution. The replication procedure is very generic but not unique as it depends on how the original dynamics are split between the local and global dynamics. As an explicit example, we apply the scheme to study thermalization of the pinned harmonic chain with velocity flips. We improve on the previous results about this system by showing that after a relatively short time period the average kinetic temperature profile satisfies the dynamic Fourier's law in a local microscopic sense without assuming that the initial data is close to a local equilibrium state. The bounds derived here prove that the above thermalization period is at most of the order L 2/3 , where L denotes the number of particles in the chain. In particular, even before the diffusive time scale Fourier's law becomes a valid approximation of the evolution of the kinetic temperature profile. As a second application of the dynamic replica method, we also briefly discuss replacing the velocity flips by an anharmonic onsite potential.

Harmonic Chain with Velocity Flips: Thermalization and Kinetic Theory

“…As shown in [2, Appendix A], its Fourier transform is with Ω = (γ/2) 1 − (2ω(k)/γ) 2 < γ/2 and µ σ (k) = γ/2 + σΩ(k). (To facilitate comparison, let us point out that the function "Ω" was denoted by "u", and only the second column of A was used in [2].) Since ω(k) = ω(−k), it follows that A t (k) = A t (−k) and thus also A t (x) = A t (−x) because A t is real-valued.…”

“…We are interested in controlling the behaviour of U t (x, k) at the diffusive scale t = O(L 2 ). We rely on the estimates derived in [2] and, for the sake of completeness, let us begin by summarizing the necessary assumptions from [2].…”

“…On the other hand, the matrix G s is a function of the temperature profile T s (x) only, and its behaviour has already been solved in [2]. As we will show next, the strong control derived for the temperature profile in [2] suffices to determine the local covariances up to order O(L −2 ) at diffusive time scales.…”

“…A different approach was chosen in [2] to study the evolution of the kinetic temperature profile, T t (x) = p 2…”

“…The goal of this paper is to clarify the physical meaning of the results in [2], and to explore its implications on the structure of general local correlations after local equilibrium has been reached. We consider the evolution of the full spatial covariance matrix of positions and momenta, and by defining a suitable Wigner function from the covariance matrix, we compute the first order corrections to the local thermal equilibrium.…”