Macroscopic energy diffusion for a chain of anharmonic oscillators
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Macroscopic energy diffusion for a chain of anharmonic oscillators Stefano Olla · Makiko Sasada
Received: 23 September 2011 / Revised: 24 October 2012 © Springer-Verlag Berlin Heidelberg 2012
Abstract We study the energy diffusion in a chain of anharmonic oscillators where the Hamiltonian dynamics is perturbed by a local energy conserving noise. We prove that under diffusive rescaling of space–time, energy fluctuations diffuse and evolve following an infinite dimensional linear stochastic differential equation driven by the linearized heat equation. We also give variational expressions for the thermal diffusivity and some upper and lower bounds. Mathematics Subject Classification
82C70 · 60K35 · 82C22
1 Introduction The deduction of the heat equation or the Fourier law for the macroscopic evolution of the energy through a diffusive space–time scaling limit from a microscopic dynamics given by Hamilton or Schrödinger equations, is one of the most important problem in non-equilibrium statistical mechanics [5]. One dimensional chains of oscillators have been used as simple models for this study. In the context of the classical (Hamiltonian) This paper has been partially supported by the European Advanced Grant Macroscopic Laws and Dynamical Systems (MALADY) (ERC AdG 246953), by Agence Nationale de la Recherche, under Grant ANR-2010-BLAN-0108 (SHEPI). We thank Professor Tadahisa Funaki for insightful discussions and his interest in this work. S. Olla (B) CEREMADE, UMR CNRS 7534, Université Paris-Dauphine, 75775 Paris Cedex 16, France e-mail: [email protected] M. Sasada Department of Mathematics, Keio University, 3-14-1, Hiyoshi, Kohoku-ku, Yokohama, Kanagawa 223-8522, Japan e-mail: [email protected]
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dynamics, it is clear that non-linear interactions are crucial for the diffusive behavior of the energy. In fact, in a chain of harmonic oscillators the energy evolution is ballistic [18]. In this linear system, the energy of each mode of vibration is conserved. Non-linearities introduce interactions between different modes and destroy these conservation laws and give a certain ergodicity to the microscopic dynamics. In order to describe the mathematical problem, let us introduce some notation we will use in the rest of the paper. We study a system of anharmonic oscillators, each is denoted by an integer i. We denote by (qi , pi ) the corresponding position and momentum (we set the mass equal to 1). Each pair of consecutive particles (i, i + 1) are connected by a spring which can be anharmonic. The interaction is described by a potential energy V¯ (qi+1 − qi ). We assume that V¯ is a non-negative smooth function satisfying ¯ Z β := e−β V (r ) dr < ∞ R
for all β > 0. Let a be the equilibrium inter-particle distance, where V¯ attains its minimum that we assume to be 0 : V¯ (a) = 0. It is convenient to work with interparticle distances as coordinates, rather than absolute particle positions, so we define r j = q j − q j−1 − a. We denote the translated function V¯ (· + a) by V (·)
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