Thermal Conductivity for a Momentum Conservative Model

Abstract

We introduce a model whose thermal conductivity diverges in dimension 1 and 2, while it remains finite in dimension 3. We consider a system of oscillators perturbed by a stochastic dynamics conserving momentum and energy. We compute thermal conductivity via Green-Kubo formula. In the harmonic case we compute the current-current time correlation function, that decay like t −d/2 in the unpinned case and like t −d/2−1 if a on-site harmonic potential is present. This implies a finite conductivity in d ≥ 3 or in pinned cases, and we compute it explicitely. For general anharmonic strictly convex interactions we prove some upper bounds for the conductivity that behave qualitatively as in the harmonic cases.Date: October 12, 2018.

Nonequilibrium stationary state of a harmonic crystal with alternating masses

“…This is expected since anharmoncity leads to interactions between phonons which helps to establish local thermal equilibrium. A simple model which incorporates phonon-phonon interactions was introduced in [13,14] where the determinsitic dynamics of the Harmonic chain is stochastically perturbed. Here we have carried out simulations with this noisy dynamics and looked at it's effect on the temperature profiles of the alternating mass chain.…”

Diffusive Propagation of Energy in a Non-acoustic Chain

“…Following [1,2] we perturb the Hamiltonian dynamics (2.1) by introducing the random momentum exchange between the neighboring sites in such a way that the total momentum and energy of the system are conserved. This is achieved by adding to the right hand side of (2.1) a local stochastic term that conserves both…”

“…This means that its hamiltonian is formally given by 2) where the energy of the oscillator x is defined…”

Energy Diffusion in Harmonic System with Conservative Noise

“…showing that thermal conductivity is infinite in dimension one and two for a system of harmonic oscillators perturbed by a conservative noise ( [5], [4]). …”

“…The key point is the derivation of a Nash type inequality which provides an estimate for convergence rates slower than exponential ( [22], [6], [30]). The diffusion coefficient is given by an infrared regularization of the thermal conductivity obtained in [4], [5], with a proper renormalization (13).…”