Nonequilibrium Steady States of Some Simple 1-D Mechanical Chains

Abstract

We study nonequilibrium steady states of some 1-D mechanical models with N moving particles on a line segment connected to unequal heat baths. For a system in which particles move freely, exchanging energy as they collide with one another, we prove that the mean energy along the chain is constant and equal to 1 2 √ T L T R where T L and T R are the temperatures of the two baths. We then consider systems in which particles are trapped, i.e., each confined to its designated interval in the phase space, but these intervals overlap to permit interaction of neighbors. For these systems, we show numerically that the system has well defined local temperatures and obeys Fourier's Law (with energy-dependent conductivity) provided we vary the masses randomly to enable the repartitioning of energy. Dynamical systems issues that arise in this study are discussed though their resolution is beyond reach.

Drastic fall-off of the thermal conductivity for disordered lattices in the limit of weak anharmonic interactions

“…It is then argued that this result extends to a disordered chain studied by Dhar and Lebowitz [12], and to a classical spins chain recently investigated by Oganesyan, Pal and Huse [23].It is generally admitted that the thermal properties of solids can be derived from molecular dynamics, but it remains to this day a widely open conjecture, at least from a mathematical point of view [9][13] [19].Indeed, the few hamiltonian systems that can be handled analytically to a large extent, also appear to have a very pathological thermal behavior. So is it for the ordered harmonic chain [26], for the Toda lattice (see Section 6.3 in [7]), and for a one-dimensional system of colliding particles [28]. As we will see soon, disordered one-dimensional harmonic chains also fall into this category.…”

“…Indeed, the few hamiltonian systems that can be handled analytically to a large extent, also appear to have a very pathological thermal behavior. So is it for the ordered harmonic chain [26], for the Toda lattice (see Section 6.3 in [7]), and for a one-dimensional system of colliding particles [28]. As we will see soon, disordered one-dimensional harmonic chains also fall into this category.…”

Local Thermal Equilibrium for Certain Stochastic Models of Heat Transport

“…It has been noted that a number of systems with integrable dynamics do not have LTE, see e.g. [DD99,BLY10,RY12]. In [EY06] and [LY10], the authors derived formulas for macroscopic observations such as energy and particle density profiles for both mechanical and stochastic models assuming LTE, and provided numerical validation for their derivations.…”

Encounter Times in Overlapping Domains: Application to Epidemic Spread in a Population of Territorial Animals

“…The sizes of both types of region are constant: the H i each occupy sites, and the R i occupy m sites. A similar geometry has been used independently in a model for heat conduction in 1D systems [18,19].Encounter times:-We first suppose that the probability p of transmission is 1. To calculate the mean first-encounter time (MFET) between the two walkers, we map their movement onto that of a single random walker in 2D, whose allowed positions are represented by the vector w = (w 1 , w 2 ); see Fig.…”

“…The sizes of both types of region are constant: the H i each occupy sites, and the R i occupy m sites. A similar geometry has been used independently in a model for heat conduction in 1D systems [18,19].…”