Abstract
We consider the motion of many confined billiard balls in interaction and discuss their transport and chaotic properties. In spite of the absence of mass transport, due to confinement, energy transport can take place through binary collisions between neighbouring particles. We explore the conditions under which relaxation to local equilibrium occurs on timescales much shorter than that of binary collisions, which characterize the transport of energy, and subsequent relaxation to local thermal equilibrium. Starting from the pseudo-Liouville equation for the time evolution of phase-space distributions, we derive a master equation which governs the energy exchange between the system constituents. We thus obtain analytical results relating the transport coefficient of thermal conductivity to the frequency of collision events and compute these quantities. We also provide estimates of the Lyapunov exponents and Kolmogorov–Sinai entropy under the assumption of scale separation. The validity of our results is confirmed by extensive numerical studies.
Highlights
Understanding the dynamical origin of the mechanisms which underly the phenomenology of heat conduction has remained one of the major open problems of statistical mechanics ever since Fourier’s seminal work [1]
Assuming relaxation to local equilibrium holds, we go on to considering the time evolution of phase-space densities and derive, from it, a master equation which governs the exchange of energy in the system [19], going from a microscopic scale description of the Liouville equation to the mesoscopic scale at which energy transport takes place. We regard this as the first milestone, namely identifying the conditions under which one can rigorously reduce the level of descrition from the deterministic dynamics at the microscopic level to a stochastic process described by a master equation at the mesoscopic level of energy exchanges
Provided we have a separation of time scales between wall and binary collisions, the advection and wall collision terms on the RHS of equation (5) will typically dominate the dynamics on the short time ∼ 1/νw, which follows every binary collision event, ensuring, thanks to the mixing within individual billiard cells, the relaxation of the phase-space distribution pN to local equilibrium well before the occurrence of the binary event, whose time scale is ∼ 1/νb
Summary
Understanding the dynamical origin of the mechanisms which underly the phenomenology of heat conduction has remained one of the major open problems of statistical mechanics ever since Fourier’s seminal work [1]. We regard this as the first milestone, namely identifying the conditions under which one can rigorously reduce the level of descrition from the deterministic dynamics at the microscopic level to a stochastic process described by a master equation at the mesoscopic level of energy exchanges This master equation is used to compute the frequency of binary collisions and to derive Fourier’s law and the macroscopic heat equation, which results from the application of a small temperature gradient between neighbouring cells. This is our second milestone : an analytic formula for heat conductivity, exact for the stochastic system, and valid for the determinisitc system at the critical geometry limit.
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