Lorentz Gas with Thermostatted Walls

Abstract

In a planar periodic Lorentz gas, a point particle (electron) moves freely and collides with fixed round obstacles (ions). If a constant force (induced by an electric field) acts on the particle, the latter will accelerate, and its speed will approach infinity (Chernov and Dolgopyat in J Am Math Soc 22: 821-858, 2009; Phys Rev Lett 99, paper 030601, 2007). To keep the kinetic energy bounded one can apply a Gaussian thermostat, which forces the particle's speed to be constant. Then an electric current sets in and one can prove Ohm's law and the Einstein relation (Chernov and Dolgopyat in Russian Math Surv 64:73-124, 2009; Chernov et al. Comm Math Phys 154:569-601, 1993; Phys Rev Lett 70:2209-2212. However, the Gaussian thermostat has been criticized as unrealistic, because it acts all the time, even during the free flights between collisions. We propose a new model, where during the free flights the electron accelerates, but at the collisions with ions its total energy is reset to a fixed level; thus our thermostat is restricted to the surface of the scatterers (the 'walls'). We rederive all physically interesting facts proven for the Gaussian thermostat in Chernov, Dolgopyat (Russian Math Surv 64:73-124, 2009) and Chernov et al. (Comm Math Phys 154:569-601, 1993; Phys Rev Lett 70:2209-2212, including Ohm's law and the Einstein relation. In addition, we investigate the superconductivity phenomenon in the infinite horizon case.

A Rigorous Derivation of Haff’s Law for a Periodic Two-Disk Fluid

“…The derivation of transport coefficients from microscopic models typically results in an expression for the transport coefficient in terms of a correlation sum typically referred to as Green-Kubo formula [35], [10,12,6,15,22,21,20,23,38,26,24]. The present work derives an equation for the cooling of a system with dissipative interactions, which is not expressed through a Green-Kubo formula.…”

“…First it is used in the derivation of the growth lemma through the one-step expansion property Lemma 4.4. For standard billiards and certain perturbations of it this property is known to be true also for the infinite horizon situation [38,26,24]. The second place where the finite horizon assumption was used is the extension of Theorem 6.4 to Theorem 6.5.…”

“…In order to keep the presentation as clear as possible we will not include lengthy proofs which are just slight modifications of proofs of similar results. Instead we will point the reader to the corresponding references, which are primarily [25,19,26,24].…”

“…Furthermore, we denote by r n (x) = distγ n (x) (F n (x), ∂γ n (x)) the distance of the pointF n (x) to the closest endpoints of the component ofF n (γ) containing it. With this notation in place we can formulate the aforementioned growth lemma forF , whose proof can be found in [25,20,24], where the particular formulation given below can be found in [24]. Lemma 4.5 (Uniform growth lemma).…”

“…Despite the fact that the statistical properties of multidimensional billiards remain a challenge, the theory of planar hyperbolic billiards is developed enough to also study perturbations of these. Popular choices are (small) external fields (typically in combination with a thermostat), which were shown in [20,21,23] and [24,26] to be models that exhibit Ohm's law and the Einstein relation between the conductivity and diffusion constant.…”