Interacting Particles

Stationary states for the noncutoff Kac equation with a Gaussian thermostat

“…Assume that b satisfies (3). For all field strengths E > 0, there exists a weak solution f to ( 5)- (2), in the sense of definition 1, that satisfies (7), i.e. has mass one, and such that moments of any order of f are finite.…”

“…This construction is known as a Gaussian isokinetic thermostat and has been applied in many fields of statistical physics and molecular dynamics, as a model for nonequilibrium steady states (see, e.g. [7][8][9] and the references therein). An example, which is relevant for the present paper, is the Lorentz gas.…”

Non-uniform ergodic properties of Hamiltonian flows with impacts

“…At the other extreme, for compact billiard tables and smooth bounded potentials, mechanical HISs limit, at high energy, to the corresponding billiard flow. The theory for intermediate energy values includes local analysis near periodic orbits [5] and near smooth convex boundaries [2,25], and, for some specific classes of HIS, hyperbolic behavior [12,[20][21][22], Liouville integrable [3,6,11,16,17] and near-integrable [15] dynamics were established. A class of quasi-integrable HISs, related to the quasi-integrable dynamics in families of polygonal right-angled corners, was introduced in [1].…”

Sinai Billiards under Small External Forces II

“…Various perturbations of Sinai billiards have been studied in [1,[18][19][20][21]29,30], see also a survey [22]. Most notably, when F is a small constant force with a Gaussian thermostat (see below), then one can rigorously prove a one-particle version of classical Ohm's law and the Einstein relation [4,5].…”