One-dimensional classical massive particle in the ideal gas
Consider a massive particle (m.p.) of mass \(M\) moving in one direction under the action of elastic collisions with equal particles of mass 1. Assume that the equal particles have no interaction. Denote by \(q_ 0\) and \(v_ 0\) the coordinate and velocity of the m.p., respectively. A collection of equal particles is denoted by \(X=\{x\}\), \(x=(q,v)\) and put \(Y=(x_ 0,X)\) where \(x_ 0=(q_ 0,v_ 0)\). The set of all possible \(Y\) is denoted by \(\Omega\). The dynamics of the system is described by the equilibrium measure \(\mu\) on \(\Omega\) : \[ d\mu =dq_ 0\sqrt{\beta M/2\pi}e^{-\beta Mv^ 2_ 0/2}dv_ 0dP(x) \] where \(P\) is the limit Gibbs measure on the phase space of all possible \(X\), i.e. the equal particles distributed on the line according to a Poisson distribution with density \(\rho\) and their velocities have independent Gaussian distribution with the density \(\sqrt{\beta /2\pi}\exp [-\beta v^ 2/2].\) The authors obtain the following results: (1) For every \(\epsilon >0\) and \(\mu\)-almost every \(Y\) one can find \(t_ 0(Y,\epsilon)\) such that for all \(t>t_ 0(Y,\epsilon)\), \(| q_ 0(t)-q_ 0| \leq t^{+\epsilon}\); (2) For \(\mu\)-almost every \(Y\) each particle of the equal particles has finitely many collisions with m.p.; (3) Assume that \(q_ 0=0\) and denote by \(\mu_ 0\) the conditional measure of \(\mu\) under \(q_ 0=0\). Then with respect to \(\mu_ 0,\) \(q_ 0(t)=n^+(t)-n^- (t)+\xi (t)\) where the random variables \(n^{\pm}(t)/\sqrt{t}\) have the same Gaussian limit distribution for \(t\to \infty\), while \(\xi(t)/\sqrt{t}\) converges in probability to zero.
scientific article; zbMATH DE number 3947374