Asymptotic and Numerical Analyses for Mechanical Models of Heat Baths

A mechanical model of a particle immersed in a heat bath is studied, in which a distinguished particle interacts via linear springs with a collection of n particles with variable masses and random initial conditions; the jth particle oscillates with frequency j^p, where p is a parameter. For p>1/2 the sequence of random processes that describe the trajectory of the distinguished particle tends almost surely, as n, to the solution of an integro-differential equation with a random driving term; the mean convergence rate is 1/n^p–1/2. We further investigate whether the motion of the distinguished particle can be well approximated by an integration scheme—the symplectic Euler scheme—when the product of time step h and highest frequency n^p is of order 1, that is, when high frequencies are underresolved. For 1/2<p<1 the numerical solution is found to converge to the exact solution at a reduced rate of |log h| h^2–1/p. These results shed light on existing numerical data.