A Matrix Model of Relaxation

Abstract

We consider a two level system, ${\mathcal{S}}_2$, coupled to a general $n$ level system, ${\mathcal{S}}_n$, via a random matrix. We derive an integral representation for the mean reduced density matrix $\rho (t)$ of ${\mathcal{S}}_2$ in the limit $n\to \infty$, and we identify a model of ${\mathcal{S}}_n$ which possesses some of the properties expected for macroscopic thermal reservoirs. In particular, it yields the Gibbs form for $\rho (\infty )$. We consider also an analog of the van Hove limit and obtain a master equation (Markov dynamics) for the evolution of $\rho (t)$ on an appropriate time scale.

20pages, LaTeX