Note on Statistical Mechanics of Random Systems

We discuss some equilibrium properties of random systems, i.e., systems whose Hamiltonian depends on some random variables y with a distribution P(y) which is independent of the dynamic state of the system. For a system of noninteracting particles which interact with randomly distributed scattering centers, the important quantity is the average density of states of a single particle per unit volume, n(E). Feynman's path‐integral formulation of quantum statistics is used to derive some properties of the average partition function for one particle ${\mathrm{〈Z}}_1〉$ , which is the Laplace transform of n(E). In particular, it is shown that ${\mathrm{〈Z}}_1〉$ is an analytic function of the density of scatterers ρ for a wide class of particle‐scattering center potentials V(r), including those with hard cores and those with negative parts. The analyticity in ρ of the equilibrium properties of these systems is very remarkable and is in contrast to the conjectured nonanalytic behavior of their transport (i.e., diffusion) coefficients. We find also upper and lower bounds on ${\mathrm{〈Z}}_1〉$ for a particle acted upon by a random potential V(r) obeying Gaussian statistics with $\mathrm{〈V(}\mathbf{r}\mathrm{)V(}\mathbf{r\prime }\mathrm{)〉\sim }\exp {\mathrm{\hspace{0.17em}[-\alpha }}^2(\mathbf{r-r\prime }{)}^2]$ . In the limit of ``white noise,'' $\mathrm{〈V(}\mathbf{r}\mathrm{)V(}\mathbf{r\prime }\mathrm{)〉\sim \delta (}\mathbf{r-r\prime })$ , ${\mathrm{〈Z}}_1〉$ is shown to diverge in two and three dimensions but remains finite in one dimension. This agrees with approximate results on the density of states. In appendices we prove the existence, in the thermodynamic limit, of the free‐energy density of a system with random scatterers and also of the frequency distribution and, thus, the free‐energy density for a random harmonic crystal.