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Markov Approach to Density Fluctuations Due to Transport and Scattering. I. Mathematical Formalism

A kinetic approach to fluctuations and correlations of stochastic processes depending on a continuous set of parameters y is presented. In particular, we consider particle densities np(y, t) which may refer to macroscopic densities in position space, or to microscopic quantities such as distributions in phase space, or occupancies of quantum states of which the labeling is continuous (as with Bloch states in solids). In a Markovian sense such processes are infinite dimensional. We describe the fluctuating particle densities in a Hilbert space: the analog of de Groot's a‐space for non‐spatial‐dependent variables. Mainly, we employ a Langevin description; i.e., we start from presumed phenomenological equations, amended with source densities ξp(y, t). A theorem is derived for the density‐density or two‐point covariance function (Λ theorem). In its general form, the theorem applies to the nonequilibrium steady state. It closely resembles the generalized g‐r theorem for finite‐dimensional processes. However, t...