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

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Abstract
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, the solutions may involve extra parts stemming from stochastic boundary conditions and constraints. Simplifications for the thermal equilibrium case and the connection with generalized Onsager's relations are discussed. A variety of expressions for the spectral intensities Σpq(y, y′, ω) are derived. In order to justify the Langevin procedure as well as to calculate the source terms and generalized Fokker‐Planck moments, we consider the master‐rate functional Wγγ′ for transitions γ→γ′ of the over‐all state. Expressions are given for transitions involving change of species and scattering due to one‐ or two‐body collisions. These determine uniquely the quantities entering into the Λ theorem. The statistical‐mechanical basis for our formalism is discussed in some detail.