Markov Approach to Density Fluctuations Due to Transport and Scattering. II. Applications

Abstract

Applications of a macroscopic and microscopic nature are given of the Markov formalism, developed in the previous paper, for finding the pair‐correlation function, two‐point covariance function, and spectral intensities from the phenomenological transport equations, transitions, and scattering. As macroscopic examples, we discuss Rayleigh diffusion, Coulomb correlations, and space‐charge‐limited flow in solids. The diffusion equation is shown to be not strictly Markovian, though correct Langevin diffusion sources are easily found. Dielectric relaxation is shown to be a macroscopic manifestation of Coulomb correlations in a plasma. The Λ theorem of the previous paper yields for the pair‐correlation functions the original Debye‐Hückel result plus a δ term. The spectra for space‐charge‐limited flow due to single‐carrier injection are treated with the Green's‐function procedure. The discrepancies between existing theories are removed since the Coulomb correlations in the steady state are shown to be not of the Debye‐Hückel type. As microscopic applications, we consider ideal gases in the presence of streaming due to fields, an extension of Lax's work. The connection with the BBGK two‐point equations for one‐ and two‐body collisions is established. Finally, we integrate over the stochastic Boltzmann equation to obtain stochastic hydrodynamic equations. These provide a microscopic basis is for the noise sources associated with electrical conduction, heat conduction, and with the Navier‐Stokes equation. Results for the hot‐electron Boltzmann gas are compared with those of Price.