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.

The Generalized Stochastic Smoluchowski Equation

“…The lifetime of the metastable states is related to the Kramers escape rate formula which can be derived from the theory of instantons [ 70 ]. To the best of our knowledge, the stochastic Smoluchowski Equation ( 9 ) was first written down in van Kampen’s book [ 71 ] (p. 349) referring to the papers of van Vliet [ 72 , 73 ] in 1971; this is the oldest reference to that equation that we have found. It then appeared in different domains of physics such as the theory of simple liquids [ 63 , 74 , 75 , 76 , 77 , 78 ], the kinetic theory of adsorbates [ 25 ], systems with long-range interactions [ 14 ], 2D Brownian vortices [ 45 ], bacterial populations undergoing chemotaxis [ 79 ], nucleation [ 48 ], self-gravitating Brownian particles [ 70 ], the BMF model [ 49 ], etc.…”

An accurate and efficient high frequency noise simulation technique for deep submicron MOSFETs

“…This contradicts recent MOSFET noise modeling approaches that have explained the behavior of and based on the hot carrier effects [27], [28]. Their primary problem is that voltage sources are used to represent the local fluctuations, assumed to be uncorrelated even though the local noise voltage sources are correlated, unlike the noise current sources [29], [30].…”

Thermal phase fluctuations in optically pumped dual-frequency vertical external-cavity surface-emitting lasers for cesium clocks based on coherent population trapping

“…where D is an isotropic diffusion coefficient. The Langevin source term ζ is characterized by the following autocorrelation: [28][29][30][31][32][33][34] ζ r, t ð Þζ r…”