Multiple time scales analysis for the Kramers-Chandrasekhar equation with a weak magnetic field

doi:10.1016/0378-4371(87)90228-7

Physica A: Statistical Mechanics and its Applications

Volume 146, Issues 1–2, 1 November 1987, Pages 201-218

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Abstract

A functional approach, similar in spirit to that used by Bogoliubov in his studies of the BBGKY hierarchy, is used to derive the Smoluchowski equation from the Kramers-Chandrasekhar equation in the presence of a weak external magnetic field. A multiple time scales expansion is then used to solve the Kramers-Chandrasekhar equation as an initial value problem, thus allowing us to calculate the reduced initial data for the Smoluchowski equation. The relationship of the functional approach and the multiple time scales is discussed.

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A matrix method, introduced by Geigenmuller, Titulaer and Felderhof, for separating fast and slow motions of a system of first order linear differential equations is shown to arise in a natural way from a multiple time scales expansion. The general method is then used to derive the Smoluchowski equation (and its initial data) for Brownian motion in a strong magnetic field from the corresponding Kramers-Chandrasekhar initial value problem.
Multiple time scales analysis for the Kramers-Chandrasekhar equation
1987, Physica A: Statistical Mechanics and its Applications
A functional approach, similar in spirit to that used by Bogoliubov in his studies of the BBGKY hierarchy, is used to derive the Smoluchowski equation (and corrections) from the Kramers-Chandrasekhar equation. The method of multiple time scales is then used to examine the initial value problem for the Kramers-Chandrasekhar equation, the reduced initial data for the Smoluchowski equation is calculated and the relationship of the function approach and the multiple time scales approach is discussed.
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^☆: Research supported in part by the National Science Foundation.

^∗∗: Current address: Department of Mathematics, State University of College at Potsdam, Potsdam, NY 13676, USA.

^§: Submitted in partial fulfillment of the requirements for the Ph.D. degree at the State University of New York, Stony Brook (1985).

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Multiple time scales analysis for the Kramers-Chandrasekhar equation with a weak magnetic field☆

Abstract

Physica

Ann. d. Physik

J. Plasma Phys.

Boulder Lectures

The Kinetic Theory of Dense Gases

J. Math. Phys.