On the theory of brownian motion

Abstract

Synopsis It is shown that the non-Gaussian-Markoff process for Brownian motion derived on a statistical mechanical basis by Prigogine and Balescu, and Prigogine and Philippot, is related through a transformation of variables to the Gaussian-Markoff process of the conventional phenomenological theory of Brownian motion. First the mathematical equivalence of the two types of processes is established by expressing the well-known formulae and equations for the random process {Vx(t), Vy(t)} which describe the motion of a charged Brownian particle in two-dimensional space under the influence of a magnetic field (Vx and Vy are the components of the velocity), in terms of the new variables ∈ = 1 2 m v x 2 + 1 2 m v y 2 and α = arccos { v x ( 2 ∈ / m ) − 1 2 } . The transformed process is called the e(t), a(t) process. It is then shown that the phenomenological theory of the Brownian motion of a strongly underdamped linear harmonic oscillator, if expressed in action-angle variables, leads under well specified conditions to the same e(t), a(t) process, i.e. to the process obtained by Prigogine e.a. in their statistical theory of irreversible processes (in which action-angle variables are used) for a system of weakly coupled harmonic oscillators.