Davis, Joel L., and Ronald Forrest Fox. “Maxwell $$\xrightarrow [{t\to\infty}]{} $$ Boltzmann.” Journal of Statistical Physics 22, no. 6 (1980): 627-645.
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This paper presents a new approach to the solution of the Brownian motion problem in which a particle subject to Brownian forces is also subject to a potential. The dynamical description of this system is written in the form of a multiplicative, stochastic, differential equation. This equation is solved and the solution is simplified so that it may be written in terms of integrals of well-understood functions. Enough of the kinematic details of the system are revealed in this way to show that the infinite-time limit of this dynamical solution is a Maxwell-Boltzmann distribution. Although there are other approaches to this problem which yield the infinite-time limit directly, these methods cannot be extended to find the solution to this problem for finite times. In this paper the solution is exhibited for all times, and the details of the approach to the infinite-time limit are elucidated.