Generalized coordinates, constrained Brownian motion

Throughout this section, we will use the notation X (t),..., X t) to denote a unspecified set of L Markov diffusion processes when discussing mathematical properties that are unrelated to the physics of constrained Brownian motion, or that are not specific to a particular set of variables. The variables refer specifically to soft coordinates, generalized coordinates for a system of N point particles, and Cartesian particle positions, respectively. The generic variables X, ..., X will be indexed by integer variables a, p,... = 1,...,L. 

Constrained Brownian motion may be described in generalized cordinates as the solution of a set off Ito SDEs for the soft generalized coordinates, . of the form... 

In this section, we begin the description of Brownian motion in terms of stochastic process. Here, we establish the link between stochastic processes and diffusion equations by giving expressions for the drift velocity and diffusivity of a stochastic process whose probability distribution obeys a desired diffusion equation. The drift velocity vector and diffusivity tensor are defined here as statistical properties of a stochastic process, which are proportional to the first and second moments of random changes in coordinates over a short time period, respectively. In Section VILA, we describe Brownian motion as a random walk of the soft generalized coordinates, and in Section VII.B as a constrained random walk of the Cartesian bead positions. 

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