Diffusion equation, constrained Brownian motion

The content of diffusion equation (2.175) for such a model is, moreover, independent of our choice of a system of 3,N coordinates for the unconstrained space. Constrained Brownian motion may thus be described by a model with a mobility and an effective potential /eff in any system of 3N coordinates for... 

Constraints may be introduced either into the classical mechanical equations of motion (i.e., Newton s or Hamilton s equations, or the corresponding inertial Langevin equations), which attempt to resolve the ballistic motion observed over short time scales, or into a theory of Brownian motion, which describes only the diffusive motion observed over longer time scales. We focus here on the latter case, in which constraints are introduced directly into the theory of Brownian motion, as described by either a diffusion equation or an inertialess stochastic differential equation. Although the analysis given here is phrased in quite general terms, it is motivated primarily by the use of constrained mechanical models to describe the dynamics of polymers in solution, for which the slowest internal motions are accurately described by a purely diffusive dynamical model. 

The theory of Brownian motion for a constrained system is more subtle than that for an unconstrained system of pointlike particles, and has given rise to a substantial, and sometimes confusing, literamre. Some aspects of the problem, involving equilibrium statistical mechanics and the diffusion equation, have been understood for decades [1-8]. Other aspects, particularly those involving the relationships among various possible interpretations of the corresponding stochastic differential equations [9-13], remain less thoroughly understood. This chapter attempts to provide a self-contained account of the entire theory. 

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. 

The anomalous diffusivity described by Eq. [13] is due entirely to the fractal nature of the diffusing particle s trajectory in free space. In fractal and multifractal porous media, the diffusing particle s trajectory is further constrained by the geometry of the pore space (Cushman, 1991 Giona et al., 1996 Lovejoy et al., 1998). As a result, when fractional Brownian motion occurs in a two-dimensional fractal porous medium, De becomes scale-dependent, as described by the following equation (Orbach, 1986 Crawford et al., 1993),... 

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