Unconstrained Brownian diffusivity

In Eq. (18.3), it is also assumed that the permeating solute partitions equally in the fluids on either membrane side. The unconstrained Brownian diffusivity in the liquid solvent (i.e., without accounting for pore density and tortuosity in the membrane) of a solute with radius is obtained experimentally and decreases with increasing solute MW. In the absence of experimental estimates, D is related to the solute equivalent spherical radius, Tj, according to the Einstein-Stokes equation, as follows ... 

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. 

Using this definition of we may generalize the diffusion equation for the distribution /( ) on the /-dimensional constraint surface to an equivalent diffusion equation for a distribution V Q) in the 3A-dimensional unconstrained space. We consider a model in which a system of 3N coordinates undergoes Brownian motion in the full unconstrained space under the influence of the mobility, defined above, as described by a diffusion equation... 

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... 

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