Page 363 - Numerical Methods for Chemical Engineering

352 7 Probability theory and stochastic simulation

This yields the famous Einstein relation
k b T
D = (7.213)
ζ
Combining this expression with Stokes’ law for the drag constant yields the Stokes–Einstein
relation, predicting the diffusivity of a sphere through a Newtonian fluid,
k b T
D = (7.214)
6πµR
The Einstein relation is a special case of a more general result known as the fluctuation-
dissipation theorem (FDT). The FDT relates the strength of the random thermal fluctuations
(here D) to the corresponding susceptibility to external perturbations (here ζ −1 ) in such a
way that ensures that the probability distribution converges to the proper equilibrium result
at steady state.
Using the Einstein relation, the random Brownian force on the particle has the statistical
properties
F R (t) = 0 F R (t)F R (0) = [2ζk b T ]δ(t) (7.215)

General formulation of SDEs; Brownian motion in multiple dimensions

Above we have considered only a single SDE, but systems of coupled SDEs can also be
solved. Consider the 3-D isotropic Brownian motion of a paticle, in which each component
of the position vector is governed by a SDE:
∂U
−1 1/2 (x)
dx =−ζ dt + (2D) dW t
∂x
∂U
−1 1/2 (y)
dy =−ζ dt + (2D) dW t (7.216)
∂y
∂U 1/2 (z)
−1
dz =−ζ dt + (2D) dW t
∂z
(x) (y) (z)
dW t , dW t , and dW t are increments of independent Wiener processes and the Einstein
relation requires ζ −1 = D/(k b T ). Defining the position, conservative force, and Wiener
update vectors,
 (x) 
    dW t
x −∂U/∂x
(c)  (y) 
y
r =   F =−∇U =  −∂U/∂y  dW t =  dW  (7.217)
 t 
z −∂U/∂z (z)
dW t
(7.216) is written more compactly as
−1 (c) 1/2
dr = ζ F dt + (2D) dW t (7.218)
For diffusion in three dimensions, in the absence of an external potential, we have
dx = (2D) 1/2 dW t (x) dy = (2D) 1/2 dW t (y) dz = (2D) 1/2 dW t (z) (7.219)

The mean-squared displacement in 3-D space is
2 2 2 2
( r) = ( x) + ( y) + ( z) (7.220)