Chemistry Reference
In-Depth Information
to cases with higher dimensions. In particular, the mean square displacement of a three-
dimensional random walker,
r
2
h
i
, is given by:
hr
2
i 56Dt
(6-30)
which is known as the Einstein relation for the self-diffusion coefficient.
C/C
0
=1
t=0
t
1
t
2
t
1
<t
2<
t
3
t
3
X
FIGURE 6.4
Spatial and time dependence of concentration of Brownian particles with an initial
concentration of C
0
at the origin.
6.5.2
Langevin Dynamics
In the previous section, it has been demonstrated that the motion of Brownian
particles can be modeled using the random walk model. But what was missing in
the analysis was that the effect of the surrounding molecules on the motion was
ignored. In this regard, Paul Langevin proposed an approach to improve the origi-
nal Brownian dynamics formalism. Since Langevin's approach
[13]
is much sim-
pler than those proposed by Einstein and Smoluchowski
[14]
, only the Langevin
dynamics is presented here.
Let us consider the motion of a Brownian particle in solvent again. The parti-
cle is continuously bombarded by a large number of solvent molecules. Langevin
