Clas Blomberg - Physics of life-Elsevier Science (2007)

Chapter 23. Brownian motion and continuation 239
The previous arguments about the Fokker–Planck equations imply that this means that there
shall be a term added to the previous expression, coming from
Ptvt ( , ( t), x) Ptvt ( , ( ), xt ( )) ( Ptvx ( , , ) v) dv t
( P( t, v,
x v F 1( x )
) ) t
dt
m
This provides a complete Fokker–Planck equation for motion in a potential as:
P
P
v
F1( x) P vP
t x m v
= ( )
g
⎡ ⎣ ⎢ v
kT
m
2P
⎤
t2
⎥
⎦
(23.12)
This equation is the natural starting point for describing Brownian motion in external fields.
It was first derived by Oscar Klein in Stockholm in the 1920s, and later re-derived by
Kramers (1940) who used the equation for describing the passage over energy barriers by
thermal fluctuations. Its usefulness has been renewed from the late 1970s.
Equation (23.12) is difficult from a mathematical point of view as the terms on the two
sides describe processes of different type: the LHS describe a flow with an external force
(these are the same kind of terms as in the Boltzmann equation), and the RHS describes
relaxation, and dissipation of kinetic energy. The equation is not self-adjoint and many
methods for partial differential equations are not applicable.
If the force is given by a potential: F 1 (x) df/dx, one obtains a stationary solution:
⎛
( x) mv
/
P0
( x, v)
Cexp
f
⎝⎜
kT B
2 2
⎞
⎠⎟
(23.13)
C is a normalisation constant. This is, of course, the ordinary equilibrium distribution as
f(x) mv 2 /2 is the sum of the potential and kinetic energies.
The problem has clear application, in particular for macromolecular transitions (see
Edholm and Blomberg, 1981; Blomberg, 1989).
23C
Brownian motion description of the passage over a potential barrier
Consider now the problem of a Brownian particle that moves in a potential of two wells with
a barrier in between. We assume that the particle at the onset is confined to one of the wells.
Because of the random force, it is possible for the particle to cross the potential barrier and go
over to the other well. Our problem is to calculate the probability rate with which this occurs.
For this, we shall use the Fokker–Planck type of equation derived in the preceding section.
P
v
P f′ ( x) P ⎡
( vP) kT
2P
⎤
g
t x m v ⎢
⎣ v
m v2
⎥
⎦
(23.14)