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

Chapter 23. Brownian motion and continuation 241
The process we are interested in, the passage of the barrier is slow with a timescale much
larger than 1/g. For this reason, we can neglect the time derivative in (23.17). Then, (23.16)
and (23.17) without time derivative makes the main set of equations for our problem. There are
several ways to get a result from these equations, but we will here use a scheme that relates
to that of Kramers (1940). With the neglect of the time derivative, (23.17) can be written as:
P
1 [ ]
g x
f/ kT B
f/
kT B
1e
e P 0
(23.17a)
First, we consider the general contributions P 0 and P 1 . P 0 is close to a Boltzmann-distribution
exp(f/k B T) in the well. P 1 is roughly the integral over this, starting from zero at the far
side of the potential maximum, increasing around the potential minimum, and then essentially
constant close to the potential maximum, where it represents the flux between wells.
Now integrate (23.16) over the potential well, where the particle is at the onset. The integral
of P 0 /t is the time change of the total probability, W, that the particle is in that well. The
corresponding integral for the P 1 -term provides the value of P 1 at the potential top. Thus:
W
t
kT B
m
P 1 ( top )
The RHS corresponds to the rate we are looking for. Now, integrate our re-written expression
(23.17a) over the position x from the potential bottom to the top. The RHS gives the difference
between the values at the top and the bottom. We then assume a situation where the
probability distribution is essentially confined to one well. Inside this well, the probability
is close to the Boltzmann-type distribution exp(f(x)/k B T), but at the top, P 0 which is
flowing over to the other well also goes down and can be put to zero. If the wells are symmetric,
this is strictly so for symmetry reasons. As P 1 increases and goes to a maximum at the
top, the integral at the LHS of (23.17a) gets its largest values from points close to the potential
top. We may there neglect the variation of P 0 . Thus, the integral over the LHS is that of
the exponential function, which has a maximum at the top. One gets:
P
1
1
≈
g
P0
b
∫
e
e
fb
/ kT B
f / kT B
The index ‘b’ refers to the potential bottom. P 0b is essentially the normalisation factor for
the distribution. It can be assumed that the potential close to the bottom can be written as a
harmonic potential: f b (x) c 1 (x x b ) 2 /2, where x b is the position of the potential minimum.
This part will dominate the distribution, and P 0b is essentially the inverse of the integral
of exp(f b (x)/k B T), equal to (c 1 /2k B T) ½ .