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

236 Part V. Stochastic dynamics
In a stricter language, this is written as a relation of differentials:
dv gv dt2
q dW
(23.5)
and instead of the singular expression (21.5), we write the solution formally as:
t
∫
0
vt () qe g ( tt
′ )
dWt (′)
(23.6)
The next step is to derive a probability distribution for the velocity and then also for the
complete Brownian motion. This can be done in many ways. One can use the expression
for the Wiener process, also the known results of the correlations, and one can also go over
a Fokker–Planck type of equation. We will consider the last possibility as this approach
will be further developed in later developments.
Thus, we want a differential equation for the probability function P(t, v). For this, we have
to consider the fact that v varies with time according to the relations above, and we see how
the distribution function varies with time: P(t t, v(t t)) A first consideration of the
time change would mean up to second order:
⎡ Ptv (, ) Ptv (, )
Pt ( t, vt ( t))
⎤ ⎡
t
⎤ ⎡
+
v
2
Ptv (, ) ⎤
( v)
⎣⎢
t
⎦⎥
⎣⎢
v
⎦⎥
⎢ v2
⎥
⎣ ⎦ 2
2
v would be given by: v dv/dt t. Thus, the second term might be equal to (P(t, v)
v) (dv/dt) t. However, this does not provide a complete relation for the change of P
because of the simple reason that the total probability, integrated over all velocities must be
equal to one. Thus, the second term must be supplemented to correct for the total probability,
and the expression should be written as:
Ptv (, ) [ vP(, t v)] Ptv (, )
Pt ( t, vt ( t)) Ptv
( , ) t
2
( v)
t
v
v2
2
2
Then, we have to take the last term into account. To get an appropriate equation, we
consider terms of order t. Again, note the equation for the velocity in the last form
above:
vg vtqW; W W( tt) W( t)