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

226 Part V. Stochastic dynamics
All information about such a process lies in the averages X(t) and the autocorrelation functions
X(t 1 ) X(t 2 ). This makes the process relatively easy to handle even if the process is not
Markovian.
If one requires that the process is Gaussian, Markovian and also has a stationary
distribution, then there is only one possibility. With one variable, its relevant probability
functions are:
1
Pxt ( , ) e
(( xx0 ) 2 / 2a)
Pxt ( , x1, tt)
2pa
1
( ) exp ⎛ 1
⎞
( xx x
gt
)
2
0 1e
2pa
1 e2gt
⎝⎜
2a
⎠⎟
(22.15)
x 0 is the average of X(t). All other probabilities are given from these. This process is called the
Ornstein–Uhlenbeck process. We will encounter this process in later example, in particular
concerning Brownian motion, Chapter 23. The velocity distribution of Brownian motion is
of this kind.
22D
Fokker–Planck equations
Transition from discrete steps to continuous equations
In the previous chapter, we considered discrete processes, characterised by certain discrete
states and probabilities of transitions between the states. The development of the probability
distribution of the respective states was described by the following type of equation, called
master equation.
dPn
() t
an ( 1) Pn1( t) bn ( 1) Pn1( t) [ an ( ) bn ( )] Pn( t)
dt
(22.16)
where P n (t) is the probability for being in state ‘n’ at time t; a(n) is the transition probability
rate for going one step downwards from state ‘n’ to state ‘n 1’, and b(n) is the transition
probability rate to go one step upwards from state ‘n’ to ‘n + 1’. The states may mean number
of particles of some kind (e.g. reacting substances) or certain positions.
These equations can be difficult to handle, and one may try other possibilities. If the state
numbers are large, it might be advantageous to go over to a continuous description, to treat
the state values as a continuous variable. A way to accomplish this is to use the first terms
in a Taylor expansion:
1
2
Fx ( 1) Fx ( ) F( x) F( x); Fx ( 1) Fx ( ) F( x) F( x)
1
2