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

Chapter 29. Noise and non-linear phenomena 315
interpretation, they are considered to be correlated. In the latter case, the correlation is half
the correlation at a time immediately after t (see van Kampen, 1981, 1992).
With such rules, eq. (29.1) is well defined for external noise, see above, which should be to
be independent of the basic dynamics. This means that the two terms on the RHS of eq. (29.1)
are independent. For internal noise, i.e. noise within the system, the situation in less clear,
and an equation such as eq. (29.1) cannot be a strict representation of the system, although
this kind of equation occurs frequently. The point is that the fluctuating force is then part of the
dynamics and should not be independent of force f(x). In the linear case, the explicit choice of
the noise term leads to the fluctuation–dissipation theorem. There is no similar, simple relation
in the non-linear case, i.e. eq. (29.1). One also sees that the relation for the average of x, x,
in the simplest case of Chapter 21, leads to the simple damping equation d x/dt g x.
However, there is no simple relation following from eq. (29.1) for such an average, which
then is influenced by the noise term. Note that, in general, f (x) f (x). An alternative is
to develop a mathematical formalism for calculating a probability distribution. This needs
some insights about details of the system, and we will not go further into that possibility.
Equations of type (29.1) may be used and motivated for qualitative results with an intrinsic
noise. However, this kind of equation can lead to completely wrong results (see the discussion
by van Kampen, 1992, Chapter 9).
Now, go back to the problem of interpreting the multiplicative noise of eq. (29.1). The
problem appears because of the singular variation of the white noise also during a brief
time interval. To see the problem and how to overcome it, introduce the Wiener process
(see Chapter 23).
t
Wt () F( t)
dt
∫
0
In the Itô description, one writes the change of this entity during a time interval t as
W W(t t) – W(t). The white noise function values at two different times are completely
uncorrelated, F(t 1 )F(t 2 ) 0 if t 1 t 2 , which means that W(t) and W as defined
here are uncorrelated. (They are defined in time intervals that do not overlap.) A derivative
of the RHS of eq. (29.1) contains the change W, and expressions containing W occur in
numerical solutions of eq. (29.1). If one maintains that W(t) and W are uncorrelated, then
eq. (29.1) is not consistent with ordinary differential and integration rules (such that a function
is the integral of its derivative), and these have to be modified. This can be done and leads
to what is called Itô rules (Itô, 1944) which provide a consistent way of treatment. This is
usually considered in mathematical literature as the most elegant way to treat the problem.
There is, as mentioned, another way to treat the problem considered by Stratonovich
(1963), by which the normal differential and integration rules are kept, but where W is
redefined to provide a consistent procedure. It is then defined for an interval around the
time t: W W(t t/2) – W(t t/2). Then W(t) and W are not uncorrelated, a fact
that is important in any integration procedure.
The white noise can be regarded as a limit of a more general (coloured) noise function
for which F c (t) gets an autocorrelation (correlation between values at two different times)