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

Chapter 29. Noise and non-linear phenomena 319
It is straightforward to calculate the probability for a stationary situation, when the time
derivative is zero. Then, it is valid that:
W X
W
(
X ) PX ( ) W
( X ) PX ( ), PX ( ) ⎡ ( ) ⎤
1 1 i.e. 1
⎢
P
W
( X1)
⎥
( X)
⎣ ⎦
(29.6)
This can be used repeatedly to get a general relation for a stationary probability density
from which one can calculate (numerically) averages and correlation functions.
The complete time-dependent equation is more intricate, but it should be possible to treat
numerically, again by a recurrence method. One should limit the number, the values of X,
say by X max , and start with an initial probability distribution with probability values for various
X. Then, at each time step, these values are changed according to the basic reaction
rules, and we get a sequence of density functions as a development in time. Such a method
may look cumbersome, but should not provide any difficulty on a modern computer.
If the species X also changes by, say, two units by reactions of type A ↔ 2X, then the step
method above (29.6) does not work, and some more intricate method is necessary. The proposed
development for the time-dependent situation still works.
If there is a very slow development, manifested by a slowly varying exponential function,
and a small eigenvalue, one might find appropriate approximation methods. Compare the
method for the barrier passage in Chapter 23.
The situation gets worse when there is more than one varying reacting main species. It is
still possible to write down a master equation for a probability density as P(X, Y, t), with
expressions for increase and decrease of the different species. Still, for that situation, there
is no simple iteration procedure such as eq. (29.6), and no simple method even for the
stationary distribution.
Otherwise, it is possible to make what can be called “simulation”, calculation of successive
states of even a complicated situation by direct use of random changes. At each time step,
each species can increase or decrease by probabilities given by the rate functions, and we
thus get a general picture of a kind of direct stochastic “computer experiment”.
This method can always be used, and it also provides values for averages and correlations
as well as possibilities to calculate probability densities (at least for stationary situation).
We will use this kind of method for a highly non-linear coupled reaction scheme of two
species based on the so-called Brusselator model.
Assume a relatively small system and rewrite the scheme to represent the numbers of the
reacting species. Put X x/n, Y y/n, A a/n, B b/n 2 with n equal to 100. x, y, a, b shall
now represent number of molecules. Further, put, a 100, b around 20,000, that is A 1,
B around 2, the bifurcation value. The stationary values of x and y are both equal to 100. Next,
assume a random scheme where the probabilities to increase or decrease the molecule
numbers during one time unit is equal to the terms of the reaction scheme: The probability
rates are: to increase x by one unit: a x 2 y/n 2 , to decrease x: (b/n 2 1)x, to increase y: bx/n 2 ,
do decrease y: x 2 y/n 2 . The rules are used to simulate the process. We get the results below
(Figures 29.2 and 29.3).