We wrote kinetic equation for the average x as
What happens if we want to measure x for one system in the ensemble? Then we write
We can speak only about average properties of f(t). We will calculate and . We will average over equilibrium ensemble.
Averaging (6), we obtain:
But we know ! Result:
This could be rewritten as
This is possible only if for But
and depends only on . We obtain:
Since is not zero, it is a -function:
Then we have
and since this is ,
Sometimes the function f(t) is called random noise. Equation (9) shows that random noise is -correlated. In other words, all harmonics are present in its Fourier transform with equal weights (equation (10)). Such noise is called white noise (remember rainbow?).
Colored noise: not all harmonics are equal:
How can it be so? What's wrong in our derivation?
We started from the equation
Another interpretation: the -function in equation (9) is the -function only approximately. In fact it is a sharp peak with the width about the characteristic time of the molecular processes.
© 1997 Boris Veytsman and Michael Kotelyanskii