Up: Time Correlation Functions. Random Previous: Spectral Representation

Subsections

Random Force and Its Applications

Stochastic Equation

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
 (6)
with random variable x. The function f(t) is random force.

Properties of Random Force

We can speak only about average properties of f(t). We will calculate and . We will average over equilibrium ensemble.

Averaging (6), we obtain:

or
 (7)
Equation (6) can be solved. The solution is
 (8)
Correlation function:

But we know ! Result:

This could be rewritten as

This is possible only if for But

and depends only on . We obtain:

From symmetry

Since is not zero, it is a -function:

Then we have

and since this is ,

We obtained:
 (9)
or, after Fourier transform
 (10)

White Noise and Colored Noise

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:

and .

How can it be so? What's wrong in our derivation?

We started from the equation
 (11)
This is a consequence of the fact that x is the slowest mode in the system and we averaged out everything else! If this is not true, equation (11) does not work, and noise is no longer white!

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.

Up: Time Correlation Functions. Random Previous: Spectral Representation

© 1997 Boris Veytsman and Michael Kotelyanskii
Tue Oct 28 22:16:24 EST 1997