   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