Subsections

# Spectral Representation

## Definitions

We applied Fourier transform for space-dependent problems--and we were successful. Let us do the same for time-dependent problems:
 (1)
Inverse transform:

## Correlation Function

The product of x's is

Or, since only x's are random,

Symmetry with respect to time: depends only on t-t'. This means that in the integral . This means that .
Conclusion:
should be proportional to -function

Definition:
Let
 (2)
Then

We see that is indeed the Fourier transform of . This is exactly the function measured in inelastic scattering, relaxation experiments (NMR, ESR, dielectric spectroscopy, shear rheometry,...). Inverse:

If t=0, we obtain

But . We obtained:

## Relaxation in Fourier Space

We obtained for correlation function

Take Fourier transform:
 (3)
This is a Lorentzian!

Relaxation with one characteristic time is Lorentzian.

Many characteristic times--more complex formulae.

## Example: Velocity Autocorrelation Function

Consider a large particle surrounded by small particles (a colloidal particle among molecules). Equation for average velocity:

with friction coefficient . For a spherical particle with radius r and mass m in liquid with viscosity
 (4)
The average square

We immediately obtain autocorrelation function:
• In the real time:

• After Fourier transform:    (5)
Equations (3) and (5) are special cases of fluctuation-dissipation theorem.

In particular, if

## Example: Diffusion

Consider particles with concentration c. Diffusion equation with diffusion coefficient D:

Fourier transform in space:

We obtain:

Let us now consider equilibrium. In equilibrium the fluid is homogeneous. For :

with .

We immediately obtain:

Or, in Fourier space

This is dynamic structure factor.

Next: Random Force and Its Up: Time Correlation Functions. Random Previous: Time Correlation Functions. Random

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