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