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We applied Fourier transform for space-dependent problems--and we were successful. Let us do the same for time-dependent problems:
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 .
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:
We obtained for correlation function
Take Fourier transform:
Relaxation with one characteristic time is Lorentzian.
Many characteristic times--more complex formulae.
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
We immediately obtain autocorrelation function:
In particular, if
Consider particles with concentration c. Diffusion equation with diffusion coefficient D:
Fourier transform in space:
Let us now consider equilibrium. In equilibrium the fluid is homogeneous. For :
We immediately obtain:
Or, in Fourier space
This is dynamic structure factor.
© 1997 Boris Veytsman and Michael Kotelyanskii