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In inhomogeneous systems G depends on the value of x in all points . G is a functional --it depends on a function .
For inhomogeneous system the free energy is
Assumptions and simplifications:
Then we make Fourier transform:
Term with external field:
Integrals with quadratic & gradient terms contain
This is non-zero only if . Then
and we obtain
We represented free energy as a sum over Fourier components--this means are normal coordinates!
We immediately obtain for fluctuations
This is exactly the function measured in scattering experiments!
What happens near critical points? Here is small.
Is called correlation function. It is just
Integrating , we obtain:
with correlation radius
This function is called Ornstein-Zernicke function.
What happens at ? Correlation radius
Above we obtained the formula
This works in thermodynamic limit . In other words . Near Tc this condition becomes very stringent!
Landau theory works if fluctuations are small. Average fluctuation in the volume is
Comparing this to the equilibrium value below critical point , we obtain that if
Landau theory works, if we are both:
Why does Landau theory work well for polymers? Coefficient g does not depend on molecular weight N, but and . We obtain , i.e. Ginsburg number is small.
© 1997 Boris Veytsman and Michael Kotelyanskii