Subsections

Inhomogeneous Systems

Problem

In inhomogeneous systems G depends on the value of x in all points . G is a functional --it depends on a function .

Analogy with multi variable case:
we have variables ,,..., and want to minimize G.
Idea:
Introduce normal coordinates that diagonalize G
Trick:
Use Fourier transform--it leads to normal coordinates in linear problems

Landau Hamiltonian and Normal Coordinates

For inhomogeneous system the free energy is

Assumptions and simplifications:
1.
We are above critical point, so a>0 (for negative a we expand around M0)
2.
We neglect the term dM4, This is the most important approximation!!!

Then we make Fourier transform:

and

Term with external field:

Mathematical result

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!

Fluctuations in Fourier Space. Scattering

We immediately obtain for fluctuations

This is exactly the function measured in scattering experiments!

What happens near critical points? Here is small.

1.
Short wavelength fluctuations do not depend on the temperature:

2.
Long wavelength fluctuations grow:

3.
The characteristic wavelength tends to infinity:
This is called critical opalescence

Correlation in Real Space. Ornstein-Zernicke Function

The function

Is called correlation function. It is just

Integrating , we obtain:

This function is called Ornstein-Zernicke function.

What happens at ? Correlation radius

1.
At we have exponential decay with distance
2.
At correlations decay as 1/r and are independent of the temperature!

Above we obtained the formula

This works in thermodynamic limit . In other words . Near Tc this condition becomes very stringent!

Ginsburg Number

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

or, since
 (4)
is Ginsburg number.

Landau theory works, if we are both:

1.
Close to critical point, so (we used expansion!)
2.
Still not close enough, so .

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.

Next: Universality and Scaling Invariance Up: Fluctuations in Inhomogeneous Systems. Previous: Fluctuations in Homogeneous Systems

© 1997 Boris Veytsman and Michael Kotelyanskii
Tue Oct 28 22:10:23 EST 1997