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- Problem
- Landau Hamiltonian and Normal Coordinates
- Fluctuations in Fourier Space. Scattering
- Correlation in Real Space. Ornstein-Zernicke Function
- Ginsburg Number

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

For *inhomogeneous* system the free energy is

- 1.
- We are above critical point, so
*a*>0 (for negative*a*we expand around*M*)_{0} - 2.
- We
*neglect*the term*dM*, This is the most important approximation!!!^{4}

Then we make Fourier transform:

Quadratic term:

Gradient term: and Term with external field:Mathematical result

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 meansWe 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:

The function

Is calledWhat 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 inLandau theory works if fluctuations are small. Average fluctuation in the volume is

Comparing this to the equilibrium value(4) |

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.

© 1997

Tue Oct 28 22:10:23 EST 1997