We neglected the term *dM ^{4}*. If we include it, free energy is no
longer a sum of independent terms--fluctuations interact!

Close enough to critical point only interactions of long wavelength fluctuations determine the properties. But the long wavelength limit is described by Landau Hamiltonian!

In other words, near critical point we start from Landau expression, calculate fluctuations and their interactions--and obtain everything!

**Universality:**- Close enough to critical point
() the behavior of the system is determined
only by the symmetry of order parameter.
In particular, for

*scalar*order parameter all systems can be described by partition function(5)

Fluctuations with the wavelength do not ``feel'' the critical point. The only fluctuations that do, are those with

**Scaling Hypothesis:**- The fluctuations at all length scales look
*exactly the same.*Change of the length scale can be compensated by the change of the temperature **Renormalization Group Approach:**- We change length scale in (5) and require to be invariant! By this method people obtained approximation for (and received Nobel Prize).

© 1997

Tue Oct 28 22:10:23 EST 1997