Disordered systems §

Models like Curie-Weiss may be too simple to capture the complexity of real solids, for example the couplings may depend on time, be pair depandant (two different molecules) or may be subjected to noise.

The key idea is to make the coupling $J_{ij}$ random variables, and then to study the typical behaviour of the system average the free energy for all realization of the couplings.

We have two different kind of averages:

• Annealed $f^A:={\beta }^{-1}\ln \mathbb{E}Z$
• Quenced $f^Q:={\beta }^{-1}\mathbb{E}\ln Z$

which correspond to two different physical cases annealing: slow cool down to make the system more homogeneous quenching: abrupt cool down to improve solid resistance.

It follow from Jensen’s inequality that the annealed free energy is a lower bound on the quenched.

$$f^Q\ge f^A$$

The computation of the annealed free energy is much simpler, since averaging a logarithm is hard. One trick (not rigourus!) is the replica trick, which uses the identity

$$\ln Z=\underset{n\to \infty }{\lim }\frac{Z^n-1}{n}$$

Then instad of computing the average of a logarithm, one computes the $n$-th momenths.