# Introduction to Mathematical Physics/N body problems and statistical equilibrium/Spin glasses

Assume that a spin glass system \index{spin glass}(see section{secglassyspin}) has the energy:

$H=\sum J_{ij}S_iS_j$

Values of variable $S_i$ are $+1$ if the spin is up or $-1$ if the spin is down. Coefficient $J_{ij}$ is $+1$ if spins $i$ and $j$ tend to be oriented in the same direction or $-1$ if spins $i$ and $j$ tend to be oriented in opposite directions (according to the random position of the atoms carrying the spins). Energy is noted:

$H_J=\sum J_{ij}S_iS_j$

where $J$ in $H_J$ denotes the $J_{ij}$ distribution. Partitions function is:

$Z_J=\sum_{[s]}e^{-\beta H_J[s]}$

where $[s]$ is a spin configuration. We look for the mean $\bar f$ over $J_{ij}$ distributions of the energy:

$\bar f=\sum_JP[J]f_J$

where $P[j]$ is the probability density function of configurations $[J]$, and where $f_J$ is:

$f_J=-\ln Z_J.$

This way to calculate means is not usual in statistical physics. Mean is done on the "chilled" $J$ variables, that is that they vary slowly with respect to the $S_i$'s. A more classical mean would consist to $\sum_J P[J]\sum_{[s]}e^{-\beta H_J[s]}$ (the $J$'s are then "annealed" variables). Consider a system $S_j^n$ compound by $n$ replicas\index{replica} of the same system $S_J$. Its partition function $Z_J^n$ is simply:

$Z_J^n=(Z_J)^n$

Let $f_n$ be the mean over $J$ defined by:

$f_n=-\frac{1}{n}\ln \sum_J P[J](Z_J)^n$

As:

$\ln Z=\lim_{n\rightarrow 0}\frac{Z^n-1}{n}$

we have:

$\lim_{n\rightarrow 0}f_n=\lim_{n\rightarrow 0}\ln (\sum_J P[J][1+n \ln Z_J])$

Using $\sum_JP[J]=1$ and $\ln(1+x)=x+O(x)$ one has:

$\bar f=\lim_{n\rightarrow 0}f_n.$

By using this trick we have replaced a mean over $\ln Z$ by a mean over $Z^n$; price to pay is an analytic prolongation in zero. Calculations are then greatly simplified [ph:sping:Mezard87].

Calculation of the equilibrium state of a frustrated system can be made by simulated annealing method .\index{simulated annealing} An numerical implementation can be done using the Metropolis algorithm\index{Metropolis}. This method can be applied to the travelling salesman problem (see [ma:compu:Press92] \index{travelling salesman problem}).