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

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

Values of variable are if the spin is up or if the spin is down. Coefficient is if spins and tend to be oriented in the same direction or if spins and tend to be oriented in opposite directions (according to the random position of the atoms carrying the spins). Energy is noted:

where in denotes the distribution. Partitions function is:

where is a spin configuration. We look for the mean over distributions of the energy:

where is the probability density function of configurations , and where is:

This way to calculate means is not usual in statistical physics. Mean is done on the "chilled" variables, that is that they vary slowly with respect to the 's. A more classical mean would consist to (the 's are then "annealed" variables). Consider a system compound by replicas\index{replica} of the same system . Its partition function is simply:

Let be the mean over defined by:

As:

we have:

Using and one has:

By using this trick we have replaced a mean over by a mean over ; 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}).