By Eric Bertin
Introduction.- Equilibrium Systems.- Nonequlibrium Systems.- References
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Extra info for A Concise Introduction to the Statistical Physics of Complex Systems
101) which defines the correlation length ξ . The latter diverges for T → Tc . 4 Phase Transitions 23 correlation between spins. A natural idea is to look for an approach that could reduce in some way the intensity of correlations, in order to make calculations tractable. This is basically the principle of the renormalization group (RG) approach, in which one progressively integrates out small scale degrees of freedom. The idea is that at the critical point, structures are present at all scales, from the lattice spacing to the system size.
In D > 1, analytical solutions are not known, and results have thus been obtained through numerical simulations. A fully connected version, called the Sherrington– Kirkpatrick model, has been proposed and solved, but the techniques involved are already rather difficult, even at the level of the fully connected model. The main qualitative picture emerging from these models is that below a given energy level, the phase space decomposes into a lot of valleys, or metastable states, from which it takes a very long time to escape.
To this purpose, one considers an Ising-like model in Ddimensions, where the spins are placed at each node of a lattice. Spins on neighboring sites (i,j) interact through a coupling constant Ji j , drawn from a distribution P(J). As the couplings Ji j are kept constant in time, one speaks about quenched disorder. This model is called the Edwards–Anderson model . In D = 1, the Edwards–Anderson model is equivalent to the standard Ising model, up to a redefinition of the spins. In D > 1, analytical solutions are not known, and results have thus been obtained through numerical simulations.
A Concise Introduction to the Statistical Physics of Complex Systems by Eric Bertin