Wikipedia / Random energy model
Wikipedia / Random energy model
In the statistical physics of disordered systemss, the random energy model is a toy model of a system with quenched disorder, such as a spin glass, having a first-order phase transition. spins (i.e.. The energies of such states are independent and identically distributed Gaussian random variables. Many properties of this model can be computed exactly. Its simplicity makes this model suitable for pedagogical introduction of concepts like quenched disorder and replica symmetry.
Thermodynamic quantities
.
.
, that is, condensation does not occur. When this is true, we say that it has the self-averaging property.
Condensation
, the Boltzmann distribution states.
, and since the entropy per particle at that point is zero, the Boltzmann distribution is concentrated on a sub-exponential number of states. This is a phase transition called condensation.
Participation
Define the participation ratioThe participation ratio measures the amount of condensation in the Boltzmann distribution. It can be interpreted as the probability that two randomly sampled states are exactly the same state. Indeed, it is precisely the Simpson index, a commonly used diversity index.
, the participation ratio is a random variable determined by the energy levels.
, the system is not in the condensed phase, and so by asymptotic equipartitionwhich decays exponentially to zero.
where the expectation is taken over all random energy levels.
Comparison with other disordered systems
-spin infinite-range model limit.
More precisely, if the Hamiltonian of the model is defined by
limit.
Derivation of thermodynamical quantities
is the energy associated with it. The final extensive variables like the free energy need to be averaged over all realizations of the disorder, just as in the case of the Edwards–Anderson model is given by
denotes the average over all realizations of the disorder. Moreover, the joint probability distribution factorizes:
It can be seen that the probability of a given spin configuration only depends on the energy of that state and not on the individual spin configuration.
The entropy of the REM is given by
and the entropy per spin vanishes in the thermodynamic limit.