In the statistical physics of disordered systems, 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.
[1][2] It concerns the statistics of a collection of
spins (i.e. degrees of freedom
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.
Critical inverse temperature
When this is true, we say that it has the self-averaging property.
, the Boltzmann distribution of the system is concentrated at energy-per-particle
, the Boltzmann distribution of the system is concentrated at
, 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.
The participation ratio measures the amount of condensation in the Boltzmann distribution.
, the participation ratio is a random variable determined by the energy levels.
, the system is not in the condensed phase, and so by asymptotic equipartition, the Boltzmann distribution is asymptotically uniformly distributed over
where the expectation is taken over all random energy levels.
-spin sets interact with a random, independent, identically distributed interaction constant, becomes the random energy model in a suitably defined
[3] More precisely, if the Hamiltonian of the model is defined by where the sum runs over all
is an independent Gaussian variable of mean 0 and variance
, the Random-Energy model is recovered in the
As its name suggests, in the REM each microscopic state has an independent distribution of energy.
refers to the individual spin configurations described by the state and
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.
over all possible realizations, we find that the probability that a given configuration of the disordered system has an energy equal to
denotes the average over all realizations of the disorder.
Moreover, the joint probability distribution of the energy values of two different microscopic configurations of the spins,
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.
However this expression only holds if the entropy per spin,
, the system remains "frozen" in a small number of configurations of energy
and the entropy per spin vanishes in the thermodynamic limit.