Random energy model

In statistical physics of disordered systems, the random energy model is a toy model of a system with quenched disorder. It concerns the statistics of a system of $N$ particles, such that the number of possible states for the systems grow as $2^N$, while the energy of such states is a Gaussian stochastic variable. The model has an exact solution. Its simplicity makes this model suitable for pedagogical introduction of concepts like quenched disorder and replica symmetry.

Comparison with other disordered systems

The $r$-spin Infinite Range Model, in which all $r$-spin sets interact with a random, independent, identically distributed interaction constant, becomes the Random-Energy Model in a suitably defined $r\to\infty$ limit.^[1]

More precisely, if the Hamiltonian of the model is defined by

$H(\sigma)=\sum_{\{i_1,\ldots,i_r\}}J_{i_1,\ldots i_r}\sigma_{i_1}\cdots\sigma_{i_r},$

where the sum runs over all ${N\choose r}$ distinct sets of $r$ indices, and, for each such set, $\{i_1,\ldots,i_r\}$, $J_{i_1,\ldots,i_r}$ is an independent Gaussian variable of mean 0 and variance $J^2r!/(2 N^{r-1})$, the Random-Energy model is recovered in the $r\to\infty$ limit.

Derivation of thermodynamical quantities

As its name suggests, in the REM each microscopic state has an independent distribution of energy. For a particular realization of the disorder, $P(E) = \delta(E - H(\sigma))$ where $\sigma=(\sigma_i)$ refers to the individual spin configurations described by the state and $H(\sigma)$ 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. Averaging $P(E)$ over all possible realizations, we find that the probability that a given configuration of the disordered system has an energy equal to $E$ is given by

$[P(E)] = \sqrt{\dfrac{1}{N\pi J^{2}}}\exp\left(-\dfrac{E^{2}}{J^{2}N}\right),$

where $[\ldots]$ 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, $\sigma$ and $\sigma'$ factorizes:

$[P(E,E')]=[P(E)]\,[P(E')].$

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.^[2]

The entropy of the REM is given by^[3] $S(E) = N\left[\log 2 - \left(\dfrac{E}{NJ}\right)^{2}\right]$ for $|E| < NJ\sqrt{\log 2}$. However this expression only holds if the entropy per spin, $\lim_{N\to\infty}S(E)/N$ is finite, i.e., when $|E|< -N J \sqrt{\log 2}.$ Since $(1/T)=\partial S/\partial E$, this corresponds to $T>T_c=1/(2\sqrt{\log 2})$. For $T<T_c$, the system remains "frozen" in a small number of configurations of energy $E\simeq -N J \sqrt{\log 2}$ and the entropy per spin vanishes in the thermodynamic limit.

References

1. Derrida, Bernard (14 July 1980). "Random Energy Model: Limit of a Family of Disordered Models". Phys. Rev. Letters 45 (2). Bibcode:1980PhRvL..45...79D. doi:10.1103/PhysRevLett.45.79.
2. Nishimori, Hidetoshi (2001). Statistical Physics of Spin Glasses and Information Processing: An Introduction (PDF). Oxford: Oxford University Press. p. 243. ISBN 9780198509400.
3. Derrida, Bernard (1 September 1981). "Random-energy model: An exactly solvable model of disordered systems". Phys. Rev. B 24 (5). Bibcode:1981PhRvB..24.2613D. doi:10.1103/PhysRevB.24.2613. Retrieved 24 March 2011.