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.

Derivation of thermodynamical quantities

The entropy of the REM is given by<ref name="RE Model: Exact Solution"{{cite journal|last=Derrida|first=Bernard|title=Random-energy model: An exactly solvable model of disordered systems|date=01|year=1981|month=September|volume=24|series=Phys. Rev. B|issue=5|doi=10.1103/PhysRevB.24.2613|url=http://link.aps.org/doi/10.1103/PhysRevB.24.2613%7Caccessdate=24 March 2011}}><\ref>

Suppose a system is described by a total energy given by a sum of random energy

$E=\sum _{i}^{N}E_{i}$

suppose that these are independent and identical random variables with average ${\bar {E}}$ and standard deviation $\sigma$, then by the central limit theorem the energy E will be a random variable with gaussian distribution with mean $N{\bar {E}}$ and standard deviation ${\sqrt {N}}\sigma$.