Parallel tempering in physics and statistics, is a computer simulation method typically used to find the lowest free energy state of a system of many interacting particles at low temperature. That is, the one expected to be observed in reality. It addresses the problem that at high temperature one may have a stable state different from low temperature, whereas simulations at low temperature may become "stuck" in a metastable state. It does this by using the fact that the high temperature simulation may visit states typical of both stable and metastable low temperature states. More specifically, parallel tempering (also known as replica exchange MCMC sampling), is a simulation method aimed at improving the dynamic properties of Monte Carlo method simulations of physical systems, and of Markov chain Monte Carlo (MCMC) sampling methods more generally. The replica exchange method was originally devised by Swendsen and Wang then extended by Geyer and later developed, among others, by Hukushima and Nemoto, Giorgio Parisi, Sugita and Okamoto formulated a molecular dynamics version of parallel tempering: this is usually known as replica-exchange molecular dynamics or REMD. Essentially, one runs N copies of the system, randomly initialized, at different temperatures. Then, based on the Metropolis criterion one exchanges configurations at different temperatures. The idea of this method is to make configurations at high temperatures available to the simulations at low temperatures and vice versa. This results in a very robust ensemble which is able to sample both low and high energy configurations. In this way, thermodynamical properties such as the specific heat, which is in general not well computed in the canonical ensemble, can be computed with great precision. Background Typically a Monte Carlo simulation using a Metropolis–Hastings update consists of a single stochastic process that evaluates the energy of the system and accepts/rejects updates based on the temperature T. At high temperatures updates that change the energy of the system are comparatively more probable. When the system is highly correlated, updates are rejected and the simulation is said to suffer from critical slowing down. If we were to run two simulations at temperatures separated by a ΔT, we would find that if ΔT is small enough, then the energy histograms obtained by collecting the values of the energies over a set of Monte Carlo steps N will create two distributions that will somewhat overlap. The overlap can be defined by the area of the histograms