Glauber dynamics

In statistical physics, Glauber dynamics[1] is a way to simulate the Ising model (a model of magnetism) on a computer.

The Metropolis acceptance criterion includes the Boltzmann weight,

, but it always flips a spin in favor of lowering the energy, such that the spin-flip probability is:

.Although both of the acceptance probabilities approximate a step curve and they are almost indistinguishable at very low temperatures, they differ when temperature gets high.

For an Ising model on a 2d lattice, the critical temperature is

However, at thermal equilibrium, these two algorithms should give identical results.

In general, at equilibrium, any MCMC algorithm should produce the same distribution, as long as the algorithm satisfies ergodicity and detailed balance.

, meaning that transition between the states of the system is always possible despite being very unlikely at some temperatures.

Detailed balance, which is a requirement of reversibility, states that if you observe the system for a long enough time, the system goes from state

In equilibrium, the probability of observing the system at state A is given by the Boltzmann weight,

So, the amount of time the system spends in low energy states is larger than in high energy states and there is more chance that the system is observed in states where it spends more time.

more frequently, counterbalancing the lower intrinsic probability of transition.

Therefore, both, Glauber and Metropolis–Hastings algorithms exhibit detailed balance.