Comparison of Langevin and Monte Carlo dynamics

The Monte Carlo and Langevin dynamical methods of simulating the thermodynamics of physical systems are compared by calculating relaxation times according to the two dynamics for a system which is analytically tractable, namely a single (planar) spin in a potential which has either a single minimum or two minima separated by a barrier. With no restriction on the maximum allowed spin reorientation per Monte Carlo step the Langevin method is faster than the Monte Carlo method for a single minimum potential. However a careful choice of restriction can make the Monte Carlo method as efficient as the Langevin method. For the double-well potential the Monte Carlo method with no restriction is the most efficient. One is forced to use a finite-time step size when numerically solving the Langevin equation and the departures this produces from the equilibrium Boltzmann distribution are studied.