Comparison of Langevin and Monte Carlo dynamics

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Abstract
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.

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