Mapping the Monte Carlo Scheme to Langevin Dynamics: A Fokker-Planck Approach

Abstract

We propose a general method of using the Fokker-Planck equation (FPE) to link the Monte Carlo (MC) and the Langevin micromagnetic schemes. We derive the drift and diffusion FPE terms corresponding to the MC method and show that it is analytically equivalent to the stochastic Landau-Lifshitz-Gilbert (LLG) equation of Langevin-based micromagnetics. Subsequent results such as the time-quantification factor for the Metropolis MC method can be rigorously derived from this mapping equivalence. The validity of the mapping is shown by the close numerical convergence between the MC method and the LLG equation for the case of a single magnetic particle as well as interacting arrays of particles. We also find that our Metropolis MC method is accurate for a large range of damping factors alpha, unlike previous time-quantified MC methods which break down at low alpha, where precessional motion dominates.

Analytical Solution to Transport in Brownian Ratchets via the Gambler’s Ruin Model

“…In this paper, we show an intimate relationship between this classic random walk problem and the thermally activated dynamics in arbitrary potentials. The linkage between these two disparate topics is made possible by recent advances in the time quantification (TQ) of Monte Carlo (MC) [2,3]. In particular, the evolutionary techniques for the Gambler's Ruin problem can be utilized to analyze the transition probabilities and the mean first passage time (MFPT) of the complex stochastic transport in Brownian ratchets.…”

“…Drawing from recent advances in quantifying the MCS [2,3] for the analytical expression of the ratchet current obtained through the MC approach.…”

Anisotropic Landau-Lifshitz-Gilbert models of dissipation in qubits

“…5. The existence of such a relationship between overdamped spin dynamics and the Metropolis-Hastings algorithm has been previously established [38][39][40].…”

“…In this manner the effect is akin to the trajectory ensemble approach [37] with a transition to dynamics confined to φ ≈ φ * . Due to the known relationship between these dynamics [38][39][40] this Langevin equation can also be related to a Monte Carlo O(2) model on appropriate timescales.…”

“…(16). However, Metropolis-Hastings dynamics describes a dissipative relaxation to thermal equilibrium and has been used previously to model dynamics described by a Langevin equation [38][39][40]45]. The ad hoc model of Refs.…”