Path-wise versus kinetic modeling for equilibrating non-Langevin jump-type processes

TL;DR

This paper compares direct numerical integration and path-wise stochastic simulation methods for solving master equations describing Levy-stable jump processes, demonstrating their compatibility and analyzing approximation effects.

Contribution

It provides a consistency check between explicit master equation solutions and path-wise Gillespie-based simulations for Levy jump processes.

Findings

Both methods yield compatible solutions for the master equation.

Analysis of cutoff effects on the solution accuracy.

Demonstration of the methods' consistency in modeling Levy jumps.

Abstract

We discuss two independent methods of solution of a master equation whose biased jump transition rates account for long jumps of L\'{e}vy-stable type and nonetheless admit a Boltzmannian (thermal) equilibrium to arise in the large time asymptotics of a probability density function $\rho (x,t)$. Our main goal is to demonstrate a compatibility of a {\it direct} solution method (an explicit, albeit numerically assisted, integration of the master equation) with an {\it indirect} path-wise procedure, recently proposed in [Physica {\bf A 392}, 3485, (2013)] as a valid tool for a dynamical analysis of non-Langevin jump-type processes. The path-wise method heavily relies on an accumulation of large sample path data, that are generated by means of a properly tailored Gillespie's algorithm. Their statistical analysis in turn allows to infer the dynamics of $\rho (x,t)$. However, no consistency check has been completed so far to demonstrate that both methods are fully compatible and indeed provide a solution of the same dynamical problem. Presently we remove this gap, with a focus on potential deficiencies (various cutoffs, including those upon the jump size) of approximations involved in solution protocols.