Controlling One-Dimensional Langevin Dynamics on the Lattice

Stochastic evolutions of classical field theories have recently become popular in the study of problems such as determination of the rates of topological transitions and the statistical mechanics of nonlinear coherent structures. To obtain high precision results from numerical calculations, a careful accounting of spacetime discreteness effects is essential, as well as the development of schemes to systematically improve convergence to the continuum. With a kink-bearing ${\varphi }^4$ field theory as the application arena, we present such an analysis for a 1+1-dimensional Langevin system. Analytical predictions and results from high resolution numerical solutions are found to be in excellent agreement.