NOISE AND ULTRAVIOLET DIVERGENCES IN SIMULATIONS OF GINZBURG–LANDAU–LANGEVIN TYPE OF EQUATIONS

Abstract

The time evolution of an order parameter towards equilibrium can be described by nonlinear Ginzburg–Landau (GL) type of equations, also known as time-dependent nonlinear Schrödinger equations. Environmental effects of random nature are usually taken into account by noise sources, turning the GL equations into stochastic equations. Noise sources give rise to lattice-spacing dependence of the solutions of the stochastic equations. We present a systematic method to renormalize the equations on a spatial lattice to obtain lattice-spacing independent solutions. We illustrate the method in approximation schemes designed to treat nonlinear and nonlocal GL equations that appear in real time thermal field theory and stochastic quantization.