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 GinzburgÀLandau (GL) type of equations, also known as time-dependent nonlinear Schr€ odinger equations. Environmental e®ects of random nature are usually taken into account by noise sources, turning the GL equations into stochastic equations. Noise sources give rise to latticespacing 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¯eld theory and stochastic quantization.

Finite element discretization of non-linear diffusion equations with thermal fluctuations

“…In this case, the SPDE is the fundamental equation and one needs to modify the action (free energy) with counterterms that depend on the lattice spacing in order for the final theory to have a proper continuum limit [11][12][13][14][15] . On the other hand, when a "more fundamental" theory exists, as is the case in colloidal suspension where the fundamental theory are Hamilton's equations, the requirement of having a continuum limit is desirable but not essential.…”

“…The matrix D(c) is manifestly symmetric and positive semi-definite because Γ > 0 (the semi character is due to (15)). The total number of particles, defined as N = µ V µ c µ (t) is a dynamical invariant of the equation (25).…”

“…In the high resolution limit (M → ∞) the function (39) converges weakly to (36). Note that due to (15), the relevant variables (39) satisfy…”

“…(4) has a proper continuum limit in 1D but it is divergent in D > 1 due to the so called ultraviolet catastrophe. In this latter case, renormalization group techniques have been used in order to recover a continuum limit [11][12][13][14][15] . A rigorous mathematical analysis of the renormalization of SPDEs near the critical point has been conducted recently 16,17 .…”

Dressed perturbation theory: a perturbative approach to Dyson-Schwinger and Bethe-Salpeter equations in QCD

“…The general idea behind DPT is not new. It shares many similarities with the method of the δ -expansion, also known as optimised perturbation theory, used with self-interacting scalar field theories [12], relativistic nuclear mean-field theories [13], and dynamical phase transitions in the context of Ginzburg-Landau-Langevin equations [14]. It also shares similarities with screened perturbation theory [15], that has been used in finite temperature field theory.…”