Studying nonlinear effects on the early stage of phase ordering using a decomposition method

Abstract

Nonlinear effects on the early stage of phase ordering are studied using Adomian's decomposition method for the Ginzburg-Landau equation for a nonconserved order parameter. While the long-time regime and the linear behavior at short times of the theory are well understood, the onset of nonlinearities at short times and the breaking of the linear theory at different length scales are less understood. In the Adomians decomposition method, the solution is systematically calculated in the form of a polynomial expansion for the order parameter, with a time dependence given as a series expansion. The method is very accurate for short times, which allows to incorporate the short-time dynamics of the nonlinear terms in a analytical and controllable way.

Noise and ultraviolet divergences in the dynamics of the chiral condensate in QCD

“…As already mentioned analytic solutions of such nonlinear equations are attainable only in very special circumstances, like when nonlinear terms in H[σ] are dropped. Such linear approximations are usually valid in a restricted time interval only [11] and extensive numerical simulations, usually by discretizing the equations on a space lattice, are necessary for obtaining solutions over a large time interval. In the next Section, we will obtain explicit numerical solutions of Eq.…”

“…Analytic solutions of GLL equations are attainable only in very special circumstances, like when nonlinear couplings are neglected. Linearization of the equations are usually valid in a restricted time interval only [11] and extensive numerical simulations, usually by discretizing the equations on a space lattice, are necessary for obtaining equilibrium solutions. One difficulty is related to the well-known Rayleigh-Jeans ultraviolet catastrophe of classical field theory [12], which manifest themselves via severe lattice-spacing dependence of the solutions of the GLL equations.…”

Noise and Ultraviolet Divergences in Simulations of Ginzburg–landau–langevin Type of Equations

“…7 Extensive numerical simulations, usually by discretizing the equations on a spatial lattice, are necessary for obtaining solutions over a large time interval. However, a known feature of numerical solutions of GLL equations, that apparently has been overlooked in recent literature, is that due to the noise sources numerical solutions are plagued by severe lattice-spacing dependence.…”