Spectral Energy Transfers in Domain Growth Problems

TL;DR

This paper analyzes spectral energy transfer mechanisms in domain growth models, specifically comparing conserved and non-conserved dynamics in the Cahn-Hilliard and TDGL equations, revealing distinct behaviors of Fourier modes during phase separation.

Contribution

It provides a detailed spectral analysis of energy transfer in domain growth models, highlighting differences between conserved and non-conserved dynamics.

Findings

TDGL leads to a uniform state with a dominant zero Fourier mode.

CH equation conserves the zero Fourier mode, affecting energy transfer dynamics.

Different spectral behaviors are observed in conserved versus non-conserved models.

Abstract

In the domain growth process, small structures gradually vanish, leaving behind larger ones. We investigate spectral energy transfers in two standard models for domain growth: (a) the {\it Cahn-Hilliard} (CH) equation with conserved dynamics, and (b) the {\it time-dependent Ginzburg-Landau} (TDGL) equation with non-conserved dynamics. The nonlinear terms in these equations dissipate fluctuations and facilitate energy transfers among Fourier modes. In the TDGL equation, only the $\phi(\mathbf{k} = 0, t)$ mode survives, and the order parameter $\phi(\mathbf{r},t)$ approaches a uniform state with $\phi = +1$ or $-1$. On the other hand, there is no dynamics of the $\phi(\mathbf{k} = 0, t)$ mode in the CH equation due to the conservation law, highlighting the different dynamics of these equations.