Critical dynamics of relativistic diffusion

TL;DR

This paper investigates the critical dynamics of relativistic scalar fields near phase transitions, combining theoretical calculations and lattice simulations to analyze spectral functions, dispersion relations, and universal scaling behaviors.

Contribution

It introduces a comprehensive study of relativistic diffusion near critical points, deriving universal scaling functions and analyzing dynamic critical exponents across different universality classes.

Findings

Spectral functions are well-described by Breit-Wigner shapes.

At criticality, spectral functions exhibit non-trivial power-law dispersion.

The dynamic critical exponent z is verified through scaling analysis.

Abstract

We study the dynamics of self-interacting scalar fields with $Z_2$ symmetry governed by a relativistic Israel-Stuart type diffusion equation in the vicinity of a critical point. We calculate spectral functions of the order parameter in mean-field approximation as well as using first-principles classical-statistical lattice simulations in real-time. We observe that the spectral functions are well-described by single Breit-Wigner shapes. Away from criticality, the dispersion matches the expectations from the mean-field approach. At the critical point, the spectral functions largely keep their Breit-Wigner shape, albeit with non-trivial power-law dispersion relations. We extract the characteristic time-scales as well as the dynamic critical exponent $z$, verifying the existence of a dynamic scaling regime. In addition, we derive the universal scaling functions implied by the Breit-Wigner shape with critical power-law dispersion and show that they match the data. Considering equations of motion for a system coupled to a heat bath as well as an isolated system, we perform this study for two different dynamic universality classes, both in two and three spatial dimensions.