# Dynamic finite-size scaling at first-order transitions

## TL;DR

This paper develops a dynamic finite-size scaling theory for first-order transitions, predicting behavior near phase coexistence and confirming it through numerical simulations of Ising and Potts models.

## Contribution

It introduces a novel dynamic finite-size scaling framework for first-order transitions and provides exact predictions validated by numerical experiments.

## Key findings

- Exact dynamical scaling functions derived
- Numerical simulations confirm theoretical predictions
- Two-state coarse-grained dynamics effectively describe the system

## Abstract

We investigate the dynamic behavior of finite-size systems close to a first-order transition (FOT). We develop a dynamic finite-size scaling (DFSS) theory for the dynamic behavior in the coexistence region where different phases coexist. It is characterized by an exponentially large time scale related to the tunneling between the two phases. We show that, when considering time scales of the order of the tunneling time, the dynamic behavior can be described by a two-state coarse-grained dynamics. This allows us to obtain exact predictions for the dynamical scaling functions. To test the general DFSS theory at FOTs, we consider the two-dimensional Ising model in the low-temperature phase, where the external magnetic field drives a FOT, and the 20-state Potts model, which undergoes a thermal FOT. Numerical results for a purely relaxational dynamics fully confirm the general theory.