The Order of Phase Transitions in Barrier Crossing

TL;DR

This paper investigates how the shape of polynomial potential barriers influences the order of phase transitions in noise-induced activation, providing criteria for first- and second-order transitions in spatially extended systems.

Contribution

It establishes conditions determining whether phase transitions are first or second order based on potential shape, especially near the barrier top, for systems with polynomial potentials.

Findings

Quartic potentials only exhibit second-order transitions.

Derived necessary and sufficient conditions for transition order.

Transition order is highly sensitive to potential behavior near the barrier top.

Abstract

A spatially extended classical system with metastable states subject to weak spatiotemporal noise can exhibit a transition in its activation behavior when one or more external parameters are varied. Depending on the potential, the transition can be first or second-order, but there exists no systematic theory of the relation between the order of the transition and the shape of the potential barrier. In this paper, we address that question in detail for a general class of systems whose order parameter is describable by a classical field that can vary both in space and time, and whose zero-noise dynamics are governed by a smooth polynomial potential. We show that a quartic potential barrier can only have second-order transitions, confirming an earlier conjecture [1]. We then derive, through a combination of analytical and numerical arguments, both necessary conditions and sufficient conditions to have a first-order vs. a second-order transition in noise-induced activation behavior, for a large class of systems with smooth polynomial potentials of arbitrary order. We find in particular that the order of the transition is especially sensitive to the potential behavior near the top of the barrier.