Onset of Singularities in the Pattern of Fluctuational Paths of a Nonequilibrium System

TL;DR

This paper investigates how singularities in the most probable fluctuation paths of nonequilibrium systems emerge as the system is driven away from equilibrium, revealing two distinct scenarios based on the growth rate of the optimal path.

Contribution

It identifies the conditions under which singularities appear in nonequilibrium fluctuation paths and characterizes their scaling behavior and thresholds using a resonantly modulated oscillator model.

Findings

Singularities can emerge without a threshold if the optimal path grows exponentially.

A threshold exists if the growth exceeds exponential, allowing reach to infinity in finite time.

Scaling laws for the location of singularities are derived as functions of control parameters.

Abstract

Fluctuations in systems away from thermal equilibrium have features that have no analog in equilibrium systems. One of such features concerns large rare excursions far from the stable state in the space of dynamical variables. For equilibrium systems, the most probable fluctuational trajectory to a given state is related to the fluctuation-free trajectory back to the stable state by time reversal. This is no longer true for nonequilibrium systems, where the pattern of the most probable trajectories generally displays singularities. Here we study how the singularities emerge as the system is driven away from equilibrium, and whether a driving strength threshold is required for their onset. Using a resonantly modulated oscillator as a model, we identify two distinct scenarios, depending on the speed of the optimal path in thermal equilibrium. If the position away from the stable state along the optimal path grows exponentially in time, the singularities emerge without a threshold. We find the scaling of the location of the singularities as a function of the control parameter. If the growth away from the stable state is faster than exponential, characterized by the ability to reach infinity in finite time, there is a threshold for the onset of singularities, which we study for the model.