Dimension reduction of noisy interacting systems

TL;DR

This paper introduces a systematic method for reducing the complexity of noisy interacting systems, accurately capturing phase transitions and dynamic responses, thus aiding the analysis of critical phenomena without prior knowledge of order parameters.

Contribution

The authors develop a cumulant-based dimension reduction technique that accurately reproduces stationary and dynamic properties of high-dimensional stochastic systems, including phase transitions.

Findings

Accurately reproduces the stationary phase diagram.

Captures the correct response to external perturbations.

Identifies critical signatures via susceptibility divergence.

Abstract

We consider a class of models describing an ensemble of identical interacting agents subject to multiplicative noise. In the thermodynamic limit, these systems exhibit continuous and discontinuous phase transitions in a, generally, nonequilibrium setting. We provide a systematic dimension reduction methodology for constructing low dimensional, reduced-order dynamics based on the cumulants of the probability distribution of the infinite system. We show that the low dimensional dynamics returns the correct diagnostic properties since it produces a quantitatively accurate representation of the stationary phase diagram of the system that we compare with exact analytical results and numerical simulations. Moreover, we prove that the reduced order dynamics yields also the prognostic, i.e., time dependent properties, as it provides the correct response of the system to external perturbations. On the one hand, this validates the use of our complexity reduction methodology since it retains information not only of the invariant measure of the system but also of the transition probabilities and time dependent correlation properties of the stochastic dynamics. On the other hand, the breakdown of linear response properties is a key signature of the occurrence of a phase transition. We show that the reduced response operators capture the correct diverging resonant behaviour by quantitatively assessing the singular nature of the susceptibility of the system and the appearance of a pole for real value of frequencies. Hence, this methodology can be interpreted as a low dimensional, reduced order approach to the investigation and detection of critical phenomena in high dimensional interacting systems in settings where order parameters are not known.