Checking the validity of truncating the cumulant hierarchy description of a small system

TL;DR

This paper investigates the accuracy of truncating the cumulant hierarchy in small coupled particle systems, revealing discrepancies with full Langevin simulations and highlighting limitations of the approach for finite systems.

Contribution

It provides a detailed numerical analysis showing that hierarchy truncation is unreliable for small systems, contrasting with its use in infinite or large systems.

Findings

Truncated hierarchy results differ significantly from Langevin simulations for small systems.

Hierarchy truncation introduces dependence on initial conditions and spurious states.

Discrepancies are prominent across large parameter regions.

Abstract

We analyze the behavior of the first few cumulant in an array with a small number of coupled identical particles. Desai and Zwanzig (J. Stat. Phys., {\bf 19}, 1 (1978), p. 1) studied noisy arrays of nonlinear units with global coupling and derived an infinite hierarchy of differential equations for the cumulant moments. They focused on the behavior of infinite size systems using a strategy based on truncating the hierarchy. In this work we explore the reliability of such an approach to describe systems with a small number of elements. We carry out an extensive numerical analysis of the truncated hierarchy as well as numerical simulations of the full set of Langevin equations governing the dynamics. We find that the results provided by the truncated hierarchy for finite systems are at variance with those of the Langevin simulations for large regions of parameter space. The truncation of the hierarchy leads to a dependence on initial conditions and to the coexistence of states which are not consistent with the theoretical expectations based on the multidimensional linear Fokker-Planck equation for finite arrays.