# Behavior of collective variables in complex nonlinear stochastic models of finite size

## TL;DR

This paper investigates how collective variables behave in finite-size complex systems with nonlinear stochastic dynamics, emphasizing feedback effects and ergodicity issues through numerical analysis of mean-field and local coupling models.

## Contribution

It introduces a numerical study of collective variable behavior in finite nonlinear stochastic systems with feedback mechanisms, highlighting ergodicity challenges.

## Key findings

- Feedback mechanisms influence collective variable dynamics.
- Finite size effects impact ergodicity and system behavior.
- Numerical analysis of mean-field and local coupling models.

## Abstract

We consider the behavior of a collective variable in a complex system formed by a finite number of interacting subunits. Each of them is characterized by a degree of freedom with an intrinsic nonlinear bistable stochastic dynamics. The lack of ergodicity of the collective variable requires the consideration of a feedback mechanism of the collective behavior on the individual dynamics. We explore numerically this issue within the context of two simple finite models with a feedback mechanism of the Weiss mean-field type: a global coupling model and another one with nearest neighbors coupling.