Statistical Uncertainties of Limit Cycle Systems in Langevin Bath

TL;DR

This paper investigates the inherent uncertainties in limit cycle systems within Langevin baths, revealing bounds on observable fluctuations and how these uncertainties evolve over time.

Contribution

It introduces a framework for understanding uncertainty bounds in stochastic limit cycle systems and explores their dependence on system parameters and time evolution.

Findings

Uncertainty bounds depend on system shape and periodicity.

Uncertainties increase over time when coupled to a bath.

Uncertainties are absent in deterministic limit cycles.

Abstract

We show that limit cycle systems in Langevin bath exhibit uncertainty in observables that define the limit-cycle plane, and maintain a positive lower bound. The uncertainty-bound depends on the parameters that determine the shape and periodicity of the limit cycle. In one dimension, we use the framework of canonical dissipative systems to construct the limit cycle, whereas in two dimensions, particle in central potentials with radial dissipation provide us natural examples. We also investigate how uncertainties, which are absent in deterministic systems, increase with time when the systems are attached to a bath and eventually cross the lower bound before reaching the steady state.