Nonergodic Brownian oscillator

TL;DR

This paper investigates how a classical harmonic oscillator coupled to a non-Markovian bath can exhibit both ergodic and nonergodic behaviors, with transitions resembling second-order phase transitions.

Contribution

It introduces the concept of ergodic and nonergodic configurations in a non-Markovian Brownian oscillator and analyzes the conditions and transitions between these states.

Findings

Oscillator can have ergodic or nonergodic states depending on frequency.

Explicit relaxation functions are derived for specific dissipation kernels.

Transitions between ergodic and nonergodic states resemble second-order phase transitions.

Abstract

We consider an open (Brownian) classical harmonic oscillator in contact with a non-Markovian thermal bath and described by the generalized Langevin equation. When the bath's spectrum has a finite upper cutoff frequency, the oscillator may have ergodic and nonergodic configurations. In ergodic configurations (when exist, they correspond to lower oscillator frequencies) the oscillator demonstrates conventional relaxation to thermal equilibrium with the bath. In nonergodic configurations (which correspond to higher oscillator frequencies) the oscillator in general does not thermalize, but relaxes to periodically correlated (cyclostationary) states whose statistics vary periodically in time. For a specific dissipation kernel in the Langevin equation, we evaluate explicitly relevant relaxation functions, which describe the evolution of mean values and time correlations. When the oscillator frequency is switched from a lower value to higher one, the oscillator may show parametric ergodic to nonergodic transitions with equilibrium initial and cyclostationary final states. These transitions are shown to resemble phase transitions of the second kind.