Non-ergodic Brownian oscillator

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 non-ergodic configurations.

In ergodic configurations (when  exist, they correspond to lower oscillator frequencies)

the oscillator demonstrates conventional thermalization, i.e. relaxation to thermal equilibrium with the bath.

In non-ergodic configurations (which correspond to higher oscillator frequencies)

the oscillator does not thermalize, but relaxes to a non-equilibrium

stationary state.

For a specific dissipation kernel in the Langevin equation, we evaluate explicitly relevant relaxation functions,

which describe the time evolution and properties of non-equilibrium stationary states.

When the oscillator's frequency is switched from a lower value to higher one,

the oscillator may show parametric transitions from ergodic to non-ergodic configurations.

These transitions are shown to resemble phase transitions of the second kind.