Thermal energies of classical and quantum damped oscillators coupled to reservoirs

TL;DR

This paper analyzes the thermodynamic properties of classical and quantum damped oscillators coupled to reservoirs, revealing differences in energy behavior between classical and quantum cases and providing insights into open system thermodynamics.

Contribution

It offers exact calculations of thermodynamic quantities for damped oscillators with both Ohmic and non-Ohmic damping, highlighting quantum energy divergences and classical energy invariance.

Findings

Classical mean force energy equals that of a free oscillator regardless of damping strength.

Quantum mean force energy depends on damping details and diverges for Ohmic damping.

Provides insights into steady-state thermodynamics of strongly coupled open systems.

Abstract

We consider the global thermal state of classical and quantum harmonic oscillators that interact with a reservoir. Ohmic damping of the oscillator can be exactly treated with a 1D scalar field reservoir, whereas general non-Ohmic damping is conveniently treated with a continuum reservoir of harmonic oscillators. Using the diagonalized Hamiltonian of the total system, we calculate a number of thermodynamic quantities for the damped oscillator: the mean force internal energy, mean force free energy, and another internal energy based on the free-oscillator Hamiltonian. The classical mean force energy is equal to that of a free oscillator, for both Ohmic and non-Ohmic damping and no matter how strong the coupling to the reservoir. In contrast, the quantum mean force energy depends on the details of the damping and diverges for strictly Ohmic damping. These results give additional insight into the steady-state thermodynamics of open systems with arbitrarily strong coupling to a reservoir, complementing results for energies derived within dynamical approaches (e.g. master equations) in the weak-coupling regime.