Steady states of harmonic oscillator chains and shortcomings of harmonic heat baths

TL;DR

This paper investigates the steady states of coupled harmonic oscillator systems, revealing how they evolve under Hamiltonian changes and highlighting limitations of harmonic baths in achieving thermal states, especially for complex subsystems.

Contribution

It derives formulas for steady state transformations under Hamiltonian changes and demonstrates the limitations of harmonic heat baths in reaching true thermal equilibrium for complex systems.

Findings

Steady states can be transformed via adiabatic changes in Hamiltonian.

Sudden Hamiltonian changes produce transient steady states in infinite systems.

Harmonic heat baths cannot generally achieve true thermal states except approximately under weak coupling.

Abstract

We study properties of steady states (states with time-independent density operators) of systems of coupled harmonic oscillators. Formulas are derived showing how adiabatic change of the Hamiltonian transforms one steady state into another. It is shown that for infinite systems, sudden change of the Hamiltonian also tends to produce steady states, after a transition period of oscillations. These naturally arising steady states are compared to the maximum-entropy state (the thermal state) and are seen not to coincide in general. The approach to equilibrium of subsystems consisting of n coupled harmonic oscillators has been widely studied, but only in the simple case where n=1. The power of our results is that they can be applied to more complex subsustems, where n>1. It is shown that the use of coupled harmonic oscillators as heat baths models is fraught with some problems that do not appear in the simple n=1 case. Specifically, the thermal states that are though to be achievable through hard-sphere collisions with heat-bath particles can generally NOT be achieved with harmonic coupling to the heat-bath particles, except approximately when the coupling is weak.