  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.
