Responses to applied forces and the Jarzynski equality in classical oscillator systems coupled to finite baths: An exactly solvable non-dissipative non-ergodic model

Responses of small open oscillator systems to applied external forces have been studied with the use of an exactly solvable classical Caldeira-Leggett (CL) model in which a harmonic oscillator (system) is coupled to finite $N$-body oscillators (bath) with an identical frequency ($\omega_n=\omega_o$ for $n=1$ to $N$). We have derived exact expressions for positions, momenta and energy of the system in nonequilibrium states and for work performed by applied forces. Detailed study has been made on an analytical method for canonical averages of physical quantities over the initial equilibrium state, which is much superior than numerical averages commonly adopted in simulations of small systems. The calculated energy of the system which is strongly coupled to finite bath is fluctuating but non-dissipative. It has been shown that the Jarzynski equality (JE) is valid in non-dissipative, non-ergodic open oscillator systems regardless of the rate of applied ramp force.