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Recently, the quantum counterpart of energy equipartition theorem has drawnconsiderable attention. Motivated by this, we formulate and investigate ananalogous statement for the free energy of a quantum oscillator linearlycoupled to a passive heat bath consisting of an infinite number of independentharmonic oscillators. We explicitly demonstrate that the free energy of theBrownian oscillator can be expressed in the form $F(T) = \langle f(\omega,T)\rangle $ where $f(\omega,T)$ is the free energy of an individual bathoscillator. The overall averaging process involves two distinct averages: thefirst one is over the canonical ensemble for the bath oscillators, whereas thesecond one signifies averaging over the entire bath spectrum of frequenciesfrom zero to infinity. The latter is performed over a relevant probabilitydistribution function $\mathcal{P}(\omega)$ which can be derived from theknowledge of the generalized susceptibility encountered in linear responsetheory. The effect of different dissipation mechanisms is also exhibited. Wefind two remarkable consequences of our results. First, the quantum counterpartof energy equipartition theorem follows naturally from our analysis. The secondcorollary we obtain is a natural derivation of the third law of thermodynamicsfor open quantum systems. Finally, we generalize the formalism to three spatialdimensions in the presence of an external magnetic field.

Partition of free energy for a Brownian quantum oscillator: Effect of dissipation and magnetic field