Negative specific heat from the partition function of a free damped particle

The definition of the specific heat for a damped quantum system is a subtle issue. A possible approach is based on an effective partition function defined as the ratio of the partition functions of system plus bath and of the bath alone. For the free damped particle it has been shown, however, that the ensuing specific heat may become negative for appropriately chosen environments. We argue that this quantity should be interpreted as the change of the specific heat of the heat bath when the system degrees of freedom are coupled to it. While this difference may become negative, the involved specific heats themselves are always positive and therefore thermodynamic stability is always guaranteed. For a damped harmonic oscillator, instead of negative values, under appropriate conditions one can observe a dip in the difference of specific heats as a function of temperature. Minimal models containing a single oscillator heat bath are employed to elucidate the temperature dependence of the difference of specific heats. Finally, we comment on the consequences for the interpretation of the density of states based on the effective partition function.