Quantum Brownian motion, entropy and the third law of thermodynamics

While some physical disciplines such as classical mechanics and electrodynamics underwent profound changes with the birth of quantum mechanics and relativity theory the Laws of thermodynamics and statistical mechanics per se proved impressively robust over the last century. The main reason being that the formulation of statistical mechanics rests on few pillars only, such as ergodic theory, ensemble theory, the scale of interactions, an entropic concept of temperature, and alike. The grandness of Thermodynamics is that these concepts hold independent of the details of the corresponding total system dynamics. Nevertheless, the field of statistical mechanics entails some subtle issues when going from a closed description of all degrees of freedom, including those of large environments, to a reduced description of an open systems where bath degrees of freedom are traced over. The latter situation is the one typically discussed in the majority of textbooks. But even here, pitfalls can arise in the quantum case already, as recently elucidated in Ref. [1]. Here we study the quantum thermodynamic behavior of small systems in presence of finite quantum dissipation [1]. We consider the archetype cases of a quantum damped harmonic oscillator and a quantum free quantum Brownian particle [2]. A main first finding is that quantum dissipation helps to ensure the validity of the Third Law. For the quantum oscillator, finite damping replaces the zero-coupling result of an exponential suppression of the specific heat at low temperatures by a power-law behavior. Rather intriguing is, however, the behavior of the free quantum Brownian particle. In this case, quantum dissipation is able to restore the Third Law, which knowingly is violated classically: Instead of being constant down to zero temperature, the specific heat now vanishes proportional to temperature with an amplitude that is inversely proportional to the ohmic dissipation strength. A distinct subtlety of finite quantum dissipation is the result that the various thermodynamic functions, – such as the entropy –, of the sub-system do not only depend on the dissipation strength but these depend as well on the prescription employed in their definition. Some subtleties relating to the Second Law and the measure of information in dissipative nanosystems are elucidated [3].