Does the third law of thermodynamics hold in the quantum regime? (Q2433952)

In recent years, the realization that quantum effects can play an important role in thermodynamic theory has led to an intense interest in quantum and mesoscopic thermodynamics, to the extent that some authors have questioned whether the fundamental laws of thermodynamics remain valid. In the paper is devoted to focus in depth on questions which have been raised in relation to Nernst's third law of thermodyanmics, which states that the entropy \(S\) vanishes as the temperature \(T\to 0\). In [\textit{G. W. Ford} and \textit{R. F. O'Conelle}, Physica E, 29, 82--86 (2005)] this question for the one-dimensional problem of a quantum oscillator in an arbitrary heat bath has been considered. In Sec. 2, where various approaches to the calculation of \(S\) are discussed, by carrying out a detailed calculation in which it is demonstrated explicitly how use of approaches based on an incorrect utilization of the Wigner function formalism, and also the von Neumann formula, leads to an incorrect form of a well-known result. The authors used a method based on the results of [\textit{G. W. Ford, J. T. Lewis} and \textit{R. F. O'Connell}, Phys. Rev. Lett. 55, 2273 (1985)]. In Sec. 3, a review of [\textit{X. L. Li, G. W. Ford} and \textit{R. F. O'Connell}, Phys. Rev. A 41, 5287 (1990), Phys. Rev. A 42, 4519 (1990)] results is given and the results in the three-dimensional case are explicitly written down. Note that all of these results involve just a single integral and, in Sec. 4, this integral for the case of an Ohmic bath and low temperature is evaluated. This enables to confirm Nernst's law. Conclusions are given in Sec. 5.