[EPJ sponsored lecture]

Let us elaborate on the notion of thermodynamic entropy S (Clausius 1865) and its consequences. Gibbs put forward two notions of entropy for isolated systems that I commonly will refer to as `volume entropy´ (involving the integrated density of states in modern language) and as the `surface entropy´, being proportional to the density of states, commonly also known (incorrectly) as the Boltzmann entropy. The absolute temperature, T^-1=∂S/∂U, where U is the internal thermodynamic energy, is then related to thermodynamic entropy; -- but which one to use? -- The consistency for thermodynamics, i.e. the validity for the celebrated 0-th, 1-st and 2-nd thermodynamic Law singles out the Gibbs-entropy [1].

I shall address shortcomings that relate to the thermodynamics of small systems when sticking to the (Boltzmann)-surface entropy [2, 3]. Most of all, the uncritical use of Boltzmann entropy for microcanonical systems may formally yield negative values of absolute temperatures. This is not only physically incorrect for the concept of an absolute temperature, but also would violate thermodynamic stability if the system is brought into (weak) contact with an omnipresent sort of environment of radiation source or otherwise with no upper bound in energy.  This criticism applies to the concept of absolute negative (spin) temperature and, as well, to the interpretation of recent experiments with isolated ultra-cold atomic gases [4].

Next, we address canonical entropy when describing quantum systems that interact strongly with an environment. Then, the canonical specific heat can assume negative values away from absolute zero temperature [5, 6]. Likewise, the thermodynamic entropy for a strongly coupled system, assuming a form which mimics a conditional entropy (but *not* quite) can be negative away from absolute $T=0$.

[1] S. Hilbert, P. Hänggi, and J. Dunkel, Thermodynamic Laws in Isolated Systems, arXiv:1408.5382 (2014).

[2] M. Campisi, Microcanonical phase transitions in small systems, arXiv 0709:1082; ibid, On the mechanical foundations of thermodynamics: The generalized Helmholtz theorem
Studies in History and Philosophy of Modern Physics 36, 275 (2005)

[3] J. Dunkel and S. Hilbert, Phase transitions in small systems: microcanonical vs. canonical ensembles, Physica A 370, 390 (2006).

[4] S. Braun, et al.,Negative absolute temperatures for motional degrees of freedom, Science 339, 52 (2013); J. Dunkel and S. Hilbert, Consistent Thermostatics forbids Negative Absolute  Temperatures, Nature Physics, 10: 67-72 (2014).

[5] P. Hänggi, G.L. Ingold, and P. Talkner, Finite quantum dissipation: the challenge of obtaining specific heat, New J. Phys. 10, 115008 (2008); ibid, Specific heat anomalies of open quantum systems, Phys. Rev. E 79, 061105 (2009).

[6] M. Campisi, P. Talkner, and P. Hänggi, Thermodynamics and fluctuation theorems for a strongly coupled open quantum system: an exactly solvable case, J. Phys. A: Math. Theor. (Fast Track) 42, 392002 (2009).

See also the two links:

(1) “(Quantum)-FluctuationTheorems”
http://www.physik.uni-augsburg.de/theo1/hanggi/Fluctuation.html

(2) “What is Temperature”
http://www.physik.uni-augsburg.de/theo1/hanggi/Temperature.html