A fundamental question is: What remains of thermodynamics if one goes to the extreme limit of small quantum systems with a few degrees of freedom? If it does survaive, are the many formulations of the second law (entropy of a closed system cannot decrease, heat goes from high to low temperatures, the optimal changes are adiabatic ones,..) still equivalent, or is there a "universal" formulation?

On this subject our group found many fundamental results, partly discovering and defining the subject itself.

It has been realized that there is one formulation which holds in general: Thomson's formulation (making cycles costs work) for a system starting in equilibrium. It applies to systems without bath, or systems coupled to a bath. The statement about Thomson's formulation was known before; we reproduced it to draw attention to it and eleborated on it [P58], [P62].

The first law can still be defined, provided the workscource and the bath are macroscopic [L38], [P51], [C29], [C27].

Several formulations of the second law appear to be violated. The Clauisius inequality dQ less or equal to T dS
was shown to be invalid at T=0. The physical reason is the formation of a cloud of bath modes around the central particle. Such clouds are well known in the Kondo problem and for polarons. The energy of that cloud must be attributed to the bath, and some of it can be taken, because changing the parameters of the particle changes the cloud [L38], [P51].
A test for the violation of the Clausius inequality was proposed for nanoscale electric circuits [P59] and in quantum optics [L42], [C27].

The Thomson formulation can be violated when the system is still coupled to a single heat bath, but starts out of equilibrium. Setups for such cycles were derived analytically [P51].

The rate of energy dissipation can be negative, even when starting in equilibrium [P51]. Classically this would be forbidden, because there it is, after dividing by temperature, equal to the rate of entropy production. Positivity of the latter is another formulation of the second law.

The Landauer bound for information erasure, sometimes said to be another formulation of the second law, can be violated in the quantum regime [P52].

Also in quantum optics the manipulation of the surrounding cloud can lead to surprising effects [L42], such as bath assisted work extraction and bath assisted cooling.
ITFA-2005-31: A.E. Allahverdyan and Th.M. Nieuwenhuizen,
In 1875 the founding father of statistical physics Josiah Willard Gibbs pointed at the following paradox: Take two equal volumina of different gases and mix them. Then the entropy increases by and amount k log 2 per particle. But if the gases are equal, there is not such an increase. The paradox lies in the discontinuity: there is an increase no matter how small the difference between the gases, but not when they are equal. This raises questions such as: if the gases are composed of similar balls, red ones for the first gas, blue ones for the second, then what should a color-blind experimentator conclude? In other words: the mixing entropy is not operational.
There has been a long effort to resolve the paradox, which shows a limit of phenemenological thermodynamics. It was believed to be solved by the quantum mixing entropy argument, but that was shown to create a new problem at almost the same spot.
Assuming that the translational degrees of freedom of both gases are in thermal equilibrium at the same temperature, we express the differences between the gases by their internal (spin) structure. The latter involve a few degrees of freedom. Therefore we approach the problem via quantum thermodynamics, the theory of thermodynamics for small quantum systems connected to a macroscopic bath and a macroscopic work source. In this field we notioced before that the notion of entropy is messy, the physical quantity is work. The maximal amount of work that can generally be extracted from a finite quantum system was already derived in a paper with R. Balian: the so-called ergotropy. This allows to consider the maximal amount of work that can be derived before and after mixing. The difference is the mixing work or mixing ergotropy. Unlike the mixing entropy, the mixing work is continuous when the gasses become more and more equal. And the extractable work is an operational concept, it depends on the work extraction process employed.
This solves the Gibbs paradox using quantum mechanics alone.

ITFA-2005-31: A.E. Allahverdyan and Th.M. Nieuwenhuizen,
Resolution of the Gibbs paradox via quantum thermodynamics, Phys. Rev. E, submitted, quant-ph/0507145 .

Papers

[P70] A.E. Allahverdyan and Th.M. N., Resolution of the Gibbs paradox via quantum thermodynamics, Phys. Rev. E, submitted. quant-ph/0507145

[P62] A.E. Allahverdyan, R. Balian and Th. M. N., Thomson's formulation of the second law for macroscopic and finite work sources Entropy 6, (2004) 30-37 Entropy 6, (2004) 30-37