Validity of the second law in nonextensive quantum thermodynamics

A.E. Allahverdyan and Th. M. Nieuwenhuizen [1] in their paper "A mathematical theorem as a basis for the second law: Thomson's formulation applied to equilibriium" present a proof of the second law of thermodynamics based on quantum mechanics. In this comment on their paper I offer a counterexample to their proof.

It is pointed out that the Tsallis entropy functional represented in terms of the escort distribution is not concave of the entropic index $q$ is less than unity. It is emphasized that the escort distribution is a secondary object calculated from the basic original distribution.

We consider a quantum linear oscillator coupled at an arbitrary strength to a bath at an arbitrary temperature. We find an exact closed expression for the oscillator density operator. This state is non-canonical but can be shown to be equivalent to that of an uncoupled linear oscillator at an effective temperature T_{eff} with an effective mass and an effective spring constant. We derive an effective Clausius inequality delta Q_{eff} =< T_{eff} dS, where delta Q_{eff} is the ...

Based on the Tsallis entropy, the nonextensive thermodynamic properties are studied as a q-deformation of classical statistical results using only probabilistic methods and straightforward calculations. It is shown that the constant in the Tsallis entropy depends on the deformation parameter and must be redefined to recover the usual thermodynamic relations. The notions of variance and covariance are generalised. A partial derivative formula of the entropy is established. It ...

The entropy shows an unavoidable tendency of disorder in thermostatistics according to the second thermodynamics law. This provides a minimization entropy principle for quantum thermostatistics with the von Neumann entropy and nonextensive quantum thermostatistics with special Tsallis entropy. Our goal in this work is to provide operational characterizations of general entropy measures. We present the first unified principle consistent with the second thermodynamics law in te...

We present an analysis of the foundations of the well known Clausius inequality. It is shown that, in general, the inequality is not a logical consequence of the Kelvin-Planck formulation of the second law of thermodynamics. Some thought experiments demonstrating the violation of the Clausius inequality are considered. The possibility of experimental detection of the violation is pointed out.

We propose a new form of the Second Law inequality that defines a tight bound for extractable work from the non-equilibrium quantum state. In classical thermodynamics, the optimal work is given by the difference of free energy, what according to the result of Skrzypczyk \emph{et al.} can be generalized for individual quantum systems. The saturation of this bound, however, requires an infinite bath and an ideal energy storage that is able to extract work from coherences. The n...

It is pointed out that the constraint to be imposed to the maximization of the entropy for processes outside the class of thermodynamical systems, is generally not well defined. In fact, any probability distribution can be derived from Jaynes's principle with a suitable choice of the constraint. In the case of Tsallis's non-extensive formalism, this implies that it is not possible to establish any connection between specific non-thermodynamical processes and non-extensive mec...

We proved when random-variable fluctuations obey the central limit theorem the equality of the uncertainty relation corresponds to the thermodynamic equilibrium state. The inequality corresponds to the thermodynamic non-equilibrium state. The uncertainty relation is a quantum-mechanics expression of the second law of thermodynamics originated in wave-particle duality. Formulas of mean square-deviations changes adjusted by random fluctuations under the minimal uncertainty rela...

According to the second law, the efficiency of cyclic heat engines is limited by the Carnot bound that is attained by engines that operate between two thermal baths under the reversibility condition whereby the total entropy does not increase. Quantum engines operating between a thermal and a squeezed-thermal bath have been shown to surpass this bound. Yet, their maximum efficiency cannot be determined by the reversibility condition, which may yield an unachievable efficiency...