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arXiv:cond-mat/0304066 · Physical Review Letters

Validity of the Second Law in Nonextensive Quantum Thermodynamics

Sumiyoshi Abe, A. K. Rajagopal

Published Sep 17, 2003• 86 citations
arXivHTMLPDF

Abstract

The second law of thermodynamics in nonextensive statistical mechanics is discussed in the quantum regime. Making use of the convexity property of the generalized relative entropy associated with the Tsallis entropy indexed by q, Clausius' inequality is shown to hold in the range q in (0, 2]. This restriction on the range of the entropic index, q, is purely quantum mechanical and there exists no upper bound of q for validity of the second law in classical theory.

DOI
10.1103/physrevlett.91.120601

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The second law of thermodynamics in nonextensive statistical mechanics is discussed in the quantum regime. Making use of the convexity property of the generalized relative entropy associated with the Tsallis entropy indexed by q, Clausius' inequality is shown to hold in the range q in (0, 2]. This restriction on the range of the entropic index, q, is purely quantum mechanical and there exists no upper bound of q for validity of the second law in classical theory.
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Gibbs-Boltzmann entropy leads to systems that have a strong dependence on initial conditions. In reality, most materials behave quite independently of initial conditions. Nonextensive entropy or Tsallis entropy leads to nonextensive statistical mechanics. In this paper, we calculate the Tsallis form of Clausius inequality and then determine the upper bound for extracting work from the small system in the nonextensive statistical mechanics with mutual information. In the following, we extract mutual information and adjust Maxwell's demon with quantum feedback control.
We formulate a complete theory of quantum thermodynamics in the Rényi entropic formalism exploiting the Rényi relative entropies, starting from the maximum entropy principle. In establishing the first and second laws of quantum thermodynamics, we have correctly identified accessible work and heat exchange in both equilibrium and nonequilibrium cases. The free energy (internal energy minus temperature times entropy) remains unaltered, when all the entities entering this relation are suitably defined. Exploiting Rényi relative entropies we have shown that this "form invariance" holds even beyond equilibrium and has profound operational significance in isothermal process. These results reduce to the Gibbs-von Neumann results when the Rényi entropic parameter α approaches 1. Moreover, it is shown that the universality of the Carnot statement of the second law is the consequence of the form invariance of the free energy, which is in turn the consequence of maximum entropy principle. Further, the Clausius inequality, which is the precursor to the Carnot statement, is also shown to hold based on the data processing inequalities for the traditional and sandwiched Rényi relative entropies. Thus, we find that the thermodynamics of nonequilibrium state and its deviation from equilibrium together determine the thermodynamic laws. This is another important manifestation of the concepts of information theory in thermodynamics when they are extended to the quantum realm. Our work is a substantial step towards formulating a complete theory of quantum thermodynamics and corresponding resource theory.
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 terms of general quantum entropies for both quantum thermostatistics and nonextensive quantum thermostatistics. This further reveals new features beyond the second thermodynamics law by maximization the cross entropy during irreversible measurement procedures. The present result is useful for asymptotical tasks of quantum entropy estimations and universal quantum source encoding without the state tomography. It is further applied to single-shot state transitions and cooling in quantum thermodynamics with limited information. These results should be interesting in the many-body theory and long-range quantum information processing.