Generalized symmetric nonextensive thermostatistics and q-modified structures

We formulate a convenient generalization of the q-expectation value, based on the analogy of the symmetric quantum groups and q-calculus, and show that the q ↔ q−1 symmetric nonextensive entropy preserves all of the mathematical structure of thermodynamics just as in the case of non-symmetric Tsallis statistics. Basic properties and analogies with quantum groups are discussed. PACS: 05.20.-y; 05.70.-a; 05.30.-d In the last few years there has been much interest in nonextensive classical and quantum physics. The nonextensive statistical mechanics proposed by Tsallis [1,2], has been the source of inspiration for many investigations in systems which represent multifractal properties, long-range interactions and/or long-range memory effects [3]. On the other hand, quantum groups and the derived q-deformed algebraic structure such as q-oscillators, based on the deformation of standard oscillator commutation-anticommutation relations, have created considerable interest in mathematical physics and in several applications [4]. The investigations described above are only two apparently unrelated areas in nonextensive physics. Although a complete understanding of the connection between nonextensive statistics and q-deformed structure is still lacking, many papers are devoted to the study of a deep connection between these two non-extensive formalisms [5–9]. Tsallis statistics is a q-nonsymmetric formalism i.e., not invariant under q ↔ q. Recently Abe [6] has employed a connection between Tsallis entropy and the non-symmetric Jackson derivative. Because the requirement of invariance under q ↔ q is very important in quantum groups [10], the above connection allows him to extend the Tsallis entropy to the q-symmetric one by means of a symmetric Jackson derivative. However, Abe has not extended the definition of the expectation value of an observable to the symmetric case and thus unable to formulate the thermostatistics which will preserve the Legendre transformation of standard thermodynamics in contrast to the Tsallis statistics which does. We would like to point out that Ref. [7] introduces a two-parameter modification for the entropy and for the expectation value of an observable but does not also produce a consistent formulation of thermostatistics. The purpose of this letter is to show how the q ↔ q symmetric generalization of the Tsallis entropy together with a natural generalization of the q-expectation value produces a 1 thermostatistics that preserves the mathematical structure of standard thermodynamics and show that this property is a direct consequence of the generalization from the non-symmetric q-calculus to the symmetric one. Before investigating the symmetric nonextensive thermostatistics, let us briefly review the fundamental properties of the Tsallis thermostatistics, which is based upon the following two postulates [1,2]. • A nonextensive generalization of the Boltzmann-Gibbs entropy (Boltzmann constant is set equal to unity) Sq = 1 q − 1 ( 1− W ∑ i=1 pqi ) , with W ∑