Two-Parameter Entropies in Extended Parastatistics of Nonextensive Systems
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ELEMENTARY PARTICLE PHYSICS AND FIELD THEORY TWO-PARAMETER ENTROPIES IN EXTENDED PARASTATISTICS OF NONEXTENSIVE SYSTEMS R. G. Zaripov
UDC 536.75
General expressions are given for entropies from which one-parameter and two-parameter entropies for quantum nonextensive systems follow in extended parastatistics. Keywords: nonextensivity, extended parastatistics, entropy.
INTRODUCTION The range of studies of nonextensive statistical mechanics continues to grow. In the first place, attention is being directed to the statistical principles of nonextensive systems and numerous results of an applied nature [1–4]. Of fundamental importance here are parametric entropies both in classical theory and in quantum theory. However, questions of the parastatistics of nonextensive systems based on the Bose method [5] have not been addressed. For this reason, quantum measures of the entropy and the information difference were obtained and spontaneous and induced transitions in processes of self-decay and self-organization of systems were investigated in [6–9]. The group of entropies and its representations along with other questions were examined. An extension of traditional parastatistics was presented in [10], in which the number of particles in the ith state is found in an arbitrary range of variations. The aim of the present work is to examine in detail the properties of entropies in such extended parastatistics and to introduce families of new two-parameter quantum entropies of nonextensive systems.
1. SEMINORM AND ENTROPY In line with the method of Bose quantum states [5], let us consider the set of particles N1, ..., Nm with states
G1 , ..., Gm , where
m is the number of states. In extended parastatistics the system is described by the statistics of
states Gij with i 1, ..., m and the number of particles j s, ..., r in the ith state is bounded from below by the number s and above by the number r . For s 0 traditional parastatistics is obtained [10], in which r 1 and r correspond to Fermi–Dirac statistics and Bose–Einstein statistics. In the Bose method, for traditional parastatistics the equalities r
r
j
j
Gi Gij , Ni jGij ,
(1)
Institute of Mechanics and Engineering of the Federal State Budgetary Institution of Science “Kazan Scientific Center of the Russian Academy of Sciences”, Kazan, Russia, e-mail: [email protected]. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 3–9, July, 2020. Original article submitted September 2, 2019. 1064-8887/20/6307-1103 2020 Springer Science+Business Media, LLC
1103
Gij
pij
Gi
r
, pij 1
(2)
j
with normalized distribution pij are valid, and the mean number of particles in the ith state is written r
Ni Gi
ni
jGij j r
.
(3)
Gij j
From the definition of the statistical weight, for the statistics of the states Gij the additive logarithmic measure for the entropy with bounded number j m r
Gij
i
Gi
S Á k Gij ln j
(4)
follows for the Bose entropy [5]. For a closed system in the equilibrium st
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