Detailed balance and intermediate statistics

Abstract

We present a theory of particles, obeying intermediate statistics ("anyons"), interpolating between Bosons and Fermions, based on the principle of Detailed Balance. It is demonstrated that the scattering probabilities of identical particles can be expressed in terms of the basic numbers, which arise naturally and logically in this theory. A transcendental equation determining the distribution function of anyons is obtained in terms of the statistics parameter, whose limiting values 0 and 1 correspond to Bosons and Fermions respectively. The distribution function is determined as a power series involving the Boltzmann factor and the statistics parameter and we also express the distribution function as an infinite continued fraction. The last form enables one to develop approximate forms for the distribution function, with the first approximant agreeing with our earlier investigation.

On non-extensive statistics, chaos and fractal strings

“…In particular, the statistics for quasi-particles of continous spin based on the generalized Fibonnaci series has been studied in [ 51 ] ; the statistics for quasi-particles coined f ractons with fractal spin has been developed by da Cruz [ 15 ] ; the so-called detailed-balanced statistics models associated with particles of intermediate statistics between bosons and fermions was presented in [ 49 ]; and the fractal inspired statistics model within the context of Tsallis statistics studied earlier by Oprisal [46 ] . It is warranted to find a unifying picture, if possible, of all these different statistical descriptions associated with quasi-particles of continous spin ( fractal spin ) and with intermediate statistics.…”

Fibonacci oscillators and two-parameter generalized thermostatistics

“…Similar Hamiltonians were also considered by several other authors [9][10][11][12][13][14][15][26][27][28][29][30][31][32][33][34][35][36]. By using the generalized Fibonacci basic integer ½n in (2), the mean value of the ðq 1 ; q 2 Þ-deformed occupation number f k;q 1 ;q 2 can be calculated [29] by…”

“…In this section, on the basis of the recently proposed formalism for the theory of q-deformed bosons [13][14][15], we wish to discuss the ðq 1 ; q 2 Þ-deformed statistical distribution function of the commuting Fibonacci oscillators in the framework of continued fractions. The ðq 1 ; q 2 Þ-deformed distribution function in (11) can be rewritten as…”

“…Some of the high-temperature thermostatistical characteristics such as the internal energy and the equation of state are obtained in terms of some functions of the model deformation parameters q 1 and q 2 . The results obtained in this paper will serve as a two-parameter generalization related to the high-temperature thermostatistics of earlier q-deformed boson gas studies [9][10][11][12][13][14][15].…”

“…Inspired by the one-dimensional q-deformed bosonic and fermionic algebra studies of Lavagno and Swamy [9][10][11][12][13][14][15], in the present paper, we consider a different generalization of bosonic system, which is called the commuting Fibonacci oscillators. By using the formalism of Fibonacci calculus, we particularly study the high-temperature thermostatistical properties of a gas of the commuting Fibonacci oscillators.…”