GENERALIZED STATISTICAL MECHANICS FOR NUCLEUS

- In this study, we derived the grand partition function and some thermodynamical quantities of the nucleus in the high temperature limit by adopting the ideal Fermi gas partition function of generalized Tsallis thermostatistics. The behaviour of number of neutron and internal energy are depicted as a function of temperature. Sensitivity of q entropy index to the number neutron are also analyzed.


INTRODUCTION
A nonextensive entropy definition was proposed in 1988 [1] by Tsallis, where k is a positive constant, p i is the probability of the system in the i th microstate, W is the total number of the configurations of the system and q is any real number. Since then, Tsallis thermostatistics (TT) has commonly been used for the investigation of the physical systems. As well-known, Boltzmann-Gibbs (BG) statistics is a powerful one to study a variety of the physical systems. However it falls for the systems which 1) have long-ranged interactions, 2) have long-ranged memory effects, and 3) evolve in multifractal space-time. In this manner, it has been understood that extensive BG statistics fails to study nonextensive physical systems not having these conditions. Consequently the standard BG thermostatistics is not universal and is appropriate for extensive systems. The parameter, q, in Eq.(1) is called the entropy index and measures the degree of the nonextensivity of the system under consideration. After a few years from 1988, the formalism was revised [2] introducing the unnormalized constraint to the internal energy. This generalization included the celebrated BG thermostatistics, which could be recovered when q = 1. After this work, this formalism has been commonly employed to study various physical systems. Some of them can be given such as; self gravitating systems [3,4], turbulence [5][6][7][8], anomalous diffusion [9][10][11][12], solar neutrinos [13], liquid crystals [14][15][16][17][18][19][20][21][22] etc.
The constraint to the internal energy of the system is given by The optimization of Tsallis entropy given by Eq.(1) according to this internal energy constraint results in (3) with (4) Therefore the q-expectation value of any observable is defined as where A represents any observable quantity which commutes with Hamiltonian. The qexpectation value of the observable reduces to the conventional one when q = 1. In literature, there exists some applications of nonextensive statistics to; the classical ideal gas [23], the quantum ideal gas [24], Bose-Einstein and Fermi-Dirac distribution functions [25], the ideal Fermi gas [26]. Prato [23] derived an partition function of the classical ideal gas for q < 1 by using integral representation of Gamma function in the complex plane. Using this partition function specific heat olso obtained and recovered the BG expression for q = 1. In another study related with the ideal gases [24] the ideal gases partition function, internal energy and number of particles for both q < 1 and q > 1 are derived using the transformations obtained from the gamma function representations at high and low temperature limit. Lenzi et al. [23] analyzed the generalized Bose-Einstein and Fermi-Dirac distributions within nonextensive statistics by considering the normalized constraints in the effective temperature approach. Martinez et al. [26] discussed the relevant aspects of the exact q-thermostatistical treatment for an ideal Fermi system. The grand canonical exact generalized partition function was given for arbitrary values of the entrpic index q, and ensuing statistics was derived.
In the present study, we obtained the grand partition function and related thermodynamical quantities at high temperature for the nucleus which is considered as a fermi gas. This is crude but useful model to explain the statistical features of nuclear spectra. In doing so, we use the expressions obtained in Ref. [24]. Details of this calculations are given in Section 2. In section 3, generalized internal energy, number of particles (neutrons and protons), spin for the nucleus is obtained. In last section, our results are compared to those in the literature and we present a summary and our main conclusions.

GENERALIZED GRAND PARTITION FUNCTION
Consider a system of N neutrons and Z protons. The neutrons have single-particle levels with energy s a , occupation number s n , magnetic quantum number s m 1 ; proton single-particle levels are similarly designated by s b , s z , s m 2 . A single state of whole system is defined by four constants of the motion: (6) For this system, the generalized grand partition function is given by The Hilhorst integral transformations [27] and the extension for q < 1 shown by Prato [23] are useful because it provides a connection between a thermodynamic or statistical generalized quantity and its respective standard quantity. Thus, generalized thermodynamics could easily be established. Therefore, the generalized statistical mechanics can be regarded as a Hilhorst integral transformation of respective standard quantities. From the integral representation of Gamma function the grand partition function (7) can be written, with the identifications ν=1/(q-1) and ), ( in the following form: . When the summations and integral are interchanged, Eq.(9) becomes (10) for q > 1, and using the integral representation of Gamma function in the complex plane we have [23] The contour C in the complex plane is Hankel contour. For the system that contains N neutrons and Z protons, standart number of states is defined as In order to calculate 1 Ω , we shall replace the sums in Eq.(13) by integrals over the energy ε , introducing for this purpose the single-particle level densities ) , ( ( where .
In the evaluation of integrals the terms depending on derivatives of g are neglected since they are small compared with the terms containing g. For the details of this calculations see [28,29]. Using the following definitions Introducing the number of states into standart partition function 1 Ξ in Eq.(12) and replacing it into the integral in Eq.(11) we obtain . Above integral with -z = t is integral represantation of the Wright function for ρ=1 which is in the form of (20) Ha is the Hankel contour chosen on negative real axis. Therefore, generalized partition function becomes (21) where ) is Wright function introduced by E. M. Wright in the asymtotic theory of partitions. For analytical properties and some generalizations of this function see the [30]. For large q γ Wright function can be written in the following asymptotic expression [30] .
the generalization partition function for q > 1 can be obtained as

q-EXPECTATION OF INTERNAL ENERGY, THE NUMBER OF PROTONS, NEUTRONS AND SPIN
Now we can calculate the internal energy of the physical system under consideration, using the partition functions obtained above. After similar transformations for the qexpectation value of the energy are written, the internal energy is obtained for q > 1, and for q < 1 .
Similar expressions are obtained for the q-expectation values of the number of neutrons, protons and spin for q > 1, and for q < 1. By using the integral representation of Wright function, i.e. Eq.(20), and after some algebra, the q-expectations of internal energy, the particle number (the number of protons and neutrons) and spin are obtained

RESULTS AND DISCUSSION
In this study, we derive the grand partition functions for q < 1 (eq. (23)) and q > 1 (eq. (25)), and intenal energy,number of neutrons, number of protons and spin (eq. (30)) , probably first time, using the Tsallis second choice of constraints. we focus only on the generalized number of neutron and internal energy of the nucleus because another thermodynamical quantities presents the similar behaviour to the number of neutrons and it is adequate for our present purposes. Our results are obtained by using the eqs. (30) and (31) for 24 Na. In Figs.(1) and (2), the variation of the q-expectation value of the number of neutrons on the entropic index q is plotted for β = 0.1 and 0.5 respectively. We restrict the q values between 0.9 and 0.99. The reason for this is simply that the q-expectation value of number of neutrons becomes very large out of this range of q and there is a convergence problem at q = 1; very similar problem was also reported elsewhere [31]. In this region, q-expectation value of number of neutrons is sensitive to the q entropy index.
In Fig.(3), the dependence of the q-expectation value of the number of neutrons on the inverse temperature is illustrated for some q values. Number of neutrons increases with the inverse temperature. Our results exhibit similar behaviour to that of Fermi systems obtained in Refs. [32,33]. as a Figure 4. The variation of the mean internal function of inverse temperature for some energy of 24 Na with temperature for q=0.85, some q values. 0.9 and 0.95.
Finally, we plot the generalized mean energy vs. temperature for q = 0.85, 0.9 and 0.95 in Fig.(4). The behaviour of the generalized mean energy vs. temperature is typically in similar to that obtained from Fermi-Gas model proportional to T². It is worth to note that whereas there is a convergence problem at q = 1, the relevant region could be q < 1 because for q > 1 number of neutron and internal energy have discontinuties at different q values.