Accepted Manuscript Radial distribution function within the framework of the Tsallis statistical mechanics Seyed Mohammad Amin Tabatabaei Bafghi, Mohammad Kamalvand, Ali Morsali, Mohammad Reza Bozorgmehr PII: S0378-4371(18)30535-1 DOI: https://doi.org/10.1016/j.physa.2018.04.107 Reference: PHYSA 19539 To appear in: Physica A Received date : 7 November 2017 Revised date : 15 April 2018 Please cite this article as: S.M.A.T. Bafghi, M. Kamalvand, A. Morsali, M.R. Bozorgmehr, Radial distribution function within the framework of the Tsallis statistical mechanics, Physica A (2018), https://doi.org/10.1016/j.physa.2018.04.107 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review) Highlights - A new equation is derived for the RDF in the Tsallis statistics. - The momentum and the coordinates are independent in this equation. - The correlation increases with an increase in the values of q. - Increase of the non-extensivity parameter and that of has similar effects.
*Manuscript Click here to view linked References Radial distribution function within the framework of the Tsallis statistical mechanics Seyed Mohammad Amin Tabatabaei Bafghi, 1 Mohammad Kamalvand, 2, Ali Morsali, 1 and Mohammad Reza Bozorgmehr 1 Abstract 1 Department of Chemistry, Mashhad Branch, Islamic Azad University, Mashhad, Iran 2 Department of Chemistry, Faculty of Science, Yazd University, Yazd 89195-741, Iran This study is conducted to obtain the radial distribution function (RDF) within the Tsallis statistical mechanics. To this end, probability distribution functions are applied in the first and fourth versions of the Tsallis statistics. Moreover, a closed formula is proposed for RDF. The power nature of the probability distribution in the Tsallis statistics makes it difficult to separate kinetic energy and configurational potential parts. By using the Taylor expansion around q=1 of the power distribution, it is possible to show the independency of momenta and coordinates through integrating over the phase space variables. In addition, at low densities, numerical calculations have been performed for the RDF. Our results show that the correlation increases as q values increase. Keywords: Radial distribution function, Tsallis statistical mechanics, Non-extensive. 1. Introduction The Boltzmann-Gibbs (BG) statistical mechanics is considered as a powerful theory for interpretation of the thermodynamic behavior of physical systems. Despite its relative success, the BG statistics is only suitable to describe extensive systems with short-range interactions, Markovian stochastic processes and, indeed, the systems whose phase space is ergodic [1-4]. However, among the phenomena in nature, one encounters cases in which this statistics is not able to predict and describe their behavior. Thus, another statistical mechanics is needed to interpret these phenomena. The basis of the difference in various statistics emanates from the definition of entropy and how it relates to probability functions. Nowadays, many phenomena have been known whose thermodynamic behavior, due to non- Corresponding author: kamalvand@yazd.ac.ir 1
extensive effects, is not explicable by the common BG statistical mechanics. For instance, one can mention the systems involving long-range interactions that have a microscopic long-range memory and the systems containing a multi-fractal structure [2,3]. Also, such entities as self-gravitating systems [5,6], inflationary cosmology [7,8], complex networks [9,10], the study of stochastic systems [11,12], dark energy models [13], electron trapping in a degenerate plasma [14], the nonlinear statistical coupling [15], negative heat capacity [16,17], anomalous diffusive behavior (e.g. Brownian motion with a random diffusion coefficient) [18], and high-energy proton-proton collisions [19-21], indicate that the correction of the Boltzmann entropy seems to be essential. In this regard, various extensions of BG entropy have been reported [22-26]. Among them, the Tsallis entropy is known as an efficient extension. The Tsallis statistical mechanics was introduced in 1988 to discuss multifractal systems [22]. In addition to the phenomena listed above, there are other usual phenomena to which the Tsallis statistics has successfully been applied. A few of these phenomena are ideal gas [27-30], harmonic oscillators [28,31,32], linear response theory [33,34], blackbody radiation [35,36], specific heat of 4 He [37], Bose-Einstein condensation [38,39], Ising model [40], and plasma two-dimensional turbulence [41]. The expression of [42]: the generalized entropy proposed by Tsallis is as follows where k is a positive constant, W implies the total number of possible microscopic configurations of the system, stands for the probability in the ith microstate, and q is an entropic index which demonstrates the degree of non-extensivity in the system. It should be emphasized that the entropy is reduced to the wellknown Boltzmann-Gibbs-Shannon formula [43] within the limit. By extremizing the entropy, using and as two constraints, a probability distribution [22] can be obtained with 2
where is referred to as the internal energy, is the Lagrange parameter, is related to the eigenvalue of the Hamiltonian system, and is the partition function. The formula of entropy, Eq. (1), can be adapted to fit the physical characteristics of the non-extensive systems, whereas it retains the essential property of entropy in the Second Law of thermodynamics [44]. Many modern theories proposed in the fields of liquids and fluids have greatly drawn upon the concept of radial distribution function (RDF). The RDF is a helpful function in the liquid-state theory where the thermodynamic properties of a fluid (e.g., internal energy, pressure, and chemical potential) can be calculated by using the pair-wise additive approximation and RDF [45-47]. For a system consisting of N particles at the temperature T and in the volume V, the equilibrium N-particle distribution function in the BG statistics can be defined as [47] where is the N-body interaction potential, and denotes the configuration integral,. Note that is only a function of the coordinates and not of the momenta. The N-particle correlation function is expressed in terms of the corresponding particle distributions by where is the bulk density. This is noticeable that, in non-equilibrium statistical mechanics, RDF, in addition to the coordinate, is a function of the momentum [48,49]. RDF shows the structure of a liquid in terms of the probability of finding other particles at a certain distance from the central molecule. Also, RDFs can be inferred indirectly from X-Ray spectra [50]. Over the past few decades, extensive studies have been conducted on RDF of dense fluids and liquids systems. The distribution functions calculated in the 3
Tsallis statistics apply to weakly interacting systems and ionic solutions [51]. In Ref. [52], the correlation function for Boson systems has been calculated within the non-extensive quantum statistics. As far as we know, no report has been published regarding a proper RDF within the framework of the Tsallis statistics. The main purpose of this study is to use the probability distribution functions of Tsallis in order to obtain the RDF of N-particles systems. Given that the probability distribution in the BG statistics has an exponential form, as in Eq. (4), the momenta are eliminated from the numerator and the denominator of the fraction, and the momentum portion is removed from the probability density function by integrating over all the momenta. More importantly, compared to power-law distributions in Tsallis [53-56], achievement of the same result with BG statistics requires more discussion and contemplation. This is one of the remarkable aspects of this research. The paper is organized as follows. Sec. 2 focuses on the theoretical formalism of the RDF in the first version of the Tsallis statistics. Also, a closed form is derived for the RDF in the first version. In Sec. 3, a closed formula is proposed for the RDF in the fourth version of the Tsallis statistics. Sec 4 is devoted to the numerical results of RDF under low-density conditions. Finally, Sec 5 is dedicated to the conclusion. 2. Theoretical formalism of RDF in the first version of the Tsallis statistics The probability distribution function in the first version of the Tsallis statistics is defined as [22] where (6) is the Hamiltonian of the N-particle system, is the momentum of the ith particle, and m stands for the particle mass. In the Tsallis statistics, with respect to kinetic energy and interaction potential parts, the power form of the probability distribution is not factorizable by integrating over the momenta and the positions. 4
In this context, a Taylor expansion around q=1 is applied for the power distribution function, giving An equation of this form allows the separation of momenta and coordinates. As shown in the following, taking the integral of Eq. (8) over all of the possible classical states leads to where From Eq. (10), it is straightforward to recognize that case is related to the partition function of Boltzmann-Gibbs. So, the following equation holds: According to Eq. (11), can be written as follows: 5
Putting Eq. (15) in, one gets so that Similarly, for, and respectively, the formulas below are obtained: 6
Therefore, is expressed by Similar to Eq. (22), for and cases, the following are yielded: Substitution of Eqs. (14) and (22-24) in Eq. (9) gives 7
In the present paper, a closed formula reported in Ref. [57] is employed for the m-th derivative of. This takes the form where,,, and the sum covers individual elements of for which. For example, the third-order derivative of is written as follows: 8
It is noteworthy that the other states of are not permitted. In contrast to the standard statistical mechanics, the Tsallis statistics has a power nature. By using uncommon mathematical definitions, this statistics can be displayed as an exponential form [58,59]. The q-exponential function in the first version [60] is defined as follows: where The Taylor expansion around q=1 of the is defined by In the limit of, it is verify that the quantity can be equated to Eq. (26). Hence, one gets so that 9
Consequently, a closed form is given for Eq. (8) by Once the integral of Eq. (34) is performed over the momenta and the coordinates, there emerges the following relationship: where Also, once the binomial expansion is applied, there emerges Consequently, a closed form is obtained for the partition function given by 10
The N-particle density function in the Tsallis statistics is defined as One can easily realize that Eq. (41) is independent of the momentum. Thus, the configurational part of the N-particle density function is expressed as So, there exists 11
As given below, a closed formula is derived for the N-particle density function in the first version of Tsallis: (45) where and The following closed formula is proposed for the RDF in the first version of the Tsallis statistics; hence, Finally, the RDF of the N-particle systems in the first version of Tsallis can be written as (46) 12
Due to the power form of the probability distribution in the Tsallis statistics, the possibility of separating the momenta and the coordinates seems to be low. But, from the calculations conducted in this study, one can find out that the partition function in the Tsallis statistics is separable in terms of kinetic energy and configurational potential. The independence of the momenta and the coordinates in the power distribution of Tsallis is consistent with that in the Boltzmann-Gibbs statistics. 3. RDF in the fourth version of the Tsallis statistics On the basis of the discussion in the previous section, a closed formula may be obtained for RDF in the fourth version of the Tsallis statistics. The probability distribution in the fourth version of the Tsallis statistics is given by [55] where Also, a closed form can be presented for RDF in the fourth version of the Tsallis statistics as follows: 13
(50) where and 4. Numerical results at low density At low densities, the correlation function in the BG statistics reduces to In the limit of g(r) in the first version of the Tsallis statistics takes the following form: (51) where and Hereby, the numerical calculations were done to solve Eq. (51). In this equation, we have extended our calculations up to the sevenths term. The variables are used in a reduced form in terms of the Lennard-Jones potential parameters and which are the potential well depth and the molecule diameter respectively. The reduced temperature is where is the Boltzmann constant, the reduced distance is, and the reduced density is where. In order to investigate the effect of non-extensivity on g(r), the function g(r) (in the first version) versus is plotted in Fig. 1 for the typical values of q containing 0.9, 0.95, 1.0, 1.05, and 1.10 at a reduced temperature equal to 1.5, and under low densities. As it can be seen in Fig. 1, the height of the peaks increases. 14
with an increase in the values of q. On the other hand, the rise in the height of the peaks exhibits an increase in the correlation value. At the same density and temperature, the value of entropy decreases with an increase in the correlation. As it turns out, the entropy of the system for q < 1 is larger than for BG; as a result, the system is super-extensive. Also, in the case of q > 1, the value of the system entropy is smaller than that in the BG case, and the system is sub-extensive. According to Fig. 1, when q = 0.9, the non-zero values of g(r) start from. With an increase in the value of q, the non-zero values start from greater values of. This behavior is similar to the effect of attractions, on g(r). It is well-known that an increase in leads to a shift in the non-zero values of g(r) to smaller values of. Therefore, increase of the value of q and that of has similar effects. However, we can not claim that this similarity is general for all systems with different interactions. Fig. 1. The pair correlation function g(r) (in the first version) versus for typical values of q equal to 0.9, 0.95, 1.0, 1.05, and 1.10 at a reduced temperature equal to 1.5 under a low density. 15
Similarly, with the first version at low densities, g(r) in the fourth version of the Tsallis statistics is defined as follows: (52) where and in Eq. (52) is expressed by [60] The value of is given by is self-consistently obtained by Eqs. (53) and (54). The numerical calculations have been performed to solve Eq. (52). In order to discuss the effect of the non-extensivity parameter on g(r), the function g(r) (in the fourth version) versus is plotted in Fig. 2 for the typical values of q containing 0.9, 0.95, 1.0, 1.05, and 1.10 at a reduced temperature equal to 1.5, and under low densities. According to Fig. 2, the correlation increases with an increase in the values of q. The qualitative aspect of the results is the same for both versions. In the fourth version of the Tsallis statistics, is a parameter pseudo-inverse temperature where depends on. Therefore, we do not expect that the results for both versions were the same, quantitatively. 16
Fig. 2. The pair correlation function g(r) (in the fourth version) versus for typical values of q equal to 0.9, 0.95, 1.0, 1.05, and 1.10 at a reduced temperature equal to 1.5 under a low density. 5. Conclusions In this paper, a radial distribution function (RDF) is derived in the Tsallis statistics, and a closed formula is obtained for that function in the first and fourth versions. Contrary to the Boltzmann-Gibbs statistics, the RDF within the Tsallis statistics has a more complex form. Using the q-exponential function in the Tsallis statistics and the Taylor expansion around q=1 of the power distribution and by integrating over the phase space variables, it is possible to make the momenta and the coordinates independent of each other. The independence thus set up through the Tsallis statistics is compatible with what takes place in the Boltzmann-Gibbs statistics. The effect of the non-extensivity parameter on the RDF of a Lennard- 17
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