Theoretical self-consistency in nonextensive statistical mechanics with parameter transformation
本站小编 Free考研考试/2022-01-02
Yahui Zheng,1,2, Jiulin Du31School of Science, Henan Institute of Technology, Xinxiang 453003, China 2Institute of Physics, Henan Institute of Technology, Xinxiang 453003, China 3Department of Physics, School of Science, Tianjin University, Tianjin 300072, China
Abstract Self-consistency in nonextensive statistical mechanics is studied as a recourse to parameter transformation, where different nonextensive parameters are presented for various theoretical branches. The unification between the first and third choices of the average definition and that between the normal and escort distributions are both examined. The problem of parameter inversion in the generalized H theorem is also investigated. The inconsistency between the statistical ensemble pressure and molecular dynamics pressure can be eliminated. This work also verifies the equivalence of physical temperature and gravitational temperature in nonextensive statistical mechanics. In these parameter transformations, the Tsallis entropy form is observed to remain invariant. Keywords：self-consistency;escort distribution;parameter transformation;nonextensive statistics
PDF (418KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Yahui Zheng, Jiulin Du. Theoretical self-consistency in nonextensive statistical mechanics with parameter transformation. Communications in Theoretical Physics, 2021, 73(12): 125601- doi:10.1088/1572-9494/ac256c
1. Introduction
Many complex physical and nonphysical systems that exhibit certain exceptional characteristics in nature are actually specified by various complex mechanisms. Such mechanisms include long-range interactions and correlations or long-range memory processes that can be mathematically described in the non-Euclidean phase space or space–time continuum. Hence, the classical Boltzmann–Gibbs statistics cannot be employed in such systems because they are only suitable for systems with short-range interactions in the Euclidean phase space.
Over the last three decades, nonextensive statistical mechanics (NSM) has been developed to deal with complex systems containing various complex mechanisms. It has been established based on the Tsallis entropy [1], which can be expressed as the sum of the q-logarithm (power law) of distribution probability of the system at a given microstate, where the so-called nonextensive parameter q is introduced. When this parameter tends to unity, the q-logarithm becomes the normal logarithm and the Tsallis entropy changes into the traditional Boltzmann entropy. In the general case, by using a standard maximum entropy procedure [1] one can easily deduce a certain type of power-law distribution function, which has been found in many systems. It is interesting that various complex mechanisms can be naturally connected with the nonextensive parameter, making this new theory itself possess wide applicability to the fields of science and technology. So far, NSM has been applied to many fields, such as physics, astronomy, chemistry, and life science. It has also been employed in high-energy physics [2], spin-glasses [3], cold atoms in optical lattices [4], trapped ions [5], anomalous diffusion [6, 7], astrophysical and space plasmas [8–10], biological systems [11–13], and so on.
With regard to the theoretical foundations of NSM, several mathematical structures for various branches have been constructed. These include the maximization entropy with different energy constraints [1, 14–17], the nonextensive statistical ensembles with multifarious expansion methods [18–22], and the generalized Boltzmann equation [23–25]. The theoretical structures in NSM are rigorous and have been successfully applied to many cases; however, self-inconsistencies remain. For example, a direct link between single-particle and ensemble distribution functions does not exist, and the normalized q-average definition of internal energy [15] has not been introduced into statistical ensembles. Moreover, escort distribution [26] cannot be directly calculated; however, it is extremely dependent on the prior deduction of normal distribution, which is directly calculated by the maximum entropy procedure [15]. The relationship between the two different expressions of the polytrophic indexes [27–29] and their associated pressures [30, 31] has not been investigated. Furthermore, the inversion problem of nonextensive parameters in the deduction of the distribution function by the generalized H theorem [23] has been neglected. The inconsistency between the statistical ensemble pressure and molecular dynamics pressure is entirely ignored. All these inconsistencies seem to be closely related to the selection of the definition of the average value [15]. Accordingly, this study endeavors to examine the self-consistency of the theoretical foundation of NSM. This paper proposes the use of single-parameter transformations to construct a link in the distribution functions between the ensemble theory and molecular dynamics. This work also aims to formulate the relationship between the two choices in the defining average to remove the discrepancies mentioned above.
The paper is organized as follows. Section 2 discusses the derivation of the canonical distribution function based on the equiprobability principle. Further, the normalized q-average energy is introduced, and the equivalence between statistical ensemble and maximum entropy principles is verified. Section 3 presents the deduction made on the single-particle distribution function by parameter transformation. It also explains the proof of the unification between the statistical ensemble and kinetic theory. Section 4 elaborates on the investigation of the parameter inversion problem and the transformation invariance of the Tsallis entropy. In section 5, a series of parameter transformations is discussed. Finally, the conclusions and discussions are summarized in section 6.
2. Statistical ensemble theory in NSM
This section presents the construction of the link between the maximum entropy principle and statistical ensemble with the equal probability assumption. To establish this link, the canonical ensemble, in which only the energy exchange between the observed system and heat reservoir transpires, is considered as an example. The combination of the observed system and reservoir is regarded as a composite system where long-range interactions exist.
The original form of the Tsallis entropy [1] behaves as the quantum summation of the power law of the distribution probability at a given quantum state of the composite system. Due to the power-law properties, the Tsallis entropy becomes nonextensive and nonadditive, further leading to the nonadditivity of many quantities, such as the physical internal energy [32, 33] and Gibbs free energy. In the microcanonical ensemble, all the microstates have the same probability, so the corresponding Tsallis entropy should adopt the form with equal probability. That is, the total Tsallis entropy of the composite system should be given by$\begin{eqnarray}{S}_{q0}=k{\mathrm{ln}}_{q}{{\rm{\Omega }}}_{0}\equiv k\displaystyle \frac{{{\rm{\Omega }}}_{0}^{1-q}-1}{1-q},\end{eqnarray}$where ω0 is the number of microstates of the composite system. For convenience, the Tsallis factor is generally defined in the following form:$\begin{eqnarray}{c}_{q0}\equiv \displaystyle \frac{{S}_{q0}}{k}(1-q)+1.\end{eqnarray}$The two forms above are also suitable for the observed system; their application is considered in the following discussion.
In a system with long-range interactions, energy is essentially nonadditive. With the probability independence assumption, the composition rule of microstate energy levels in the observed system can be written as [32, 33]$\begin{eqnarray}{E}_{ij(1,2)}={E}_{i1}+{E}_{j2}-(1-q)\beta ^{\prime} ({E}_{i1}-{U}_{q1}^{(3)})({E}_{j2}-{U}_{q2}^{(3)}),\end{eqnarray}$where subscripts 1 and 2 represent two different microstates. In this composition rule (equation (3)), parameter β′ is recognized as a generalized Lagrange multiplier, and U is the most probable microstate energy level in the 'equilibrium' state or the normalized q-average internal energy, which is the third energy definition choice in the NSM [15]. This energy is given by$\begin{eqnarray}{U}_{q}^{(3)}\equiv \displaystyle \frac{\displaystyle \sum {E}_{i}{p}_{i}^{q}}{\displaystyle \sum {p}_{i}^{q}},\end{eqnarray}$where pi is the probability distribution of the ith microstate.
The third term in composition rule (3) shows the correlations among different microstates originating from various complex mechanisms. The magnitude of the third term relative to that of the first term or second term on the right-hand side of (3) is 1/N, where N is the number of particles in the observed system. Hence, the third term is extremely small compared with the first two terms when N is sufficiently large. In a macroscopic system, N is considerably large; thus, the cross term in (3) may also be neglected:$\begin{eqnarray}{E}_{ij(1,2)}={E}_{i1}+{E}_{j2}.\end{eqnarray}$The total energy, E0, of the composite system can therefore be written as the sum of the system energy (Ei) at the ith microstate and reservoir energy (Er), as follows:$\begin{eqnarray}{E}_{0}={E}_{i}+{E}_{r}.\end{eqnarray}$When the system is at the ith microstate, the possible number of microstates is ωr(E0−Ei). Based on the equal probability principle, this number is proportional to the probability distribution, pi, of the system at the ith microstate. This distribution is given by$\begin{eqnarray}{p}_{i}=\displaystyle \frac{{{\rm{\Omega }}}_{r}({E}_{0}-{E}_{i})}{{{\rm{\Omega }}}_{t}({E}_{0})},\end{eqnarray}$where ωt(E0) is the total number of possible microstates in the composite system.
The energy fluctuation is defined as$\begin{eqnarray}{\rm{\Delta }}{E}_{i}={E}_{i}-{U}_{q}^{(3)},\end{eqnarray}$which must be considerably less than the most probable energy of the observed system when N is large. The quantity lnqωr can be conveniently expanded as a function of the energy fluctuation around the most probable energy of the reservoir. Typically, only the first two terms of the Taylor expansion are retained, thus yielding the following:$\begin{eqnarray}\begin{array}{l}{\mathrm{ln}}_{q}{{\rm{\Omega }}}_{r}({E}_{0}-{E}_{i})={\mathrm{ln}}_{q}{{\rm{\Omega }}}_{r}({E}_{0}-{U}_{q}^{(3)})-{\left(\displaystyle \frac{\partial {\mathrm{ln}}_{q}{{\rm{\Omega }}}_{r}}{\partial {E}_{r}}\right)}_{{E}_{r}={E}_{0}-{U}_{q}^{(3)}}({E}_{i}-{U}_{q}^{(3)})\\ \,=\,{\mathrm{ln}}_{q}{{\rm{\Omega }}}_{r}({U}_{qr}^{(3)})-{\beta }_{r}({E}_{i}-{U}_{q}^{(3)}),\end{array}\,\end{eqnarray}$where$\begin{eqnarray}{\beta }_{r}={\left(\displaystyle \frac{\partial {\mathrm{ln}}_{q}{{\rm{\Omega }}}_{r}}{\partial {E}_{r}}\right)}_{{E}_{r}={E}_{0}-{U}_{q}^{(3)}}.\end{eqnarray}$The role of this parameter is similar to the Lagrange multiplier in the constrained extreme problem [15].
When the observed system is in equilibrium with its reservoir, the total number of microstates, ω0, in the composite system reaches the maximum. To determine the equilibrium condition, the Lagrange multiplier of the observed system is marked as β and that of the reservoir as βr; the number of microstates in the observed system is ω and that in the reservoir is ωr. Accordingly,$\begin{eqnarray}{{\rm{\Omega }}}_{0}={\rm{\Omega }}{{\rm{\Omega }}}_{r},\end{eqnarray}$yields$\begin{eqnarray}{\mathrm{ln}}_{q}{{\rm{\Omega }}}_{0}={\mathrm{ln}}_{q}{\rm{\Omega }}+{\mathrm{ln}}_{q}{{\rm{\Omega }}}_{r}+(1-q){\mathrm{ln}}_{q}{\rm{\Omega }}{\mathrm{ln}}_{q}{{\rm{\Omega }}}_{r}.\end{eqnarray}$In contrast, in the equilibrium state,$\begin{eqnarray}{U}_{q0}^{(3)}={U}_{q}^{(3)}+{U}_{qr}^{(3)}.\end{eqnarray}$Under the extreme condition, that is,$\begin{eqnarray}\displaystyle \frac{\partial {\mathrm{ln}}_{q}{{\rm{\Omega }}}_{0}}{\partial {U}_{q}^{(3)}}=0,\end{eqnarray}$the following is derived:$\begin{eqnarray}\displaystyle \frac{\beta }{{c}_{q}}=\displaystyle \frac{{\beta }_{r}}{{c}_{qr}},\end{eqnarray}$which is the thermal equilibrium condition for the canonical ensemble equilibrium [32]. Then, the probability distribution (equation (7)) can be written as$\begin{eqnarray}{p}_{i}=\displaystyle \frac{1}{{\bar{Z}}_{q}^{(3)}}{\left[1-(1-q)\displaystyle \frac{\beta }{{c}_{q}}({E}_{i}-{U}_{q}^{(3)})\right]}^{\tfrac{1}{1-q}},\end{eqnarray}$where the partition function is given by$\begin{eqnarray}{\bar{Z}}_{q}^{(3)}=\displaystyle \frac{{{\rm{\Omega }}}_{t}({E}_{0})}{{{\rm{\Omega }}}_{r}({U}_{qr}^{(3)})}=\displaystyle \sum _{i}{\left[1-(1-q)\displaystyle \frac{\beta }{{c}_{q}}({E}_{i}-{U}_{q}^{(3)})\right]}^{\tfrac{1}{1-q}}.\end{eqnarray}$The probability distribution, (16), is a generalized Gibbs distribution function.
At this point, the equivalence between the distribution function (equation (16)) and that obtained using the entropy maximization method [15] is investigated. For this purpose, let us recall the original form of the Tsallis entropy [1], namely,$\begin{eqnarray}{S^{\prime} }_{q}=k\displaystyle \frac{\displaystyle \sum {p}_{i}^{q}-1}{1-q}.\end{eqnarray}$Here, to distinguish it from the entropy inferred in this section, we add a prime to it. According to (16), we have$\begin{eqnarray}{[{p}_{i}{\bar{Z}}_{q}^{(3)}]}^{1-q}=1-(1-q)\displaystyle \frac{\beta }{{c}_{q}}({E}_{i}-{U}_{q}^{(3)}).\end{eqnarray}$The foregoing yields,$\begin{eqnarray}\displaystyle \sum _{i}{[{p}_{i}{\bar{Z}}_{q}^{(3)}]}^{1-q}{p}_{i}^{q}={[{\bar{Z}}_{q}^{(3)}]}^{1-q}=\displaystyle \sum _{i}{p}_{i}^{q},\end{eqnarray}$where the definition of average (equation (4)) is applied.
Furthermore, based on (20), the following is derived:$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{\partial {S^{\prime} }_{q}}{\partial {U}_{q}^{(3)}}={[{\bar{Z}}_{q}^{(3)}]}^{-q}k\displaystyle \frac{\partial {\bar{Z}}_{q}^{(3)}}{\partial {U}_{q}^{(3)}}={[{\bar{Z}}_{q}^{(3)}]}^{-q}k\\ \,\times \displaystyle \sum {\left[1-(1-q)\displaystyle \frac{\beta }{{c}_{q}}({E}_{i}-{U}_{q}^{(3)})\right]}^{\displaystyle \frac{q}{1-q}}\displaystyle \frac{\beta }{{c}_{q}}\\ \,=\,k\beta \displaystyle \frac{\displaystyle \sum {{p}_{i}}^{q}}{{c}_{q}}.\end{array}\end{eqnarray}$Impose the following thermodynamic relationship in [15]:$\begin{eqnarray}\displaystyle \frac{\partial {S^{\prime} }_{q}}{\partial {U}_{q}^{(3)}}=k\beta .\end{eqnarray}$Accordingly, the following equivalence must exist:$\begin{eqnarray}{c}_{q}={{\rm{\Omega }}}^{1-q}\equiv \displaystyle \sum {{p}_{i}}^{q}.\end{eqnarray}$This shows that the newly defined Tsallis entropy, equation (18), is equivalent to that in equation (1), thus establishing a link between the maximum entropy principle and statistical ensemble. Hence, the following probability independence assumption is valid:$\begin{eqnarray}{p}_{ij}={p}_{i}{p}_{j},\end{eqnarray}$where subscripts i and j represent the different microstates of a complex system. The foregoing can be proved as naturally leading to the composite rule (equation (3)) provided that the following relationship is adopted:$\begin{eqnarray}\beta ^{\prime} =\displaystyle \frac{\beta }{{c}_{q}},\end{eqnarray}$where β′ is the generalized Lagrange multiplier.
Because of the self-referential property of the distribution function [17], equation (16) is exigent to apply to a real complex system; hence, its transformation to another form is necessary. To achieve this, a new partition function is defined [15]:$\begin{eqnarray}\begin{array}{l}{[{Z}_{q}^{(3)}]}^{1-q}\equiv {[{\bar{Z}}_{q}^{(3)}]}^{1-q}-(1-q)\beta {U}_{q}^{(3)}\\ \,={[{\bar{Z}}_{q}^{(3)}]}^{1-q}\left[1-(1-q)\displaystyle \frac{\beta }{{c}_{q}}{U}_{q}^{(3)}\right].\end{array}\end{eqnarray}$Substituting (26) into (16) yields$\begin{eqnarray}\begin{array}{l}{p}_{i}=\displaystyle \frac{1}{{Z}_{q}^{(3)}}{\left[1-(1-q)\displaystyle \frac{\beta }{{c}_{q}}{U}_{q}^{(3)}\right]}^{\tfrac{1}{1-q}}\\ \,\times {\left[1-(1-q)\displaystyle \frac{\beta }{{c}_{q}}({E}_{i}-{U}_{q}^{(3)})\right]}^{\tfrac{1}{1-q}}\\ \,=\displaystyle \frac{1}{{Z}_{q}^{(3)}}\left[1-(1-q)\displaystyle \frac{\beta }{{c}_{q}}{E}_{i}\right.\\ \,+{\left.{\left[(1-q)\displaystyle \frac{\beta }{{c}_{q}}{U}_{q}^{(3)}\right]}^{2}\displaystyle \frac{({E}_{i}-{U}_{q}^{(3)})}{{U}_{q}^{(3)}}\right]}^{\tfrac{1}{1-q}}.\end{array}\end{eqnarray}$With the above, the magnitudes of the terms on the right-hand side of (27) can be estimated. First, with the partition function in (26) exceeding zero,$\begin{eqnarray}(1-q)\displaystyle \frac{\beta }{{c}_{q}}{U}_{q}^{(3)}\sim O(1),\end{eqnarray}$where O(1) denotes that the order of magnitude is approximately unity. In contrast, according to the traditional fluctuation theory, for a system with N number of particles, typically,$\begin{eqnarray}{\left[(1-q)\displaystyle \frac{\beta }{{c}_{q}}{U}_{q}^{(3)}\right]}^{2}\displaystyle \frac{({E}_{i}-{U}_{q}^{(3)})}{{U}_{q}^{(3)}}\sim \displaystyle \frac{({E}_{i}-{U}_{q}^{(3)})}{{U}_{q}^{(3)}}\sim O\left(\displaystyle \frac{1}{\sqrt{N}}\right).\end{eqnarray}$Therefore, for an extremely large N, the distribution function (equation (16)) changes into$\begin{eqnarray}{p}_{i}=\displaystyle \frac{1}{{Z}_{q}^{(3)}}{\left[1-(1-q)\displaystyle \frac{\beta }{{c}_{q}}{E}_{i}\right]}^{\tfrac{1}{1-q}}.\end{eqnarray}$Equation (30) can also be derived based on the entropy maximization procedure by assuming that the Tsallis factor is irrelevant to the probability distribution in each configuration of the system [17].
The following proves the approximate composite rule in (5). According to (29),$\begin{eqnarray}\begin{array}{l}(1-q)\displaystyle \frac{\beta }{{c}_{q}}({E}_{i1}-{U}_{q1}^{(3)})({E}_{j2}-{U}_{q2}^{(3)})\\ \,\sim (1-q)\displaystyle \frac{\beta }{{c}_{q}}{U}_{q1}^{(3)}{U}_{q2}^{(3)}O\left(\displaystyle \frac{1}{N}\right)\\ \,\sim {U}_{q2}^{(3)}O\left(\displaystyle \frac{1}{N}\right)\sim {U}_{q1}^{(3)}O\left(\displaystyle \frac{1}{N}\right).\end{array}\end{eqnarray}$The foregoing shows that the order of magnitude of the cross term on the right-hand side of (3) is that of the average energy of a single particle, that is, approximately 1/N of the first or second term. Therefore, the cross term in (3) can be ignored, and rule (5) can be applied when the number of particles is extremely large.
Rule (5) is evidently based on the weak fluctuation of energy or other quantities of the observed system; in this case, the system does not experience any phase transition. Accordingly, the conclusion is that for most systems in normal states, the approximate additive rule (i.e. equation (5)) is generally acceptable. Thus, in this study, the additive rule of energy is frequently applied.
Consider the following observations pertaining to the most probable state of the statistical ensemble. This state evidently corresponds to the maximum probability leading to the ensemble distribution (equation (16)). It is also equivalent to the 'equilibrium' state, which possesses the maximum Tsallis entropy. This 'equilibrium' is not necessarily the thermal equilibrium; it is statistical equilibrium. For instance, in a self-gravitating system, the state of statistical equilibrium is a nonequilibrium steady state. Based on this simple fact, the temperature duality assumption [32, 33] can be applied to afford reasonable interpretations to the 'equilibrium' and 'nonequilibrium' properties in the system.
3. Single-particle distribution function in NSM
In this section, the single-particle distribution function in a dynamic system is deduced based on the distribution given by (30), where the system is governed by the generalized Boltzmann equation [23]. For convenience, the continuous form of equation (30) is applied:$\begin{eqnarray}F=\displaystyle \frac{1}{{Z}_{q}^{(3)}}{\left[1-(1-q)\displaystyle \frac{E}{k{T}_{q}}\right]}^{\tfrac{1}{1-q}}.\end{eqnarray}$In the above, the temperature duality relationship is used [33], as follows:$\begin{eqnarray}\beta =\displaystyle \frac{1}{kT},\,{T}_{q}={c}_{q}T,\end{eqnarray}$where Tq is the physical temperature, T is the Lagrange temperature, and k is the Boltzmann constant. The partition function in (30) is defined as$\begin{eqnarray}{Z}_{q}^{(3)}=\displaystyle \int {\left[1-(1-q)\displaystyle \frac{E}{k{T}_{q}}\right]}^{\tfrac{1}{1-q}}{\rm{d}}{\tau }_{1}{\rm{d}}{\tau }_{2}\mathrm{...}{\tau }_{N},\end{eqnarray}$where the volume element in six-dimensional phase space is ${\rm{d}}\tau ={{\rm{d}}}^{3}v{{\rm{d}}}^{3}r,$ and N is the number of particles in the system.
According to the additive rule (equation (5)), the energy level (equation (32)) in the dynamic system is explained as the sum of all particle energies, including their kinetic and potential energies with long-range interactions. To bypass the many-body problem in systems with long-range interactions, the mean field approximation is adopted. This means that the total potential energy of the observed system can be regarded as the sum of the potential energy of all particles. Therefore, the total energy of the system is$\begin{eqnarray}E=\displaystyle \sum _{1}^{N}{\varepsilon }_{i},\end{eqnarray}$where ϵi represents the total energy of a single particle, including its mean field potential energy.
The direct factorization of the canonical distribution function (equation (32)) is difficult. However, all particles can be confirmed to have the same chance to appear in any place in the system; that is, the distribution function of all particles is similar in the phase space. In contrast, note that the energy level, E, and single-particle energy, ϵi, in (35) are both stochastic variables. Their fluctuations are both affected by the interaction between the system and reservoir as well as modulated by the ubiquitous mean field of the system. Consequently, without the loss of generality, the energy of each particle can be simply assumed to fluctuate with the same 'frequency' as that of the entire system. This means that E=Nϵ1, where ϵ1 is the energy of a particle.
To derive the single-particle distribution, the following parameter transformation can be employed:$\begin{eqnarray}(1-q)N=1-\nu ,\end{eqnarray}$where a new nonextensive parameter, ν, is introduced. This transformation suggests that two different nonextensive parameters must be considered simultaneously: one is used in the statistical ensemble, and the other is employed in the molecular dynamic system. Equation (36) actually constructs the link between the nonextensive statistical ensemble and molecular dynamic system.
According to (28), at the most, the value of (1−ν) has an order of magnitude of unity. However, for most normal states of nonextensive systems, the value of (1−ν) may be considerably less than unity. For instance, with the approximation (1−ν)→+0, the calculated result with respect to the second virial coefficient of nonextensive gas fits the experimental curve well [34]. This seemingly suggests that in normal states, the value of (1−q) is inconsequential; this is another reason why the cross term in (3) can be neglected. According to the parameter transformation (equation (36)),$\begin{eqnarray}\begin{array}{l}{Z}_{q}^{(3)}={\displaystyle \int \left[1-(1-\nu )\displaystyle \frac{{\varepsilon }_{1}}{k{T}_{q}}\right]}^{\tfrac{N}{1-\nu }}{\rm{d}}{\tau }_{1}\cdots {\rm{d}}{\tau }_{N}\\ \,=\displaystyle \prod _{i=1}^{N}{\displaystyle \int \left[1-(1-\nu )\displaystyle \frac{{\varepsilon }_{i}}{k{T}_{q}}\right]}^{\tfrac{1}{1-\nu }}{\rm{d}}{\tau }_{i}={Z}_{\nu }^{N},\end{array}\end{eqnarray}$where the single-particle partition function is defined as$\begin{eqnarray}{Z}_{\nu }={\displaystyle \int \left[1-(1-\nu )\displaystyle \frac{\varepsilon }{k{T}_{q}}\right]}^{\tfrac{1}{1-\nu }}{\rm{d}}\tau .\end{eqnarray}$Then, the single-particle distribution function is$\begin{eqnarray}f=\displaystyle \frac{1}{{Z}_{\nu }}{\left[1-(1-\nu )\displaystyle \frac{\varepsilon }{k{T}_{q}}\right]}^{\tfrac{1}{1-\nu }}.\end{eqnarray}$The canonical q-distribution function can be written in the following form:$\begin{eqnarray}F=\displaystyle \frac{1}{{Z}_{\nu }^{N}}{\left[1-(1-\nu )\displaystyle \frac{\varepsilon }{k{T}_{q}}\right]}^{\tfrac{N}{1-\nu }}={f}^{N}.\end{eqnarray}$The single-particle Tsallis factor can be defined as$\begin{eqnarray}{c}_{\nu }=\displaystyle \int {f}^{\nu }{\rm{d}}\tau =\displaystyle \frac{1}{{Z}_{\nu }^{\nu }}\displaystyle \int {\left[1-(1-\nu )\displaystyle \frac{\varepsilon }{k{T}_{q}}\right]}^{\tfrac{\nu }{1-\nu }}{\rm{d}}\tau .\end{eqnarray}$From this, the following is facilely found:$\begin{eqnarray}\begin{array}{l}{c}_{q}=\displaystyle \frac{1}{{[{Z}_{q}^{(3)}]}^{q}}\displaystyle \int {\left[1-(1-q)\displaystyle \frac{N{\varepsilon }_{1}}{k{T}_{q}}\right]}^{\tfrac{q}{1-q}}{\rm{d}}{\tau }_{1}\cdots {\rm{d}}{\tau }_{N}\\ \,=\displaystyle \frac{1}{{Z}_{\nu 1}^{N-1+\nu }}\displaystyle \int {\left[1-(1-\nu )\displaystyle \frac{{\varepsilon }_{1}}{k{T}_{q}}\right]}^{\tfrac{N-1+\nu }{1-\nu }}{\rm{d}}{\tau }_{1}\cdots {\rm{d}}{\tau }_{N}\\ \,=\displaystyle \frac{1}{{Z}_{\nu 1}^{\nu }}\displaystyle \int {\left[1-(1-\nu )\displaystyle \frac{{\varepsilon }_{1}}{k{T}_{q}}\right]}^{\tfrac{\nu }{1-\nu }}{\rm{d}}{\tau }_{1}={c}_{\nu }.\end{array}\end{eqnarray}$The foregoing means that the Tsallis factor of the system is identical to the single-particle Tsallis factor.
The L-energy [33] of the system is given by$\begin{eqnarray}\begin{array}{l}{U}_{q}^{(3)}=\displaystyle \frac{\displaystyle \int N{\varepsilon }_{1}{F}^{q}{\rm{d}}{\tau }_{1}\cdots {\rm{d}}{\tau }_{N}}{\displaystyle \int {F}^{q}{\rm{d}}{\tau }_{1}\cdots {\rm{d}}{\tau }_{N}}\\ \,=\,N\displaystyle \frac{\displaystyle \int {\varepsilon }_{1}{\left[1-(1-q)\displaystyle \frac{N{\varepsilon }_{1}}{k{T}_{q}}\right]}^{\tfrac{q}{1-q}}{\rm{d}}{\tau }_{1}\cdots {\rm{d}}{\tau }_{N}}{\displaystyle \int {\left[1-(1-q)\displaystyle \frac{N{\varepsilon }_{1}}{k{T}_{q}}\right]}^{\tfrac{q}{1-q}}{\rm{d}}{\tau }_{1}\cdots {\rm{d}}{\tau }_{N}}\\ \,=\,N\displaystyle \frac{\displaystyle \int {\varepsilon }_{1}{\left[1-(1-\nu )\displaystyle \frac{{\varepsilon }_{1}}{k{T}_{\nu }}\right]}^{\tfrac{\nu }{1-\nu }}{\rm{d}}{\tau }_{1}}{\displaystyle \int {\left[1-(1-\nu )\displaystyle \frac{{\varepsilon }_{1}}{k{T}_{\nu }}\right]}^{\tfrac{\nu }{1-\nu }}{\rm{d}}{\tau }_{1}}=N\displaystyle \frac{\displaystyle \int {\varepsilon }_{1}{f}^{\nu }{\rm{d}}{\tau }_{1}}{\displaystyle \int {f}^{\nu }{\rm{d}}{\tau }_{1}}.\end{array}\end{eqnarray}$In the above expression, the employed relationship, Tq=Tν, is based on equation (42). If the single-particle average energy in the L-set of thermodynamic formalism is defined as$\begin{eqnarray}\bar{\varepsilon }=\displaystyle \frac{{U}_{q}^{(3)}}{N},\end{eqnarray}$then$\begin{eqnarray}\bar{\varepsilon }=\displaystyle \frac{\displaystyle \int \varepsilon {f}^{\nu }{\rm{d}}\tau }{\displaystyle \int {f}^{\nu }{\rm{d}}\tau }=-\displaystyle \frac{\partial }{\partial \beta }{\mathrm{ln}}_{\nu }{Z}_{\nu }\equiv -\displaystyle \frac{\partial }{\partial \beta }\displaystyle \frac{{Z}_{\nu }^{1-\nu }-1}{1-\nu }.\end{eqnarray}$The use of (45) in the calculation of single-particle average energy has been widely reported in the literature [29, 30].
To obtain the familiar single-particle distribution function, the molecular energy in (39) is regarded as the sum of kinetic and relative potential energies:$\begin{eqnarray}\varepsilon =\displaystyle \frac{{\rm{1}}}{{\rm{2}}}m{v}^{2}+m(\varphi -{\varphi }_{0}),\end{eqnarray}$where φ represents the potential, and φ0 is the system origin. Substituting (46) into (39) and multiplying by N yields$\begin{eqnarray}\begin{array}{l}{f}_{N\nu }=\displaystyle \frac{N}{{Z}_{\nu }}{\left[1-(1-\nu )\displaystyle \frac{m(\varphi -{\varphi }_{0})}{k{T}_{\nu }}\right]}^{\tfrac{1}{1-\nu }}\left[1\right.\\ \,-{\left.(1-\nu )\displaystyle \frac{m{v}^{2}/2}{k{T}_{\nu }-(1-\nu )m(\varphi -{\varphi }_{0})}\right]}^{\tfrac{1}{1-\nu }}.\end{array}\end{eqnarray}$The normalization of the above distribution function is evidently equal to N. The following relationships are further introduced:$\begin{eqnarray}\begin{array}{l}{B}_{\nu }=\displaystyle \frac{N}{{Z}_{\nu }}{\left[1-(1-\nu )\displaystyle \frac{m(\varphi -{\varphi }_{0})}{k{T}_{\nu }}\right]}^{\tfrac{1}{1-\nu }},\\ kT=k{T}_{\nu }-(1-\nu )m(\varphi -{\varphi }_{0}),\end{array}\end{eqnarray}$where T is recognized as the Lagrange temperature. Then,$\begin{eqnarray}{f}_{N\nu }={B}_{\nu }{\left[1-(1-\nu )\displaystyle \frac{m{v}^{2}}{2kT}\right]}^{\tfrac{1}{1-\nu }},\end{eqnarray}$where [25]$\begin{eqnarray}\begin{array}{l}{B}_{\nu }=n(\vec{{\rm{r}}}){(1-\nu )}^{1/2}\left(\displaystyle \frac{5-3\nu }{2}\right)\left(\displaystyle \frac{3-\nu }{2}\right)\displaystyle \frac{{\rm{\Gamma }}\left(\tfrac{1}{2}+\tfrac{1}{1-\nu }\right)}{{\rm{\Gamma }}\left(\tfrac{1}{1-\nu }\right)}\\ \,\times {\left(\displaystyle \frac{m}{2\pi kT}\right)}^{3/2}\nu \lt 1,\end{array}\end{eqnarray}$$\begin{eqnarray}{B}_{\nu }=n(\vec{{\rm{r}}}){(\nu -1)}^{3/2}\displaystyle \frac{{\rm{\Gamma }}\left(\tfrac{1}{\nu -1}\right)}{{\rm{\Gamma }}\left(\tfrac{1}{\nu -1}-\tfrac{3}{2}\right)}{\left(\displaystyle \frac{m}{2\pi kT}\right)}^{3/2}\nu \geqslant 1,\end{eqnarray}$where the n(r) is the number density of molecules, which is a function of the space positions.
Finally, according to (36) and (40), the following can be proved:$\begin{eqnarray}{S}_{q}=k\displaystyle \frac{\displaystyle \int {F}^{q}{\rm{d}}{\tau }_{1}\cdots {\rm{d}}{\tau }_{N}-1}{1-q}={S}_{\nu }+{c}_{\nu }{N}^{\nu }k{\mathrm{ln}}_{\nu }N,\end{eqnarray}$where$\begin{eqnarray}{S}_{\nu }\equiv -k\displaystyle \int {f}_{N\nu }^{\nu }{\mathrm{ln}}_{\nu }{f}_{N\nu }{\rm{d}}\tau =k\displaystyle \frac{\displaystyle \int {f}_{N\nu }^{\nu }{\rm{d}}\tau -\displaystyle \int {f}_{N\nu }{\rm{d}}\tau }{1-\nu }.\end{eqnarray}$The above expression actually defines the entropy considered in the discussion of the generalized H theorem [23]. Equation (52) shows that between the statistical ensemble and molecular dynamics, the difference in entropy is a negligible constant in physics; that is, these two entropies are equivalent. The subsequent section presents the examination of the parameter inversion problem within the generalized H theorem framework.
4. Parameter inversion problem and transformation invariance of entropy
In the study of the generalized Boltzmann equation, the generalized H function or Tsallis entropy employs the definition given in (53). The δ logarithmic form of this expression is$\begin{eqnarray}{S}_{\delta }=-k^{\prime} \displaystyle \int {f}_{N\delta }^{\delta }{\mathrm{ln}}_{\delta }{f}_{N\delta }{\rm{d}}\tau .\end{eqnarray}$In the foregoing expression, another nonextensive parameter, δ, is introduced and distinguished from the nonextensive parameter, ν, presented in the previous section. For the same reason, a new constant, k′, is also introduced. The generalized collision core function or generalized molecular chaos hypothesis is proposed [23] as follows:$\begin{eqnarray}\begin{array}{l}{R}_{\delta }({f}_{N\delta },{f^{\prime} }_{N\delta })={\exp }_{\delta }({f^{\prime} }_{N\delta }^{1-\delta }{\mathrm{ln}}_{\delta }{f^{\prime} }_{N\delta }+{f^{\prime} }_{N\delta 1}^{1-\delta }{\mathrm{ln}}_{\delta }{f^{\prime} }_{N\delta 1})\\ \,-\,{\exp }_{\delta }({f}_{N\delta }^{1-\delta }{\mathrm{ln}}_{\delta }{f}_{N\delta }+{f}_{N\delta 1}^{1-\delta }{\mathrm{ln}}_{\delta }{f}_{N\delta 1}),\end{array}\end{eqnarray}$where the generalized δ exponential function is defined as$\begin{eqnarray}{\exp }_{\delta }(x)={{e}_{\delta }}^{x}\equiv {[1+(1-\delta )x]}^{\tfrac{{\rm{1}}}{1-\delta }}.\end{eqnarray}$The collision balance can be verified as leading to the following single-particle distribution function [23]:$\begin{eqnarray}{f}_{N\delta }={B}_{2-\delta }{\left[1-(\delta -1)\displaystyle \frac{m{v}^{2}}{2k^{\prime} T^{\prime} }\right]}^{\tfrac{1}{\delta -1}},\end{eqnarray}$where Tˊ, as a function of space coordinates, is similar to the Lagrange temperature, T. This distribution function (equation (57)) differs from (49); hence, the composite relationship between the nonextensive parameter and unity also varies. Specifically, (1−ν) is present in (49), whereas (Σ−1) exists in (57); this phenomenon is called parameter inversion.
For the (1–ν) distribution to lead to the Tsallis entropy in (53) is extremely unusual. In contrast, the Tsallis entropy in (54) leads to the (Σ−1) distribution through the generalized H theorem, which is an inevitable result of the hypothesis in (55) [23, 35]. To reconcile the relationship between the two distributions, (49) and (57), the following transformation is introduced [23, 35]:$\begin{eqnarray}\delta -1=1-\nu .\end{eqnarray}$
However, the application of this parameter transformation has no particular physical motivation. To demonstrate this clearly, the following polytropic index related to (57) is introduced [27]:$\begin{eqnarray}n=\displaystyle \frac{1}{\delta -1}+\displaystyle \frac{3}{2}.\end{eqnarray}$The gas pressure related to (57) can be written as [31]$\begin{eqnarray}P=\displaystyle \frac{2}{2+5(\delta -1)}n(\vec{{\rm{r}}})k^{\prime} T^{\prime} .\end{eqnarray}$In contrast, according to (45), the ν-average of molecular kinetic energy is [29, 30]$\begin{eqnarray}\begin{array}{l}\overline{{\varepsilon }_{k}}=\displaystyle \frac{1}{2}m\overline{{v}^{2}}=\displaystyle \frac{\displaystyle \int \displaystyle \frac{1}{2}m{v}^{2}{\left[1-(1-\nu )\displaystyle \frac{m{v}^{2}}{2kT}\right]}^{\tfrac{\nu }{1-\nu }}{{\rm{d}}}^{3}\vec{v}}{\displaystyle \int {\left[1-(1-\nu )\displaystyle \frac{m{v}^{2}}{2kT}\right]}^{\tfrac{\nu }{1-\nu }}{{\rm{d}}}^{3}\vec{v}}\\ \,=\displaystyle \frac{3}{2+3(1-\nu )}kT,\end{array}\end{eqnarray}$which yields the gas pressure$\begin{eqnarray}P=\displaystyle \frac{1}{3}n(\vec{{\rm{r}}})m\overline{{v}^{2}}=\displaystyle \frac{2}{2+3(1-\nu )}n(\vec{{\rm{r}}})kT.\end{eqnarray}$The polytropic index is [28]$\begin{eqnarray}n=\displaystyle \frac{1}{1-\nu }+\displaystyle \frac{1}{2}.\end{eqnarray}$The transformation (equation (58)) neither unifies the polytropic indexes (equations (59) and (63)) nor the pressure expressions (equations (60) and (62)).
The foregoing differences in gas pressures and polytropic indexes are fundamentally derived from the fact that the ν-average of (45) is adopted in the series of (62) and (63), whereas the following average is employed in the series of (59) and (60) [27]:$\begin{eqnarray}\overline{{\varepsilon }_{k}}=\displaystyle \int {\varepsilon }_{k}{f}_{N\delta }{{\rm{d}}}^{3}\vec{v}.\end{eqnarray}$Based on the comparison between (61) and (64), the following escort distribution is introduced:$\begin{eqnarray}{f}_{N\delta }=N\displaystyle \frac{{({f}_{N\nu })}^{\nu }}{\displaystyle \int {({f}_{N\nu })}^{\nu }{{\rm{d}}}^{3}\vec{v}}=N\displaystyle \frac{{\left[1-(1-\nu )\displaystyle \frac{m{v}^{2}}{2kT}\right]}^{\tfrac{\nu }{1-\nu }}}{\displaystyle \int {\left[1-(1-\nu )\displaystyle \frac{m{v}^{2}}{2kT}\right]}^{\tfrac{\nu }{1-\nu }}{{\rm{d}}}^{3}\vec{v}}.\end{eqnarray}$Therefore, to eliminate the variations between the two nonextensive parameters above, the following single-parameter transformation is proposed:$\begin{eqnarray}\delta =\displaystyle \frac{1}{\nu }.\end{eqnarray}$This relationship is similar to the multiplicative duality assumption, which is employed to establish a dual description formalism of Tsallis statistics, as reported in the literature [36–38]. By applying the above parameter transformation (equation (66)), indexes (59) and (63) are evidently perfectly unified. Moreover, with the relationships$\begin{eqnarray}\nu k^{\prime} =k,\,T^{\prime} =T,\end{eqnarray}$the pressure expressions (60) and (62) are also unified. In addition, as an alternative to (66), by applying (67) and$\begin{eqnarray}\begin{array}{l}{B}_{2-\delta }=\displaystyle \frac{N}{\displaystyle \int {\left[1-(1-\nu )\displaystyle \frac{m{v}^{2}}{2kT}\right]}^{\tfrac{\nu }{1-\nu }}{{\rm{d}}}^{3}\vec{v}}\\ \,=\displaystyle \frac{N}{\displaystyle \int {\left[1-(\delta -1)\displaystyle \frac{m{v}^{2}}{2k^{\prime} T}\right]}^{\tfrac{1}{\delta -1}}{{\rm{d}}}^{3}\vec{v}},\end{array}\end{eqnarray}$the two distribution functions, (57) and (65), are found to be completely identical.
The following proves the transformation invariance of Tsallis entropy. First, the ν-form of Tsallis entropy in (53) is introduced:$\begin{eqnarray}{S}_{\nu }=-k\displaystyle \int {{f}_{N\nu }}^{\nu }{\mathrm{ln}}_{\nu }{f}_{N\nu }{\rm{d}}\tau .\end{eqnarray}$Based on the relationship$\begin{eqnarray}\displaystyle \int {({f}_{N\nu })}^{\nu }{\rm{d}}\tau ={N}^{\nu }{c}_{\nu },\end{eqnarray}$and the escort definition in (65), the following is derived:$\begin{eqnarray}{S}_{\nu }=k\delta \displaystyle \frac{{N}^{1-\delta }{{c}_{\nu }}^{\delta }\displaystyle \int {({f}_{N\delta })}^{\delta }{\rm{d}}\tau -{c}_{\nu }{N}^{\nu }}{1-\delta }.\end{eqnarray}$By introducing the relationships$\begin{eqnarray}k\delta =k^{\prime} ,\,T^{\prime} =T,\end{eqnarray}$the following is obtained$\begin{eqnarray}{S}_{\nu }={{c}_{v}}^{\delta }{N}^{1-\delta }{S}_{\delta }+k^{\prime} \displaystyle \frac{{{c}_{v}}^{\delta }{N}^{2-\delta }-{c}_{\nu }{N}^{\nu }}{1-\delta }.\end{eqnarray}$Although an additional coefficient exists, Sν and Sδ have the same mathematical form. Furthermore, the second term on the right-hand side of (73) can be proved to vanish when ν→1. This indicates that the Tsallis entropy is invariant with the parameter transformation (equation (66)); that is, the two entropies, Sν and Sδ, are identical in physics. In the representation of Sν, the parameter inversion problem apparently remains, whereas this problem naturally disappears in the representation of Sδ.
In the next section, a parameter transformation series is presented to afford a more thorough discussion regarding self-consistency in the NSM theory.
5. Another parameter transformation series
Consider another parameter transformation series. First, the invariance of the Tsallis entropy transformation in the statistical ensemble is proved. For this, the following integral form of q-entropy is employed:$\begin{eqnarray}{S}_{q}=k\displaystyle \frac{\displaystyle \int {F}^{q}{{\rm{d}}}^{N}\tau -1}{1-q}.\end{eqnarray}$To the above, introduce the escort distribution,$\begin{eqnarray}\rho =\displaystyle \frac{{F}^{q}}{\displaystyle \int {F}^{q}{{\rm{d}}}^{N}\tau }=\displaystyle \frac{{F}^{q}}{{c}_{q}}.\end{eqnarray}$And the parameter transformation,$\begin{eqnarray}q=\displaystyle \frac{1}{\eta }.\end{eqnarray}$Accordingly, the following is derived:$\begin{eqnarray}{S}_{q}=k\eta \displaystyle \frac{{c}_{q}-{{c}_{q}}^{\eta }\displaystyle \int {\rho }^{\eta }{{\rm{d}}}^{N}\tau }{\eta -1}=k\eta {{c}_{q}}^{\eta }\displaystyle \frac{\displaystyle \int {\rho }^{\eta }{{\rm{d}}}^{N}\tau -{{c}_{q}}^{1-\eta }}{1-\eta }.\end{eqnarray}$As a recourse to the definition$\begin{eqnarray}{S}_{\eta }=k\displaystyle \frac{\displaystyle \int {\rho }^{\eta }{{\rm{d}}}^{N}\tau -1}{1-\eta },\end{eqnarray}$the following is found:$\begin{eqnarray}{S}_{q}=\eta {{c}_{q}}^{\eta }{S}_{\eta }-\eta k{{c}_{q}}^{\eta }{\mathrm{ln}}_{\eta }{c}_{q}.\end{eqnarray}$This relationship (equation (79)) also shows that the Tsallis entropy for the statistical ensemble is invariant with the parameter transformation (equation (76)).
According to (75), the internal energy of the system can be defined as$\begin{eqnarray}{U}^{(1)}\equiv \displaystyle \int \rho E{{\rm{d}}}^{N}\tau =\displaystyle \frac{\displaystyle \int {F}^{q}E{{\rm{d}}}^{N}\tau }{\displaystyle \int {F}^{q}{{\rm{d}}}^{N}\tau }={U}_{q}^{(3)}.\end{eqnarray}$This indicates the unification of the first and third choices of the internal energy constraints [15]. In addition, if the continuous form of (16) is employed, then the distribution function, ρ, is$\begin{eqnarray}\begin{array}{l}\rho =\displaystyle \frac{{F}^{q}}{{c}_{q}}=\displaystyle \frac{1}{{c}_{q}{[{\bar{Z}}_{q}^{(3)}]}^{q}}{\left[1-(1-q)\displaystyle \frac{(E-{U}_{q}^{(3)})}{k{T}_{q}}\right]}^{\tfrac{q}{1-q}}\\ \,=\displaystyle \frac{1}{{c}_{q}{[{\bar{Z}}_{q}^{(3)}]}^{q}}{\left[1-\displaystyle \frac{(\eta -1)}{\eta }\displaystyle \frac{(E-{U}^{(1)})}{k{T}_{q}}\right]}^{\tfrac{1}{\eta -1}}.\end{array}\end{eqnarray}$In contrast, by considering (23) and (16), the following is derived:$\begin{eqnarray}\begin{array}{l}{c}_{q}=\displaystyle \frac{1}{{[{\bar{Z}}_{q}^{(3)}]}^{q}}\displaystyle \int {\left[1-(1-q)\displaystyle \frac{(E-{U}_{q}^{(3)})}{k{T}_{q}}\right]}^{\tfrac{q}{1-q}}{{\rm{d}}}^{N}\tau \\ \,=\,\displaystyle \frac{1}{{[{\bar{Z}}_{q}^{(3)}]}^{q}}\displaystyle \int {\left[1-\displaystyle \frac{(\eta -1)}{\eta }\displaystyle \frac{(E-{U}^{(1)})}{k{T}_{q}}\right]}^{\tfrac{1}{\eta -1}}{{\rm{d}}}^{N}\tau \\ \,=\,{[{\bar{Z}}_{q}^{(3)}]}^{-q}{\bar{Z}}_{\eta }^{(1)}.\end{array}\end{eqnarray}$Substituting (82) into (81) yields$\begin{eqnarray}\rho =\displaystyle \frac{1}{{\bar{Z}}_{\eta }^{(1)}}{\left[1-\displaystyle \frac{(\eta -1)}{\eta }\displaystyle \frac{(E-{U}^{(1)})}{k{T}_{q}}\right]}^{\tfrac{1}{\eta -1}}.\end{eqnarray}$In the above distribution, with the definition of the new distribution and new internal energy, the so-called physical temperature must be re-explained. For this, the combination of (82) and (20) yields$\begin{eqnarray}{\bar{Z}}_{\eta }^{(1)}=\displaystyle \int {\left[1-\displaystyle \frac{(\eta -1)}{\eta }\displaystyle \frac{(E-{U}^{(1)})}{k{\rm{\Theta }}}\right]}^{\tfrac{1}{\eta -1}}{{\rm{d}}}^{N}\tau ={\bar{Z}}_{q}^{(3)},\end{eqnarray}$where the physical temperature is represented by an undetermined parameter, Θ. In addition,$\begin{eqnarray}\begin{array}{l}{c}_{\eta }\equiv \displaystyle \int {\rho }^{\eta }{{\rm{d}}}^{N}\tau =\displaystyle \frac{1}{{[{\bar{Z}}_{\eta }^{(1)}]}^{\eta }}\\ \,\times \displaystyle \int {\left[1-\displaystyle \frac{(\eta -1)}{\eta }\displaystyle \frac{(E-{U}^{(1)})}{k{\rm{\Theta }}}\right]}^{\tfrac{\eta }{\eta -1}}{{\rm{d}}}^{N}\tau \\ \,=\,{[{\bar{Z}}_{\eta }^{(1)}]}^{-\eta }{\bar{Z}}_{q}^{(3)}={[{\bar{Z}}_{\eta }^{(1)}]}^{1-\eta }.\end{array}\end{eqnarray}$Therefore, the following exists:$\begin{eqnarray}{S}_{\eta }=k\displaystyle \frac{{[{\bar{Z}}_{\eta }^{(1)}]}^{1-\eta }-1}{1-\eta }=k{\mathrm{ln}}_{\eta }{\bar{Z}}_{\eta }^{(1)}.\end{eqnarray}$Moreover,$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{\partial {S}_{\eta }}{\partial {U}^{(1)}}=k{[{\bar{Z}}_{\eta }^{(1)}]}^{-\eta }\displaystyle \frac{\partial }{\partial {U}^{(1)}}\\ \,\times \displaystyle \int {\left[1-\displaystyle \frac{(\eta -1)}{\eta }\displaystyle \frac{(E-{U}^{(1)})}{k{\rm{\Theta }}}\right]}^{\tfrac{\eta }{\eta -1}}{{\rm{d}}}^{N}\tau \\ \,=\,k{[{\bar{Z}}_{\eta }^{(1)}]}^{1-\eta }\displaystyle \frac{1}{k{\rm{\Theta }}}=\displaystyle \frac{{[{\bar{Z}}_{\eta }^{(1)}]}^{1-\eta }}{{\rm{\Theta }}}.\end{array}\end{eqnarray}$If the similarity of the thermodynamic relationship to (22) is imposed, then$\begin{eqnarray}\displaystyle \frac{\partial {S}_{\eta }}{\partial {U}^{(1)}}=\displaystyle \frac{{\rm{1}}}{T},\end{eqnarray}$and the following exists:$\begin{eqnarray}{\rm{\Theta }}=T{c}_{\eta }\equiv {T}_{\eta }.\end{eqnarray}$Thus, the self-consistent distribution in the present representation is as follows:$\begin{eqnarray}\rho =\displaystyle \frac{1}{{\bar{Z}}_{\eta }^{(1)}}{\left[1-\displaystyle \frac{(\eta -1)}{\eta }\displaystyle \frac{(E-{U}^{(1)})}{k{T}_{\eta }}\right]}^{\tfrac{1}{\eta -1}}.\end{eqnarray}$
The above distribution function can also be deduced by the standard maximization entropy procedure. To be specific, the following function is constructed:$\begin{eqnarray}\begin{array}{l}F\left\{\rho \right\}=\displaystyle \frac{\displaystyle \int {\rho }^{\eta }{{\rm{d}}}^{N}\tau -\eta \displaystyle \int \rho {{\rm{d}}}^{N}\tau }{1-\eta }-1-\alpha \left(\displaystyle \int \rho {{\rm{d}}}^{N}\tau -1\right)\\ \,-\beta \displaystyle \int \rho (E-{U}^{(1)}){{\rm{d}}}^{N}\tau ,\end{array}\end{eqnarray}$where β (given by β=1/kT) is the ordinary Lagrange multiplier. The maximum of (91) can easily be found to exactly lead to (90) using the following definitions:$\begin{eqnarray}{\bar{Z}}_{\eta }^{(1)}=\displaystyle \int {\left[1-\displaystyle \frac{(\eta -1)}{\eta }\beta ^{\prime} (E-{U}^{(1)})\right]}^{\tfrac{1}{\eta -1}}{{\rm{d}}}^{N}\tau ,\end{eqnarray}$$\begin{eqnarray}\beta ^{\prime} =\displaystyle \frac{\beta }{1+(1-\eta )\alpha }\equiv \displaystyle \frac{1}{k{T}_{\eta }}.\end{eqnarray}$Similar to (16), the distribution in (90) is also self-referential, which is inapplicable to real systems. Accordingly, a new partition function is defined:$\begin{eqnarray}{[{\bar{Z}}_{\eta }^{(1)}]}^{1-\eta }\equiv {[{Z}_{\eta }^{(1)}]}^{1-\eta }\left[1-\displaystyle \frac{(\eta -1)}{\eta }\displaystyle \frac{\beta }{{c}_{\eta }}{U}^{(1)}\right].\end{eqnarray}$Therefore, (90) transforms into$\begin{eqnarray}\begin{array}{l}\rho =\displaystyle \frac{1}{{Z}_{\eta }^{(1)}}{\left[1-\displaystyle \frac{(\eta -1)}{\eta }\displaystyle \frac{{U}^{(1)}}{k{T}_{\eta }}\right]}^{\tfrac{1}{\eta -1}}\\ \,\times \,{\left[1-\displaystyle \frac{(\eta -1)}{\eta }\displaystyle \frac{(E-{U}^{(1)})}{k{T}_{\eta }}\right]}^{\tfrac{1}{\eta -1}}\\ \,=\,\displaystyle \frac{1}{{Z}_{\eta }^{(1)}}\left[1-\displaystyle \frac{(\eta -1)}{\eta }\displaystyle \frac{E}{k{T}_{\eta }}\right.\\ \,+\,{\left.{\left[\displaystyle \frac{(\eta -1)}{\eta }\displaystyle \frac{{U}^{(1)}}{k{T}_{\eta }}\right]}^{2}\displaystyle \frac{(E-{U}^{(1)})}{{U}^{(1)}}\right]}^{\tfrac{1}{\eta -1}}\\ \,\approx \,\displaystyle \frac{1}{{Z}_{\eta }^{(1)}}{\left[1-\displaystyle \frac{(\eta -1)}{\eta }\displaystyle \frac{E}{k{T}_{\eta }}\right]}^{\tfrac{1}{\eta -1}}.\end{array}\end{eqnarray}$Moreover, with the approximation$\begin{eqnarray}\displaystyle \frac{{[{Z}_{\eta }^{(1)}]}^{1-\eta }}{\eta }\approx {c}_{\eta },\end{eqnarray}$(94) leads to$\begin{eqnarray}{[{\bar{Z}}_{\eta }^{(1)}]}^{1-\eta }={[{Z}_{\eta }^{(1)}]}^{1-\eta }+(1-\eta )\beta {U}^{(1)}.\end{eqnarray}$This relationship is ordinarily utilized to define the appropriate freedom energy [15].
The subsequent discussions present the available distribution function (equation (95)). To deduce the single-particle distribution function, the assumptions that E=Nϵ1 and ϵ1 is the total energy of a specific particle are retained. With the foregoing, the second parameter transformation in this series is presented as$\begin{eqnarray}(\eta -1)N=\sigma -1.\end{eqnarray}$Furthermore,$\begin{eqnarray}\begin{array}{l}{Z}_{\eta }^{(1)}=\displaystyle \int {\left[1-\displaystyle \frac{(\eta -1)}{\eta }\displaystyle \frac{N\varepsilon }{k{T}_{\eta }}\right]}^{\tfrac{1}{\eta -1}}{{\rm{d}}}^{N}\tau \\ \,=\displaystyle \int {\left[1-\displaystyle \frac{(\sigma -1)}{1+(\sigma -1)/N}\displaystyle \frac{\varepsilon }{k{T}_{\eta }}\right]}^{\tfrac{N}{\sigma -1}}{{\rm{d}}}^{N}\tau .\end{array}\end{eqnarray}$In experiments, distinguishing between (36) and (98) is difficult. Hence, the following is reasonable: regardless of whether (Σ−1) is positive or negative, its absolute value is sufficiently small. Accordingly, in the limit of a large number of particles (N), (Σ−1)/N→0. This means that$\begin{eqnarray}{Z}_{\eta }^{(1)}={Z}_{\sigma }^{N},\end{eqnarray}$where$\begin{eqnarray}{Z}_{\sigma }=\displaystyle \int {\left[1-(\sigma -1)\displaystyle \frac{\varepsilon }{k{T}_{\eta }}\right]}^{\tfrac{1}{\sigma -1}}{\rm{d}}\tau .\end{eqnarray}$Therefore, the single-particle distribution function is$\begin{eqnarray}{f}_{\sigma }=\displaystyle \frac{1}{{Z}_{\sigma }}{\left[1-(\sigma -1)\displaystyle \frac{\varepsilon }{k{T}_{\eta }}\right]}^{\tfrac{1}{\sigma -1}}.\end{eqnarray}$The following is also derived:$\begin{eqnarray}\rho =\displaystyle \frac{1}{{Z}_{\eta }^{(1)}}{\left[1-\displaystyle \frac{(\eta -1)}{\eta }\displaystyle \frac{E}{k{T}_{\eta }}\right]}^{\tfrac{1}{\eta -1}}={f}_{\sigma }^{N}.\end{eqnarray}$
In addition,$\begin{eqnarray}\begin{array}{l}{c}_{\eta }\equiv \displaystyle \int {\rho }^{\eta }{{\rm{d}}}^{N}\tau =\displaystyle \int {f}_{\sigma }^{\eta N}{{\rm{d}}}^{N}\tau =\displaystyle \int {f}_{\sigma }^{N-1+\sigma }{{\rm{d}}}^{N}\tau \\ \,=\displaystyle \int {f}_{\sigma }^{\sigma }{\rm{d}}\tau \equiv {c}_{\sigma }.\end{array}\end{eqnarray}$This means that, similar to (42), the η-Tsallis factor of the system is identical to Σ-one of a single particle in the transformation (i.e. (98)); thus, Tη=TΣ. Moreover, the following is found:$\begin{eqnarray}{S}_{\eta }=k\displaystyle \frac{\displaystyle \int {\rho }^{\eta }{{\rm{d}}}^{N}\tau -1}{1-\eta }=Nk\displaystyle \frac{\displaystyle \int {f}_{\sigma }^{\sigma }{\rm{d}}\tau -1}{1-\sigma }.\end{eqnarray}$Similarly, the distribution function of the gas system is defined as$\begin{eqnarray}{f}_{N\sigma }=\displaystyle \frac{N}{{Z}_{\sigma }}{\left[1-(\sigma -1)\displaystyle \frac{\varepsilon }{k{T}_{\sigma }}\right]}^{\tfrac{1}{\sigma -1}},\end{eqnarray}$where the number of particles (N) is employed for normalization. Then, in view of (46), the following is derived:$\begin{eqnarray}\begin{array}{l}{f}_{N\sigma }=\displaystyle \frac{N}{{Z}_{\sigma }}{\left[1-(\sigma -1)\displaystyle \frac{m(\varphi -{\varphi }_{0})}{k{T}_{\sigma }}\right]}^{\tfrac{1}{\sigma -1}}\left[1\right.\\ \,-{\left.(\sigma -1)\displaystyle \frac{m{v}^{2}/2}{k{T}_{\sigma }-(\sigma -1)m(\varphi -{\varphi }_{0})}\right]}^{\tfrac{1}{\sigma -1}}.\end{array}\end{eqnarray}$The introduction of the following relationships,$\begin{eqnarray}\begin{array}{l}{B}_{2-\sigma }=\displaystyle \frac{N}{{Z}_{\sigma }}{\left[1-(\sigma -1)\displaystyle \frac{m(\varphi -{\varphi }_{0})}{k{T}_{\sigma }}\right]}^{\tfrac{1}{\sigma -1}},\\ k{T}_{\sigma }-(\sigma -1)m(\varphi -{\varphi }_{0})=kT,\end{array}\end{eqnarray}$yields$\begin{eqnarray}{f}_{N\sigma }={B}_{2-\sigma }{\left[1-(\sigma -1)\displaystyle \frac{m{v}^{2}/2}{kT}\right]}^{\tfrac{1}{\sigma -1}},\end{eqnarray}$where B2−Σ is given by (50) and (51) by setting 2−Σ=ν. Then, Σ-entropy can be written as$\begin{eqnarray}{S}_{\sigma }=k\displaystyle \frac{\displaystyle \int {f}_{N\sigma }^{\sigma }{\rm{d}}\tau -\displaystyle \int {f}_{N\sigma }{\rm{d}}\tau }{1-\sigma }=-k\displaystyle \int {f}_{N\sigma }^{\sigma }{\mathrm{ln}}_{\sigma }{f}_{N\sigma }{\rm{d}}\tau .\end{eqnarray}$Based on (110), the distribution function (equation (109)) can evidently be re-deduced following the generalized H theorem procedure [24]; moreover, the parameter inversion naturally disappears.
Accordingly, the following is facilely derived:$\begin{eqnarray}{S}_{\eta }={S}_{\sigma }+{c}_{\sigma }{N}^{\sigma }k{\mathrm{ln}}_{\sigma }N.\end{eqnarray}$Furthermore, this relationship (equation (111)) indicates the equivalence of the two entropies, Sη and SΣ.
In experiments, the two distributions, (57) and (109), are indistinguishable; hence, they can be confirmed as equivalent. Consequently, δ-entropy and Σ-entropy are also equivalent.
6. Certain applications
6.1. Entropy of box-confined gas
When N particles of gas consisting of free molecules is confined in a box of volume V, the total thermodynamic probability is proportional to VN. For this system, the Tsallis q-entropy is$\begin{eqnarray}{S}_{q}=k\displaystyle \frac{{\left(\tfrac{V}{{V}_{0}}\right)}^{N(1-q)}-1}{1-q},\end{eqnarray}$where V0 is the least volume that a gas molecule occupies during the average time interval between two collisions that must be related to the average free path of molecules.
By considering the parameter transformation (equation (30)), the following is obtained:$\begin{eqnarray}{S}_{\nu }=Nk\displaystyle \frac{{\left(\tfrac{V}{{V}_{0}}\right)}^{1-\nu }-1}{1-\nu }.\end{eqnarray}$If parameter ν is maintained to be irrelevant to the number of particles, then the above result shows that the Tsallis entropy is extensive, although it is nonadditive.
6.2. Gas pressure from statistical ensemble to molecular dynamics
By considering the gas pressure for nonextensive ideal gas based on the ensemble theory [33], the following is derived:$\begin{eqnarray}P=\displaystyle \frac{NkT{({Z}_{q})}^{1-q}}{V}=nkT{({Z}_{q})}^{1-q},\end{eqnarray}$where n represents the density of the number of particles, and Zq is the ensemble partition function, which is given by [33]$\begin{eqnarray}{Z}_{q}=\displaystyle \frac{{\rm{\Gamma }}\left(\displaystyle \frac{2-q}{1-q}\right)}{{\rm{\Gamma }}\left(\displaystyle \frac{2-q}{1-q}+\displaystyle \frac{3N}{2}\right)}\displaystyle \frac{{V}^{N}}{N!{h}^{3N}}{\left[\displaystyle \frac{2\pi mk{T}_{q}}{(1-q)}\right]}^{3N/2}.\end{eqnarray}$For convenience, the single-particle partition function is defined as$\begin{eqnarray}{Z}_{1}=b(N,q)V{[k{T}_{q}]}^{3/2},\end{eqnarray}$where b is an indefinite constant related to the number of particles and nonextensive parameter.
Because of the approximation relationship$\begin{eqnarray}{Z}_{q}\approx {{Z}_{1}}^{N},\end{eqnarray}$the following is obtained:$\begin{eqnarray}P=nkT{({Z}_{1})}^{1-\nu }.\end{eqnarray}$In the above equation, the pressure differs from that in (62), which is calculated in molecular dynamics. Having derived the approximate relationship (i.e. (117)), the pressure is recalculated based exactly on definition (45), which yields the average molecular kinetic energy, as follows:$\begin{eqnarray}\overline{\displaystyle \frac{{\rm{1}}}{{\rm{2}}}m{v}^{2}}=\displaystyle \frac{\displaystyle \int \displaystyle \frac{{\rm{1}}}{{\rm{2}}}m{v}^{2}{f}^{\nu }{\rm{d}}\tau }{\displaystyle \int {f}^{\nu }{\rm{d}}\tau }.\end{eqnarray}$In the foregoing, the distribution function adopts the form of (39).
It can be proved that (appendix)$\begin{eqnarray}\displaystyle \frac{{\rm{1}}}{{\rm{2}}}\overline{m{v}^{2}}=\displaystyle \frac{3}{2}k{T}_{\nu }-\displaystyle \frac{3}{2}(1-\nu )\bar{\varepsilon }.\end{eqnarray}$In view of definition (38), the following can be proved:$\begin{eqnarray}{{Z}_{\nu }}^{1-\nu }={c}_{\nu }\left[1-\displaystyle \frac{1-\nu }{k{T}_{\nu }}\bar{\varepsilon }\right].\end{eqnarray}$Substituting the foregoing into (120) yields$\begin{eqnarray}P=\displaystyle \frac{1}{3}n\overline{m{v}^{2}}=nkT{{Z}_{\nu }}^{1-\nu },\end{eqnarray}$which is considerably similar to (118).
Interestingly, the combination of these two equations, (121) and (122), leads to$\begin{eqnarray}P=nk{T}_{\nu }-n(1-\nu )\bar{\varepsilon }.\end{eqnarray}$On the right-hand side of (123), the second term is proportional to the energy density of the system. This formula is applicable to all complex systems and may be useful in cosmology.
Furthermore, for self-gravitating gas, the total molecular energy consists of the gravitational potential and kinetic energies; in this regard, equation (46) is available. If the average of (119) is confined in the local gas region, the following is derived based on (40):$\begin{eqnarray}\bar{\varepsilon }=\displaystyle \frac{{\rm{1}}}{{\rm{2}}}\overline{m{v}^{2}}+m(\varphi -{\varphi }_{0}).\end{eqnarray}$Substituting (124) into (120) and considering (46) [39, 40] yields$\begin{eqnarray}\displaystyle \frac{{\rm{1}}}{{\rm{2}}}\overline{m{v}^{2}}=\displaystyle \frac{3}{2}kT-\displaystyle \frac{3}{2}(1-\nu )\displaystyle \frac{{\rm{1}}}{{\rm{2}}}\overline{m{v}^{2}}.\end{eqnarray}$Therefore,$\begin{eqnarray}P=\displaystyle \frac{1}{3}n\overline{m{v}^{2}}=\displaystyle \frac{2}{2+3(1-\nu )}nkT,\end{eqnarray}$which is completely identical to (62). This suggests that the pressure formula (equation (126)) is generally applicable to self-gravitating gaseous systems.
6.3. Statistical ensemble interpretation of gravitational heat capacity
Note that equation (120) is entirely derived based on the discussion of statistical ensemble and parameter transformation. Let this equation be applied to a self-gravitating gaseous system in which the inside radiation energy (produced due to gravitational contraction or unclear fusion) cannot be ignored. Accordingly, the total energy can be written as$\begin{eqnarray}{E}_{t}=E+{E}_{r},\end{eqnarray}$where Et and Er are the total energy and total radiation energy inside the considered system, respectively; E is the sum of the total kinetic and relative potential energies of molecules in the self-gravitating system.
In general terms, the radiation fields in the spaces among molecules are in a statistical equilibrium with the thermal motion of particles [41]. For convenience, the following relationship is assumed:$\begin{eqnarray}{E}_{r}=\lambda E,\end{eqnarray}$where λ is determined by the statistical properties of the system. Considering the mean field approximation and based on the virial theorem of a pure gaseous self-gravitating system, the energy, E, is$\begin{eqnarray}\begin{array}{l}E=-\displaystyle \frac{1}{2}N\overline{m{v}^{2}}-Nm{\varphi }_{0}=-\displaystyle \frac{3}{2}Nk{T}_{\nu }\\ \,+\displaystyle \frac{3}{2}(1-\nu )N\bar{\varepsilon }-Nm{\varphi }_{0}.\end{array}\end{eqnarray}$In contrast, ϵ is defined as the sum of kinetic energy, relative potential energy, and radiation energy stored in a single molecule [41]. Accordingly, the total energy of the self-gravitating system is$\begin{eqnarray}{E}_{t}=N\bar{\varepsilon }=(1+\lambda )E.\end{eqnarray}$Ultimately, the following is derived:$\begin{eqnarray}E=-\displaystyle \frac{{\rm{1}}}{1-\tfrac{3}{2}(1-\nu )(1+\lambda )}\displaystyle \frac{3}{2}Nk{T}_{\nu }-\displaystyle \frac{Nm{\varphi }_{0}}{1-\tfrac{3}{2}(1-\nu )(1+\lambda )}.\end{eqnarray}$Then, the physical heat capacity is$\begin{eqnarray}{C}_{{\rm{VP}}}=\displaystyle \frac{{\rm{d}}E}{{\rm{d}}{T}_{\nu }}=-\displaystyle \frac{{\rm{1}}}{1-\tfrac{3}{2}(1-\nu )(1+\lambda )}\displaystyle \frac{3}{2}Nk.\end{eqnarray}$When λ=1, the foregoing is exactly the same as the gravitational heat capacity [39], originating from the molecular dynamics analysis. This confirms that the physical temperature, Tν, is equivalent to the gravitational temperature, Tg [39].
7. Conclusions and discussion
In summary, to reconcile the various branches of NSM and to establish the complete formalism of statistical mechanics with perfect self-consistency, two correlated series of parameter transformations are proposed. The first transformation series is q–ν–δ, where the q–ν transformation (equation (36)) constructs the link between the statistical ensemble and molecular dynamics, and the ν–δ transformation (equation (66)) resolves the parameter inversion problem in the generalized H theorem. The ν–δ transformation also perfectly verifies the unification of the pressure expressions (i.e. (60) and (62)) and the consistency of polytropic indexes (i.e. (59) and (63)).
The second transformation series is q–η–Σ, where the q–η transformation (equation (76)) unifies the first and third choices of energy constraints [15]. Moreover, the η–Σ transformation (equation (98)) harmonizes the statistical ensemble and molecular dynamics, where the Σ-entropy is compatible with the parameter inversion mechanism of the generalized H theorem.
To demonstrate the relationship between the two transformation series clearly, a parameter transformation map is shown in figure 1. In the figure, the Tsallis entropies are specified in different parameter formalisms. Along the longitudinal direction of the figure, the q–η and ν–δ transformations are performed in the statistical ensemble and molecular dynamics, respectively. Along the horizontal direction, the q–ν and η–Σ transformations employ the third and first choices of the average definition, respectively.
The foregoing demonstrates that the application of the parameter transformation technique establishes the self-consistent formalism of NSM. Interestingly, the Tsallis entropy remains invariant in its mathematical form in the parameter transformations. Moreover, parameter transformation can also be employed to resolve the factorization problem of the grand partition function in quantum statistics. For related works, refer to recent papers [42, 43], where the analytical and concise nonextensive quantum statistical formula was derived.
,1,2, Jiulin Du31School of Science, Henan Institute of Technology, Xinxiang 453003, 
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