Nonadditive Tsallis $q$-statistics has successfully been applied for a plethora of systems in natural sciences and other branches of knowledge. Nevertheless, its foundations have been severely criticised by some authors based on the standard additive Boltzmann-Gibbs approach thereby remaining a quite controversial subject. In order to clarify some polemical concepts, the distribution function for an ideal gas with a finite number of point particles and its $q$-index are analytically determined. The two-particle correlation function is also derived. The degree of correlation diminishes continuously with the growth of the number of particles. The ideal finite gas system is usually correlated, becomes less correlated when the number of particles grows, and is finally, fully uncorrelated when the molecular chaos regime is reached. It is also advocated that both approaches can be confronted through a careful kinetic spectroscopic experiment. The analytical results derived here suggest that Tsallis q-statistics may play a physical role more fundamental than usually discussed in the literature.
Qualms regarding "Non-extensive Hamilton systems follow Boltzmann's principle no Tsallis statistics--phase transitions, second law of Thermodynamics" by D.H.E. Gross, Physica A, 305 (2002) 99
Comment on ``Nonextensive hamiltonian systems follow Boltzmann's principle not Tsallis statistics-phase transition, second law of thermodynamics'' by Gross
Momentum distribution and correlation function of free particles in the Tsallis statistics using conventional expectation value and equilibrium temperature
The general formalisms of the $q$-dual statistics, the Boltzmann-Gibbs statistics, and three versions of the Tsallis statistics known as Tsallis-1, Tsallis-2, and Tsallis-3 statistics have been considered in the canonical ensemble. We have rigorously proved that the probability distribution of the Tsallis-1 statistics is invariant under the uniform energy spectrum translation at a fixed temperature. This invariance demonstrates that the formalism of the Tsallis-1 statistics is consistent with the fundamentals of the equilibrium statistical mechanics. The same results we have obtained for the p…
From the Tsallis unnormalized (or Tsallis-2) statistical mechanical formulation, Büyükkiliç {\it et al.} [Phys. Lett. A 197, 209 (1995)] derived the expressions for the single-particle distribution functions (known as the phenomenological Tsallis distributions) for particles obeying the Maxwell-Boltzmann, Bose-Einstein and the Fermi-Dirac statistics using the factorization approximation. In spite of the fact that this paper was published long time ago, its results are still extensively used in many fields of physics, and it is considered that it was this paper that established the connection b…
我们表明，Tsallis系综的幂律分布提供了一个力学模型的非广泛的平衡热力学小相互作用的哈密顿系统，即，利用玻尔兹曼的原始命名法，我们证明了它是一个精确的正交线。这意味着热微分允许平均动能的倒数作为积分因子。一个直接的结果是归一化函数的对数可以用熵来识别，而不是q变形的对数。注意到这种熵与Renyi熵而不是Tsallis熵一致，它是非加性的，当幂指数趋于无穷大时趋于标准正则熵，并且与[S. Abe et.等. Phys. Lett. A 281，126（2001）]。还证明了热微分也可以用非广延热力学中的拉格朗日乘子作为积分因子，而相关熵由普通非广延熵给出。在这项工作中提出的机械方法是完全一致的信息理论的方法的基础上最大化的Renyi熵。
We prove a proposition that the entropy of the system composed of finite $N$ molecules of ideal gas is the $q$-entropy or Havrda-Charvát-Tsallis entropy, which is also known as Tsallis entropy, with the entropic index $q=\frac{D(N-1)-4}{D(N-1)-2}$ in $D$-dimensional space. The indispensable infinity assumption used by Boltzmann and others in their derivation of entropy formulae is not involved in our derivation, therefore our derived formula is exact. The analogy of the $N$-body system brings us to obtain the entropic index of a combined system $q_C$ formed from subsystems having different ent…
We investigate the overdamped stochastic dynamics of a particle in an asymptotically flat external potential field, in contact with a thermal bath. For an infinite system size, the particles may escape the force field and diffuse freely at large length scales. The partition function diverges and hence the standard canonical ensemble fails. This is replaced with tools stemming from infinite ergodic theory. Boltzmann-Gibbs statistics, even though not normalized, still describes integrable observables, like energy and occupation times. The Boltzmann infinite density is derived heuristically using…
In this paper, we study the thermodynamics of quantum harmonic oscillator in the Tsallis framework and in the presence of a minimal length uncertainty. The existence of the minimal length is motivated by various theories such as string theory, loop quantum gravity, and black-hole physics. We analytically obtain the partition function, probability function, internal energy, and the specific heat capacity of the vibrational quantum system for $1<q<\frac{3}{2}$ and compare the results with those of Tsallis and Boltzmann-Gibbs statistics without the minimal length scale.
The statistical mechanics of a cloud of particles interacting via their gravitational potentials is an old problem which encounters some issues when the traditional Boltzmann-Gibbs statistics is applied. In this article, we consider the generalized statistics of Tsallis and analyze the statistical and thermodynamical implications for a self-gravitating gas, obtaining analytical and convergent expressions for the equation of state and specific heat in the canonical as well as microcanonical ensembles. Although our results are comparable in both ensembles, it turns out that only in the canonical…
We studied the effects of the nonextensivity on the phase transition for the system of small volume $V$ in the $ϕ^4$ theory in the Tsallis nonextensive statistics of entropic parameter $q$ and temperature $T$, when the deviation from the Boltzmann-Gibbs statistics, $|q-1|$, is small. We calculated the condensate and the mass to the order $q-1$ with the normalized $q$-expectation value under the massless free particle approximation. The following facts were found. The condensate $Φ$ divided by $v$, $Φ/v$, at $q$ is smaller than that at $q'$ for $q>q'$ as a function of $T_{\mathrm{ph}}/v$ which …
On the extended Kolmogorov-Nagumo information-entropy theory, the q -> 1/q duality and its possible implications for a non-extensive two dimensional Ising model
