Rigid rotators and diatomic molecules via Tsallis statistics

We obtain an analytic expression for the specific heat of a system of N rigid rotators exactly in the high temperature limit, and via a perturbative approach in the low temperature limit. We then evaluate the specific heat of a diatomic gas with both translational and rotational degrees of freedom, and conclude that there is a mixing between the translational and rotational degrees of freedom in nonextensive statistics.     

An insight to the nonextensive parameter in the actual gas

The ideal gas has been reinvestigated in the framework of Tsallis nonextensive statistics, which can be called nonextensive gas. According to the modified thermodynamic relationships, and applying the nonextensive gas model to analyze actual gas, the relationship between the nonextensive parameter and the second virial coefficient can be deduced. On the other hand, this coefficient can also be expressed as the integration of interaction potential between the molecules of actual gas. This indicates that the nonextensive parameter may be related to the interaction potential. Our further analysis to the relation seems to imply that the nonextensive parameter is irrelevant to the thermodynamic temperature of the gas.     

谐振势阱中旋转广义玻色系统的热力学性质

以非广延Tsallis统计理论为基础,导出了广义玻色-爱因斯坦统计分布表达式,并用其分别讨论了三维和二维谐振势阱约束的旋转广义玻色气体的热力学性质.结合系统粒子数、玻色-爱因斯坦凝聚(BEC)临界温度、基态粒子占据率和比热等物理量的解析表达式,分析了非广延参数和势阱旋转频率等因素对系统热力学性质的影响.     

Nonextensive statistics of the classical relativistic ideal gas

Investigating the canonical ensemble of a classical relativistic ideal gas in the Tsallis nonextensive framework we evaluate the specific heat in the extreme relativistic case in a closed form by directly employing the third constraint scenario. The canonical ensemble of N particles in D dimensions is well defined for the choice of the deformation parameter in the range . In the instance of a classical relativistic ideal gas with arbitrarily massive particles a perturbative scheme in the nonextensivity parameter (1−q) is developed by employing an infinite product expansion of the q-exponential, and a direct transformation of the internal energy from the second to the third constraint picture. All thermodynamic quantities may be uniformly evaluated to any desired perturbative order.     

Equilibrium statistical mechanics for incomplete nonextensive statistics

The incomplete nonextensive statistics in the canonical and microcanonical ensembles is explored in the general case and in a particular case for the ideal gas. By exact analytical results for the ideal gas it is shown that taking the thermodynamic limit, with z=q/(1−q) being an extensive variable of state, the incomplete nonextensive statistics satisfies the requirements of equilibrium thermodynamics. The thermodynamical potential of the statistical ensemble is a homogeneous function of the first degree of the extensive variables of state. In this case, the incomplete nonextensive statistics is equivalent to the usual Tsallis statistics. If z is an intensive variable of state, i.e. the entropic index q is a universal constant, the requirements of the equilibrium thermodynamics are violated.     

Creating kappa-like distributions from a Galton board

The Galton board is a classical device demonstrating geometrically the formation of normal distributions. In view of long-range interactions, governed by nonextensive statistics, the concept of the Galton board is extended providing numerically the corresponding power-law distributions. Within nonextensive statistics kappa-distributions, which are linked to q-Gaussians, are a consequence from entropy generalization. In this way the transition from normal distributions to kappa-like distributions (where kappa is the entropic index) is available within a Galton board concept where both the generalized distributions for positive and negative kappa-like values are reproduced. Based on two similar numerical approaches it is shown how positive kappa-like exponents are related to memory effects and negative kappa-like exponents to enhanced interactions of increased order in the system, as compared to the normal distribution case.     

Nonextensive quantum method for itinerant-electron ferromagnetism: Factorization approach

Magnetic and thermodynamical properties of itinerant-electron (metallic) ferromagnets described by the Hubbard model have been discussed with the use of the generalized Fermi-Dirac (GFD) distribution for nonextensive quantum systems. We have derived the GFD distribution within the superstatistics, which is equivalent to that obtained by the maximum-entropy method to the Tsallis entropy with the factorization approximation. By using the Hartree-Fock approximation to the electron-electron interaction in the Hubbard model, we have calculated magnetic moment, energy, specific heat and Curie-Weiss-type spin susceptibility, as functions of the temperature and entropic index q expressing the degree of the nonextensivity: q=1.0 corresponds to the Boltzmann-Gibbs statistics. It has been shown that by increasing the nonextensivity of |q−1|, the temperature dependence of magnetic moment becomes more significant and the low-temperature electronic specific heat is very much increased. This is attributed to enlarged Stoner excitations in the GFD distribution, which is elucidated by an analysis with the use of the generalized Sommerfeld expansion. We discuss the difference and similarity between the effects of the nonextensivity on metallic and insulating ferromagnets.     

Generalized statistics framework for rate distortion theory

Variational principles for the rate distortion (RD) theory in lossy compression are formulated within the ambit of the generalized nonextensive statistics of Tsallis, for values of the nonextensivity parameter satisfying 0<q<1 and q>1. Alternating minimization numerical schemes to evaluate the nonextensive RD function, are derived. Numerical simulations demonstrate the efficacy of generalized statistics RD models.     

Nonadditive conditional entropy and its significance for local realism

Based on the form invariance of the structures given by Khinchin's axiomatic foundations of information theory and the pseudoadditivity of the Tsallis entropy indexed by q, the concept of conditional entropy is generalized to the case of nonadditive (nonextensive) composite systems. The proposed nonadditive conditional entropy is classically nonnegative but can be negative in the quantum context, indicating its utility for characterizing quantum entanglement. A criterion deduced from it for separability of density matrices for validity of local realism is examined in detail by employing a bipartite spin-1/2 system. It is found that the strongest criterion is obtained in the limit q→∞.     

Nonextensive black-body distribution function and Einstein's coefficients A and B

The nonextensive statistics of a boson gas within the dilute gas approximation is applied to black-body radiation. Einstein's spontaneous emission coefficient A₂₁, stimulated emission coefficient B₂₁, and absorption coefficient B₁₂ are studied in connection with this new statistical mechanics. Relations including nonextensivity between these coefficients are obtained.     

Nonextensive effects on the phase structure of quantum hadrodynamics

In this Letter, we investigate nonextensive effects on phase transition in nuclear matter in the context Walecka many-body field theory. A difference is observed when the results calculated for the nuclear matter at vanishing baryon density is compared to those obtained through the standard Fermi-Dirac distribution. It is observed a dependence between the nonextensive parameter q and the coupling constants of the phase transition. A numerical relation for this thermodynamical dependence is also proposed.     

Nonextensive formalism and continuous Hamiltonian systems

A recurring question in nonequilibrium statistical mechanics is what deviation from standard statistical mechanics gives rise to non-Boltzmann behavior and to nonlinear response, which amounts to identifying the emergence of “statistics from dynamics” in systems out of equilibrium. Among several possible analytical developments which have been proposed, the idea of nonextensive statistics introduced by Tsallis about 20 years ago was to develop a statistical mechanical theory for systems out of equilibrium where the Boltzmann distribution no longer holds, and to generalize the Boltzmann entropy by a more general function Sq while maintaining the formalism of thermodynamics. From a phenomenological viewpoint, nonextensive statistics appeared to be of interest because maximization of the generalized entropy Sq yields the q-exponential distribution which has been successfully used to describe distributions observed in a large class of phenomena, in particular power law distributions for q>1. Here we re-examine the validity of the nonextensive formalism for continuous Hamiltonian systems. In particular we consider the q-ideal gas, a model system of quasi-particles where the effect of the interactions are included in the particle properties. On the basis of exact results for the q-ideal gas, we find that the theory is restricted to the range q<1, which raises the question of its formal validity range for continuous Hamiltonian systems.     

Direct numerical simulation methods of hypersonic flat-plate boundary layer in thermally perfect gas

High-temperature effects alter the physical and transport properties of air such as vibrational excitation in a thermally perfect gas,and this factor should be considered in order to compute the flow field correctly.Herein,for the thermally perfect gas,a simple method of direct numerical simulation on flat-plat boundary layer is put forward,using the equivalent specific heat ratio instead of constant specific heat ratio in the N-S equations and flux splitting form of a calorically perfect gas.The results calculated by the new method are consistent with that by solving the N-S equations of a thermally perfect gas directly.The mean flow has the similarity,and consistent to the corresponding Blasius solution,which confirms that satisfactory results can be obtained basing on the Blasius solution as the mean flow directly in stability analysis.The amplitude growth curve of small disturbance is introduced at the inlet by using direct numerical simulation,which is consistent with that obtained by linear stability theory.It verified that the equation established and the simulation method is correct.     

Exchange effect on the electronic specific heat

By improving the evaluation of the contribution from the first order exchange graphs reported recently by Tsai and Isihara, the electronic specific heat of an electron gas is evaluated to ordere ². It is shown that the specific heat contains a logarithmic term.     

Nonextensive statistical mechanics of ionic solutions

Classical mean-field Poisson–Boltzmann theory of ionic solutions is revisited in the theoretical framework of nonextensive Tsallis statistics. The nonextensive equivalent of Poisson–Boltzmann equation is formulated revisiting the statistical mechanics of liquids and the Debye–Hückel framework is shown to be valid for highly diluted solutions even under circumstances where nonextensive thermostatistics must be applied. The lowest order corrections associated to nonadditive effects are identified for both symmetric and asymmetric electrolytes and the behavior of the average electrostatic potential in a homogeneous system is analytically and numerically analyzed for various values of the complexity measurement nonextensive parameter q.     

The influence of fractality on the time evolution of the diffusion process

In the literature, the deviations from standard behaviors of the solutions of the kinetic equation and the analogous diffusion equation are put forward by investigations which are carried out in the frame of fractional mathematics and nonextensive physics. On the other hand, the physical origins of the order of derivative namely α in fractional mathematics and the entropy index q in nonextensive physics are a topic of interest in scientific media. In this study, the solutions of the diffusion equation which have been obtained in the framework of fractional mathematics and nonextensive physics are revised. The diffusion equation is solved by the cumulative diminuation/growth method which has been developed by two of the present authors and physical nature of the parameters α and q are enlightened in connection with fractality of space and the memory effect. It has been emphasized that the mathematical basis of deviations from standard behavior in the distribution functions could be established by fractional mathematics where as the physical mechanism could be revealed using the cumulative diminuation/growth method.     

Nonextensive statistical mechanics of q-bosons based on the q-deformed entropy

Based on the q ↔ q⁻¹ symmetric deformed entropy, we develop a general framework for nonextensive statistical mechanics of ensembles of q-deformed systems. Applying this doubly deformed formalism to q-bosons, a correction to the Planck law is evaluated in the weak deformation regime and its properties are discussed. It is found that at high temperature the dominant part of the correction comes from the deformation of the oscillator dynamics, whereas at low temperature the deformation of the entropy gives a leading contribution. This suggests the nonextensive approach to q-deformed ensembles might be important at low temperature.