A non-extensive statistical mechanical approach to define the equilibrium value function in the kinetics of voltage-gated ion channels

We present a model for the steady-state (or equilibrium) behaviors of the voltage-gated ion channels in cell membranes using the non-extensive or generalized statistical mechanics. The equilibrium value function in the gating kinetics of batrachotoxin-modified sodium channels from a squid optic nerve in planar bilayers are calculated for different values of entropic index (q) which characterizes the degree of non-extensivity of Tsallis’ entropy and the fractal structure of the channels. It is found that in the limit q→1, the results of calculation reduce to the results described by the well-known Boltzmann statistics or the extensive physics. For the non-extensive case (q≠1), a small deviation with respect to the Boltzmann curve which was observed in a great variety of physical systems occurred.     

非对易相空间中谐振子体系热力学性质的探讨

2000年以来, 有关非对易空间的各种物理问题一直是研究的热点, 并在量子力学、场论、凝聚态物理、天体物理等各领域中已被广泛地探讨. 采用统计物理方法讨论非对易效应对谐振子体系热力学性质的影响. 先以对易相空间中确定二维和三维谐振子的配分函数求出谐振子体系的热力学函数; 非对易相空间中的坐标和动量通过坐标-坐标和动量-动量之间的线性变换而以对易相空间中的坐标和动量来表示; 最终以非对易相空间中求出配分函数来讨论非对易效应对谐振子体系热力学性质的影响. 结果显示, 在非对易相空间中谐振子体系的配分函数和熵表达式均包含因非对易引起的修正项. 从分析结果得出如下结论: 非对易效应对谐振子的配分函数和熵函数等微观状态函数有一定的影响, 但对谐振子体系的内能、热容量等宏观热力学函数没有影响. 研究结果只是对应于满足玻尔兹曼统计的经典体系, 对于满足费米-狄拉克和玻色-爱因斯坦统计的量子体系需进一步推广研究.     

Non-extensive thermostatistics approach to metal melting entropy

A generalized Mott relation of metal melting entropy is derived by means of non-extensive solid and liquid quantum entropy that we calculate from grand partition functions of the localized ordered quantum solid and of the disordered quantum Boltzmann liquid. For each of the 18 elements considered, the entropic parameter _qm$q_m$, depending on particle correlations, is deduced such that a better agreement is obtained between calculated non-extensive metal melting entropy and available experimental data. The non-extensive entropic parameter makes the difference between normal and anomalous metals. Therefore, those elements not reported here should also belong to one of the two classes. Possible applications to condensed matter, Earth, and other solar planets seismology are mentioned.     

The stability of a non-extensive relativistic Fermi system

Based on the statistical theory of non-extensive relativity,and using theoretical analysis and numerical simulation,the non-extensive mechanical stability of ultra-relativistic free Fermi gas is investigated.The expressions of the stability conditions under high and low temperatures are given,and the mechanisms of the influences of temperature,ultrarelativistic effect,and non-extensive parameter q on stability are analysed.Our results show that at high temperature and under the condition of q < 1,the stability of a non-extensive system is weaker than that of an extensive system,and the relativistic effect reduces system stability as compared with a non-relativistic system.However,under the condition of q > 1,the stability of the non-extensive system is stronger than that of the extensive system,and the relativistic effect strengthens the system stability as compared with the non-relativistic system.In addition,under the condition of low temperature,the variation of the stability of the non-extensive system with temperature has a turning point.     

Thermodynamics and relativistic kinetic theory for <Emphasis Type="Italic">q</Emphasis>-generalized Bose–Einstein and Fermi–Dirac systems

The thermodynamics and covariant kinetic theory are elaborately investigated in a non-extensive environment considering the non-extensive generalization of Bose–Einstein (BE) and Fermi–Dirac (FD) statistics. Starting with Tsallis’ entropy formula, the fundamental principles of thermostatistics are established for a grand canonical system having q-generalized BE/FD degrees of freedom. Many particle kinetic theory is set up in terms of the relativistic transport equation with q-generalized Uehling–Uhlenbeck collision term. The conservation laws are realized in terms of appropriate moments of the transport equation. The thermodynamic quantities are obtained in a weak non-extensive environment for a massive pion–nucleon and a massless quark–gluon system with non-zero baryon chemical potential. In order to get an estimate of the impact of non-extensivity on the system dynamics, the q-modified Debye mass and hence the q-modified effective coupling are estimated for a quark–gluon system.     

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.     

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.     

Thermal entanglement of Hubbard dimers in the nonextensive statistics

The thermal entanglement of the Hubbard dimer (two-site Hubbard model) has been studied with the nonextensive statistics. We have calculated the auto-correlation (O_q), pair correlation (L_q), concurrence (Γ_q) and conditional entropy (R_q) as functions of entropic index q and the temperature T. The thermal entanglement is shown to considerably depend on the entropic index. For q<1.0, the threshold temperature where Γ_q vanishes or R_q changes its sign is more increased and the entanglement may survive at higher temperatures than for q=1.0. Relations among L_q, Γ_q and R_q are investigated. The physical meaning of the entropic index q is discussed with the microcanonical and superstatistical approaches. The nonextensive statistics is applied also to Heisenberg dimers.     

The Minimal Length Uncertainty and the Nonextensive Thermodynamics

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.     

Is the expression H=1/(3-q) valid for real financial data?

Non-extensive thermodynamics is one of the most intriguing physics new frontiers. A large number of researchers have been successfully finding connections between the new concepts introduced by this new field and other complex systems already presented. In particular, Borland [Phys. Rev. E 57 (1998) 6634–6642] has introduced a very interesting relation between the entropic index q that arises in the non-extensive entropy and the well-known Hurst exponent H used to measure long-range dependence in complex systems. In this paper, we provide statistical support to Borland results and test the validity of these results in real financial data.     

On the non-extensive nature of the isothermal depolarization relaxation currents in cement mortars

The study of isothermal depolarization current relaxation during the cement hardening process follows a generalized exponential function, which explicitly introduces hierarchically constrained dynamics and macroscopic interactions. The interactions are associated to the non-extensive entropy parameter q, which varies during hardening evolution. Combining ideas of Levy and Tsallis statistics we argue that a Levy walk type mechanism can organize the geometry of the cement mortar heterogeneous system. The estimation of non-extensive q parameter safe concludes that cement's hardening is a subextensive process.     

Thermodynamic non-additivity in disordered systems with extended phase space

Non-additivity effects in coupled dynamic-stochastic systems are investigated. It is shown that there is a mapping of the replica approach to disordered systems with finite replica indexn on Tsallis non-extensive statistics, if the average thermodynamic entropy of the dynamic subsystem differs from the information entropy for the probability distribution in the stochastic subsystem. The entropic indexq is determined by the entropy difference ΔS. In the case of incomplete information, the entropic indexq=1−n is shown to be related to the degree of lost information.     

Molecular constants of SiH3C ≡ CH and SiD3C ≡ CH

A normal co-ordinate analysis of silylacetylene and silylacetylene-d ₃ has been carried out following Wilson'sF-G matrix method. The potential energy constants obtained therefrom have been used to evaluate rotational distortion constants and mean square amplitudes of vibration for these molecules. Thermodynamic functions, such as heat content, free energy, entropy heat capacity for the ideal gaseous state at one atmosphere pressure and with the usual rigid rotor harmonic oscillator approximation, have also been calculated for 12 temperatures from 100°K to 1000°K.     

Dissipative contributions of internal multiplicative noise: I. mechanical oscillator

Langevin equations for closed systems with multiplicative fluctuations must also include appropriate dissipative terms that ensure eventual equilibration of the system. We consider an oscillator coupled to a heat bath and show that a particular nonlinear coupling to a harmonic heat bath leads to a fluctuating frequency and to nonlinear dissipative terms. We also analyze the effects of the multiplicative fluctuations and of the corresponding nonlinear dissipation on the temporal evolution of the average oscillator energy. We find that the rate of equilibration of this system can be significantly different from that of an oscillator with only additive fluctuations and linear dissipation.     

On non-extensive nature of thermal conductivity

In this paper we study non-extensive nature of thermal conductivity. It is observed that there is similarity between non-extensive entropic index and fractal dimension obtained for the silica aerogel thermal conductivity data at low temperature.      

Systematic analysis of pT -distributions in p + p collisions

A systematic analysis of transverse momentum distribution of hadrons produced in ultra-relativistic p + p collisions is presented. We investigate the effective temperature and the entropic index from the non-extensive thermodynamic theory of strong interaction. We conclude that the existence of a limiting effective temperature and of a limiting entropic index is in accordance with experimental data.     

Transition to sub-Planck structures through the superposition of q-oscillator stationary states

We investigate the superposition of four different quantum states based on the q-oscillator. These quantum states are expressed by means of Rogers-Szegö polynomials. We show that such a superposition has the properties of the quantum harmonic oscillator when q→1, and those of a compass state with the appearance of chessboard-type interference patterns when q→0.     

Coherent states and squeezed states of realq-deformed quantum oscillators

A detailed physical characterisation of the coherent states and squeezed states of a realq-deformed oscillator is attempted. The squeezing andq-squeezing behaviours are illustrated by three different model Hamiltonians, namely i) Batemann Hamiltonian ii) harmonic oscillator with time dependent mass and frequency and iii) a system with constant mass and time-dependent frequency.