Tsallis Thermostatistics Research Articles

Application for Tsallis thermostatistics to half-width method used in thermoluminescence glow peaks in analysis of thermal desorption spectra

Several methods have been developed in the analysis of the thermoluminescence glow peaks, and in particular when applied to thermoluminescence (TL) dosimetry and TL dating of archaeological and geological minerals. Some of these methods are identical to those developed for the study of thermal desorption. Such a method is called half-width method where a shape factor is introduced. Recently TL decay and thermal desorption process have been studied using a nonextensive formalism, namely Tsallis thermostatistics. In the present work, we consider the half-width method because it is the easiest one to estimate the activation energy. Importantly, we give the generalized form of the activation energy for all order of kinetics. We also investigate the influence of the nonextensivity on the shape factor parameter and on the activation energy.

A nonextensive statistical approach to the kinetics of phase transformation

Some limitations of Johnson–Mehl–Avrami–Kolmogorov (JMAK) equation, which is used widely for describing kinetics of phase transformation, were demonstrated using probabilistic analysis and Monte Carlo simulations [Acta Mater. 48 (2000) 4217]. As well-known, JMAK equation predicts correctly the real transformed fraction only if the number of the growing nuclei in the controlled volume is large. However JMAK equation seems to fail, if the number of growing nuclei is small, no matter how large the volume of the controlled volume is. In this study, we propose a different equation for describing kinetics of phase transformation, using a nonextensive formalism, namely Tsallis thermostatistics which has been commonly employed to study the various physical systems for a decade.

Application for Tsallis thermostatistics to half-width method used in thermoluminescence glow peaks in analysis of thermal desorption spectra

Several methods have been developed in the analysis of the thermoluminescence glow peaks, and in particular when applied to thermoluminescence (TL) dosimetry and TL dating of archaeological and geological minerals. Some of these methods are identical to those developed for the study of thermal desorption. Such a method is called half-width method where a shape factor is introduced. Recently TL decay and thermal desorption process have been studied using a nonextensive formalism, namely Tsallis thermostatistics. In the present work, we consider the half-width method because it is the easiest one to estimate the activation energy. Importantly, we give the generalized form of the activation energy for all order of kinetics. We also investigate the influence of the nonextensivity on the shape factor parameter and on the activation energy.

On the dependence of the Navier–Stokes equations on the distribution of molecular velocities

In this work we introduce a completely general Chapman–Enskog procedure in which we divide the local distribution into an isotropic distribution with anisotropic corrections. We obtain a recursion relation on all integrals of the distribution function required in the derivation of the moment equations. We obtain the hydrodynamic equations in terms only of the first few moments of the isotropic part of an arbitrary local distribution function. The incompressible limit of the equations is completely independent of the form of the isotropic part of the distribution, whereas the energy equation in the compressible case contains an additional contribution to the heat flux. This additional term was also found by Boghosian [Braz. J. Phys. 29 (1999) 91] and by Potiguar and Costa [Physica A 303 (2002) 457] in the derivation of the Navier–Stokes equations for Tsallis thermostatistics, and is the only additional term allowed by the Curie principle.

Orientational ordering in the nematic phase of a thermotropic liquid crystal: a nonextensive approach

In this study, the dependence of fourth-rank order parameter on second-rank order parameter for para-azoxyanisole is investigated by using Tsallis statistics. Also, orientational distribution function is studied within Tsallis thermostatistics using the maximum entropy method, and the generalized distribution function is plotted for various values of the entropic index. The effects of the nonextensivity are illustrated for various values of the entropic index.

The effect of nonextensivity on the time evolution of the SU(1,1) coherent states driven by a damped harmonic oscillator

In this study, we have investigated the nonextensive effects on the time evolution of Kanai–Caldirola oscillator that is well-known in quantum optics. For this purpose, the Hamiltonian of Kanai–Caldirola oscillator has been generalized within Tsallis thermostatistics. By using SU(1,1) coherent states, the classical equation of motion of generalized Kanai–Caldirola oscillator has been obtained and the effect of nonextensivity has been clearly illustrated in the figures.

A generalized method for handling first- and second-order thermal desorption and thermoluminescence

Abstract The influence of the nonextensivity on first- and second-order thermal desorption is investigated. A theoretical model for analyzing first- and second-order desorption spectra and glow curves were been presented [J. Appl. Phys. 51 (1980) 2594]. In addition, thermoluminescence decay has been recently investigated within Tsallis thermostatistics (TT). In this manner, our study considers this model and the effect of the nonextensivity is shown for various values of the entropic index q .

Mean-field theory of anisotropic potential of rank L=4 and nonextensive formalism

In this study, we employ the potential U ij =− f ( r ij ) P L (cos θ ij ) with rank L =4 to study on the nematic–isotropic phase transition, using generalized mean-field theory. We use a mean-field theory ( P 4 model) with this potential and generalize it using a nonextensive thermostatistics, namely Tsallis thermostatistics (TT). It is well known that mean-field theories have some insufficiencies. So we could expect similar insufficiencies from the P 4 model. Therefore to improve the P 4 model, if necessary, one could use TT. In this study, we give the possible contributions of TT to the nematic–isotropic transition. TT includes a parameter which is called the entropy index and it is a measure of the nonextensivity of the system. In this framework, the variations of P ̄ 4 and P ̄ 2 vs. temperature have been showed for various values of the entropic index q . The entropy changes with respect to the entropic index have been also reported.

Nonextensive thermostatistical approach to the thermoluminescence decay

In this study, thermoluminescence decay is investigated within Tsallis thermostatistics (TT). We believe that this is the first attempt to handle the thermoluminescence decay process within TT. The generalized form of glow intensity vs temperature is examined and the effect of the nonextensitivity is shown for various values of the entropic index q. The results are interpreted in terms of the entropic index q and the occupied trap density n.

The effects of the nonextensivity on the dimerization process and nematic ordering

The influence of the dimerization process on the nematic ordering is investigated by using a nonextensive thermostatistics, namely Tsallis thermostatistics (TT). A theoretical model taking into account the dimerization influence on the nematic scalar order parameter has been recently presented (Phys. Lett. A 256 (1999) 309). Our study considers this model and the theoretical predictions for the nematic order parameter are improved by using TT.

Tsallis thermostatistics for finite systems: a Hamiltonian approach

The derivation of the Tsallis generalized canonical distribution from the traditional approach of the Gibbs microcanonical ensemble is revisited (Phys. Lett. A 193 (1994) 140). We show that finite systems whose Hamiltonians obey a generalized homogeneity relation rigorously follow the nonextensive thermostatistics of Tsallis. In the thermodynamical limit, however, our results indicate that the Boltzmann–Gibbs statistics is always recovered, regardless of the type of potential among interacting particles. This approach provides, moreover, a one-to-one correspondence between the generalized entropy and the Hamiltonian structure of a wide class of systems, revealing a possible origin for the intrinsic nonlinear features present in the Tsallis formalism that lead naturally to power-law behavior. Finally, we confirm these exact results through extensive numerical simulations of the Fermi–Pasta–Ulam chain of anharmonic oscillators.

Multiparameter generalization of nonextensive statistical mechanics.

We show that the stochastic interpretation of Tsallis's thermostatistics given recently by Beck [Phys. Rev. Lett 87, 180601 (2001)] leads naturally to a multiparameter generalization. The resulting class of distributions is able to fit experimental results, which cannot be reproduced within Boltzmann's or Tsallis's formalism.

Extended phase-space dynamics for the generalized nonextensive thermostatistics.

We apply a variant of the Nosé thermostat to derive the Hamiltonian of a nonextensive system that is compatible with the canonical ensemble of the generalized thermostatistics of Tsallis. This microdynamical approach provides a deterministic connection between the generalized nonextensive entropy and power-law behavior. For the case of a simple one-dimensional harmonic oscillator, we confirm by numerical simulation of the dynamics that the distribution of energy H follows precisely the canonical q statistics for different values of the parameter q. The approach is further tested for classical many-particle systems by means of molecular dynamics simulations. The results indicate that the intrinsic nonlinear features of the nonextensive formalism are capable of generating energy fluctuations that obey anomalous probability laws. For q<1 a broad distribution of energy is observed, while for q>1 the resulting distribution is confined to a compact support.

Generalized parallel sampling

We develop a generalized version of the parallel tempering algorithm, based upon the non-extensive thermostatistics of Tsallis and coworkers. The effectiveness of the method is demonstrated on a simple one-dimensional problem and on a Lennard-Jones cluster.

Generalization of the Maier–Saupe theory of the nematics within Tsallis thermostatistics

In this study, one of the mean-field theories in nematics, the Maier–Saupe theory (MST), is generalized within Tsallis thermostatistics (TT). The variation of the order parameter versus temperature has been investigated and compared with experimental data for p-azoxyanisole (PAA). So far, we believe that this is the first attempt at the application of TT in liquid crystals. It is well known that MST fails to explain the experimental data for some of the nematics, one of which is PAA. However, generalized MST is able to account for the experimental data for a long range of temperatures. Also, in this study, the effect of nonextensivity is shown for various values of the entropic index.

Enhanced sampling in numerical path integration: An approximation for the quantum statistical density matrix based on the nonextensive thermostatistics

Here we examine a proposed approximation for the quantum statistical density matrix motivated by the nonextensive thermostatistics of Tsallis and co-workers. The approximation involves replacing the physical potential energy with an effective one, corresponding to a generalized nonextensive statistical ensemble. We examine the convergence properties of averages calculated using the effective potential, and introduce a related method for enhanced sampling in numerical path integration. As a necessary measure, path integral energy estimators are introduced for potentials that involve explicit temperature dependence. This sampling method is found to be effective for path integral simulations involving broken ergodicity.

Nonextensive thermostatistical investigation of the blackbody radiation

Thermodynamical quantities of the blackbody radiation such as free energy, entropy, total radiation energy, specific heat are calculated within the Tsallis thermostatistics where factorization method is incorporated. It is shown that basic thermodynamical relation of the blackbody radiation is form invariant with respect to the nonextensivity entropic index q. Furthermore, the nonextensive thermodynamical quantities related to the blackbody radiation are separately obtained in terms of q and the standard thermodynamical quantities of the blackbody radiation. It is indicated that the formulation may give a way to determine q which determines the degree of the nonextensivity that is one of the aims of the present study.

Thermostatistics of extensive and non-extensive systems using generalized entropies

We describe in detail two numerical simulation methods valid to study systems whose thermostatistics is described by generalized entropies, such as Tsallis. The methods are useful for applications to non-trivial interacting systems with a large number of degrees of freedom, and both short- and long-range interactions. The first method is quite general and it is based on the numerical evaluation of the density of states with a given energy. The second method is more specific for Tsallis thermostatistics and it is based on a standard Monte Carlo Metropolis algorithm along with a numerical integration procedure. We show here that both methods are robust and efficient. We present results of the application of the methods to the one-dimensional Ising model both in a short-range and in a long-range (non-extensive) case. We show that the thermodynamic potentials for different values of the system size N and different values of the non-extensivity parameter q can be described by scaling relations which are an extension of the ones holding for the Boltzmann–Gibbs statistics ( q=1). Finally, we discuss the differences in using standard or non-standard mean value definitions in the Tsallis thermostatistics formalism and present a microcanonical ensemble calculation approach of the averages.

Modified exponential function: the connection between nonextensivity and q-deformation

Tsallis thermostatistics has deep-rooted connection with quantum group formalism. Assuming that the modification of the standard exponential function considered in Tsallis thermostatistics has the same functional form as the one appearing in the q-calculus formalism and using the appropriate internal energy constraint, we derive the temperature dependent connection between the nonextensivity parameter and deformation parameter.

Quantum corrections for general partition functions

The classical limit for quite general partition functions is obtained using coherent states. In this framework a general procedure is presented to obtain all the quantum corrections to the classical limit. In particular, the first- and second-order corrections are worked out explicitly, and the classical limit for the Tsallis thermostatistics is determined. The results of this work generalize the ones obtained by Wigner (Phys. Rev. 40 (1932) 749) for usual statistical mechanics.