The present work investigates the Lesche stability (experimental robustness), the thermodynamic stability, the Legendre structure of thermodynamics, and derives the Maximum Entropy distribution of the one--parametric ``nonextensive Gaussian'' entropy. We show that this entropy definition fulfills both stability conditions for all values of its parameter ($q\in\mathbb{R}$). The entropy maximizer contains the Lambert $W$--function, which allows the preservation of the Legendre transformations.
Macroscopic nonextensive thermodynamics is studied without recourse to microscopic statistical mechanics. It is shown that if entropy is nonextensive, the concept of physical temperature introduced through the generalized zeroth law of thermodynamics necessarily leads to modifications of the first law of thermodynamics and some of thermodynamic relations including Clausius definition of thermodynamic entropy. It is also shown, by applying this generalized Clausius entropy to a composite nonextensive system, how the nonextensive entropy and the quantity of heat consistently behave in an arbitra…
Gibbs-Boltzmann entropy leads to systems that have a strong dependence on initial conditions. In reality, most materials behave quite independently of initial conditions. Nonextensive entropy or Tsallis entropy leads to nonextensive statistical mechanics. In this paper, we calculate the Tsallis form of Clausius inequality and then determine the upper bound for extracting work from the small system in the nonextensive statistical mechanics with mutual information. In the following, we extract mutual information and adjust Maxwell's demon with quantum feedback control.
The concepts of quantity of heat and work are deduced in the non-extensive statistical mechanics context, following steps in parallel to those employed in the extensive statistical mechanics.
Within the nonextensive framework, it is shown that zeroth law of thermodynamics determines not only the mapping between Lagrange multipliers and intensive variables, but also the mapping between nonextensive and extensive entropy. The form of constraints decides the form of the extensive entropy, standard averages lead to Boltzmann-Shannon-Gibbs entropy while normalised biased averages lead to Renyi entropy. The mapping between Lagrange multipliers and intensive variables is also discussed in the more general context of composable entropy.
The problem of temperature in nonextensive statistical mechanics is studied. Considering the first law of thermodynamics and a "quasi-reversible process", it is shown that the Tsallis entropy becomes the Clausius entropy if the inverse of the Lagrange multiplier, $beta$, associated with the constraint on the internal energy is regarded as the temperature. This temperature is different from the previously proposed "physical temperature" defined through the assumption of divisibility of the total system into independent subsystems. A general discussion is also made about the role of Boltzmann's …
The form invariance of pseudoadditivity is shown to determine the structure of nonextensive entropies. Nonextensive entropy is defined as the appropriate expectation value of nonextensive information content, similar to the definition of Shannon entropy. Information content in a nonextensive system is obtained uniquely from generalized axioms by replacing the usual additivity with pseudoadditivity. The satisfaction of the form invariance of the pseudoadditivity of nonextensive entropy and its information content is found to require the normalization of nonextensive entropies. The proposed prin…
We briefly review central concepts concerning nonextensive statistical mechanics, based on the nonadditive entropy $S_q=k\frac{1-\sum_{i}p_i^q}{q-1} (q \in {\cal R}; S_1=-k\sum_{i}p_i \ln p_i)$. Among others, we focus on possible realizations of the $q$-generalized Central Limit Theorem, including at the edge of chaos of the logistic map, and for quasi-stationary states of many-body long-range-interacting Hamiltonian systems.
A family of nonlinear ordinary differential equations with arbitrary order is obtained by using nonextensive concepts related to the Tsallis entropy. Applications of these equations are given here. In particular, a connection between Tsallis entropy and the one-dimensional correlated anomalous diffusion equation is established. It is also developed explicitly a WKB-like method for second order equations and it is applied to solve approximately a class of equations that contains as a special case the Thomas-Fermi equation for an atom. It is expected that the present ideas can be useful in the d…
We perform a throughout numerical study of the average sensitivity to initial conditions and entropy production for two symplectically coupled standard maps focusing on the control-parameter region close to regularity. Although the system is ultimately strongly chaotic (positive Lyapunov exponents), it first stays lengthily in weak-chaotic regions (zero Lyapunov exponents). We argue that the nonextensive generalization of the classical formalism is an adequate tool in order to get nontrivial information about this complex phenomenon. Within this context we analyze the relation between the powe…
The joint eigenvalue distributions of random-matrix ensembles are derived by applying the principle maximum entropy to the Renyi, Abe and Kaniadakis entropies. While the Renyi entropy produces essentially the same matrix-element distributions as the previously obtained expression by using the Tsallis entropy, and the Abe entropy does not lead to a closed form expression, the Kaniadakis entropy leads to a new generalized form of the Wigner surmise that describes a transition of the spacing distribution from chaos to order. This expression is compared with the corresponding expression obtained b…
Ordinary Boltzmann-Gibbs entropy is inadequate to be used in systems depending on a control parameter that yield different mean energy values. Such systems fail to give the correct comparison between the off-equilibrium and equilibrium entropy values. Klimontovich's S theorem solves this problem by renormalizing energy and making use of escort distributions. Since nonextensive thermostatistics is a generalization of Boltzmann-Gibbs entropy, it too exhibits this same deficiency. In order to remedy this, we present the nonextensive generalization of Klimontovich's S theorem. We show that this ge…
Using Rényi entropy, a possible thermostatistics for nonextensive systems is discussed. We show that it is possible to get the $q$-exponential distribution function for nonextensive systems having nonadditive energy but additive entropy. It is also shown that additive energy, as a approximation within nonextensive statistics, is not suitable for discussing fundamental problems for interacting systems.
We introduce a nonextensive entropic measure $S_χ$ that grows like $N^χ$, where $N$ is the size of the system under consideration. This kind of nonextensivity arises in a natural way in some $N$-body systems endowed with long-range interactions described by $r^{-α}$ interparticle potentials. The power law (weakly nonextensive) behavior exhibited by $S_χ$ is intermediate between (1) the linear (extensive) regime characterizing the standard Boltzmann-Gibbs entropy and the (2) the exponential law (strongly nonextensive) behavior associated with the Tsallis generalized $q$-entropies. The functiona…
Recently, Gross claims that Boltzmann entropy $S=k\ln W$ is valid for any system at equilibrium, so that Tsallis entropy is useless in this case. I comment on some arguments forwarded to reach this conclusion and argue that the additive energy formalism of nonextensive statistics is not appropriate for the fundamental study of the theory for nonadditive systems.
Based on the Tsallis entropy, the nonextensive thermodynamic properties are studied as a q-deformation of classical statistical results using only probabilistic methods and straightforward calculations. It is shown that the constant in the Tsallis entropy depends on the deformation parameter and must be redefined to recover the usual thermodynamic relations. The notions of variance and covariance are generalised. A partial derivative formula of the entropy is established. It verifies important relations from which most of the nonextensive thermodynamics relations can be recovered. This leads t…
We have proposed a new stochastic interpretation of the sudiffusion described by the Sharma-Mittal entropy formalism which generates a nonlinear subdiffusion equation with natural order derivatives. We have shown that the solution to the diffusion equation generated by Gauss entropy (which is the particular case of Sharma-Mittal entropy) is the same as the solution of the Fokker-Planck (FP) equation generated by the Langevin generalised equation where the `long memory effect' is taken into account. The external noise which pertubates the subdiffusion coefficient (occuring in the solution of FP…
The Shannon-Khinchin axioms for the ordinary information entropy are generalized in a natural way to the nonextensive systems based on the concept of nonextensive conditional entropy, and a complete proof of the uniqueness theorem for the Tsallis entropy is presented.
The general formalism for the nonextensive statistics based on the Landsberg-Vedral entropy was derived. The formula for the first law of thermodynamics and the exact relations of the thermodynamic quantities to their ensemble averages were obtained. It was found that under the transformation $q\to 2-q$ the probabilities of microstates of the nonextensive statistics based on the Landsberg-Vedral entropy formally resemble the corresponding probabilities of the Tsallis statistics with escort probabilities. However, the nonextensive statistics with the Landsberg-Vedral entropy does not require in…
The second law of thermodynamics in nonextensive statistical mechanics is discussed in the quantum regime. Making use of the convexity property of the generalized relative entropy associated with the Tsallis entropy indexed by q, Clausius' inequality is shown to hold in the range of q between zero and two. This restriction on the range of the entropic index, q, is purely quantum mechanical and there exists no upper bound of q for validity of the second law in classical theory.
We study the nonextensive thermodynamics for open systems. On the basis of the maximum entropy principle, the dual power-law q-distribution functions are re-deduced by using the dual particle number definitions and assuming that the chemical potential is constant in the two sets of parallel formalisms, where the fundamental thermodynamic equations with dual interpretations of thermodynamic quantities are derived for the open systems. By introducing parallel structures of Legendre transformations, other thermodynamic equations with dual interpretations of quantities are also deduced in the open…
We propose a simple modification, the Gaussian truncation, of the probability density function which was obtained by Beck (2001) to fit the experimental distribution of fluid particle acceleration component from fully developed fluid turbulence inspired by the framework of nonextensive statistical mechanics with the underlying gamma distribution of the model parameter. The modified distribution is given a phenomenological treatment and provides a good fit to new experimental results on the acceleration of fluid particle reported by Crawford, Mordant, and Bodenschatz. We compare the results wit…
We have studied finite $N$-body $D$-dimensional nonextensive ideal gases and harmonic oscillators, by using the maximum-entropy methods with the $q$- and normal averages ($q$: the entropic index). The validity range, specific heat and Tsallis entropy obtained by the two average methods are compared. Validity ranges of the $q$- and normal averages are $0 < q < q_U$ and $q > q_L$, respectively, where $q_U=1+(ηDN)^{-1}$, $q_L=1-(ηDN+1)^{-1}$ and $η=1/2$ ($η=1$) for ideal gases (harmonic oscillators). The energy and specific heat in the $q$- and normal averages coincide with those in the Boltzmann…
The nonextensive statistics based on Tsallis entropy have been so far used for the systems composed of subsystems having same $q$. The applicability of this statistics to the systems with different $q$'s is still a matter of investigation. The actual difficulty is that the class of systems to which the theory has been applied is limited by the usual nonadditivity rule of Tsallis entropy which, in reality, has been established for the systems having same $q$ value. In this paper, we propose a more general nonadditivity rule for Tsallis entropy. This rule, as the usual one for same $q$-systems, …
We have discussed dynamical properties of the Tsallis entropy and the generalized Fisher information in nonextensive systems described by the Langevin model subjected to additive and multiplicative noise. Analytical expressions for the time-dependent Tsallis entropy and generalized Fisher information have been obtained with the use of the $q$-moment approach to the Fokker-Planck equation developed in a previous study [H. Hasegawa, Phys. Rev. E {\bf 77}, 031133 (2008)]. Model calculations of the information entropies in response to an applied pulse and sinusoidal inputs have been presented.
We show that the dynamical and entropic properties at the chaos threshold of the logistic map are naturally linked through the nonextensive expressions for the sensitivity to initial conditions and for the entropy. We corroborate analytically, with the use of the Feigenbaum renormalization group(RG) transformation, the equality between the generalized Lyapunov coefficient $λ_{q}$ and the rate of entropy production $K_{q}$ given by the nonextensive statistical mechanics. Our results advocate the validity of the $q$-generalized Pesin identity at critical points of one-dimensional nonlinear dissi…
This is a study of composition rule and temperature definition for nonextensive systems containing different $q$ subsystems. The physical meaning of the multiplier $β$ associated with the energy expectation in the optimization of Tsallis entropy is investigated for the formalism with normalized expectation given by escort probability. This study is carried out for two possible cases: the case of the approximation of additive energy; and the case of nonadditive energy prescribed by an entropy composition rule for different $q$ systems.
We discuss the generalized von Neumann (Tsallis) entropy and the generalized Fisher information (GFI) in nonextensive quantum systems, by using the interpolation approximation (IA) which has been shown to yield good results for the quantal distributions within $O(q-1)$ and in high- and low-temperature limits, $q$ being the entropic index [H. Hasegawa, Phys. Rev. E 80 (2009) 011126]. Three types of GFIs which have been proposed so far in the nonextensive statistics, are discussed from the viewpoint of their metric properties and the Cramér-Rao theorem. Numerical calculations of the $q$- and tem…
We propose a new way of defining entropy of a system, which gives a general form which may be nonextensive as Tsallis entropy, but is linearly dependent on component entropies, like Renyi entropy, which is extensive. This entropy has a conceptually novel but simple origin and is mathematically easy to define by a very simple expression, though the probability distribution resulting from optimizing it gives rather complex, which is compared numerically with the other entropies. It may, therefore, appear as the right candidate in a physical situation where the probability distribution does not s…
The nonextensive statistics based on the $q$-entropy $S_q=-\frac{\sum_{i=1}^v(p_i-p_i^q)}{1-q}$ has been so far applied to systems in which the $q$ value is uniformly distributed. For the systems containing different $q$'s, the applicability of the theory is still a matter of investigation. The difficulty is that the class of systems to which the theory can be applied is actually limited by the usual nonadditivity rule of entropy which is no more valid when the systems contain non uniform distribution of $q$ values. In this paper, within the framework of the so called incomplete information th…
We show that using nonextensive entropy can lead to spontaneous symmetry breaking when a parameter changes its value from that applicable for a symmetric domain, as in field theory. We give the physical reasons and also show that even for symmetric Dirichlet priors, such a defnition of the entropy and the parameter value can lead to asymmetry when entropy is maximized.
Within the Tsallis thermodynamics' framework, and using scaling properties of the entropy, we derive a generalization of the Gibbs-Duhem equation. The analysis suggests a transformation of variables that allows standard thermodynamics to be recovered. Moreover, we also generalize Einstein's formula for the probability of a fluctuation to occur by means of the maximum statistical entropy method. The use of the proposed transformation of variables also shows that fluctuations within Tsallis statistics can be mapped to those of standard statistical mechanics.
It is shown that the distribution derived from the principle of maximum Tsallis entropy is a superposable Levy-type distribution. Concomitantly, the leading order correction to the limit distribution is also deduced. This demonstration fills an important gap in the derivation of the Levy-stable distribution from the nonextensive statistical framework.
Within Tsallis' nonextensive statistics, a model is elaborated to address self-similar time series as a thermodynamic system. Thermodynamic-type characteristics relevant to temperature, pressure, entropy, internal and free energies are introduced and tested. Stability conditions of time series analysis are discussed in details on the basis of Van der Waals model.
It is pointed out that the Tsallis entropy functional represented in terms of the escort distribution is not concave of the entropic index $q$ is less than unity. It is emphasized that the escort distribution is a secondary object calculated from the basic original distribution.
We have discussed the Tsallis entropy in finite $N$-unit nonextensive systems, by using the multivariate $q$-Gaussian probability distribution functions (PDFs) derived by the maximum entropy methods with the normal average and the $q$-average ($q$: the entropic index). The Tsallis entropy obtained by the $q$-average has an exponential $N$ dependence: $S_q^{(N)}/N \simeq \:e^{(1-q)N \:S_1^{(1)}} $ for large $N$ ($\gg \frac{1}{(1-q)} >0$). In contrast, the Tsallis entropy obtained by the normal average is given by $S_q^{(N)}/N \simeq [1/(q-1)N]$ for large $N$ ($\gg \frac{1}{(q-1)} > 0)$. $N$ dep…
We present a simple and general argument showing that a class of dynamical correlations give rise to the so-called Tsallis nonextensive statistics. An example of a system having such a dynamics is given, exhibiting a non-Boltzmann energy distribution. A relation with prethermalization processes is discussed.
The classical Gaussian ensembles of random matrices can be constructed by maximizing Boltzmann-Gibbs-Shannon's entropy, S_{BGS} = - \int d{\bf H} [P({\bf H})] \ln [P({\bf H})], with suitable constraints. Here we construct and analyze random-matrix ensembles arising from the generalized entropy S_q = (1 - \int d{\bf H} [P({\bf H})]^q)/(q-1) (thus S_1=S_{BGS}). The resulting ensembles are characterized by a parameter q measuring the degree of nonextensivity of the entropic form. Making q -> 1 recovers the Gaussian ensembles. If q \ne 1, the joint probability distributions P(\bf H) cannot be fact…
We show that if one uses the invariant form of the Boltzmann-Shannon continuous entropy, it is possible to obtain the generalized Pareto-Tsallis density function, using an appropriate "prior" measure m_{q}(x) and a "Boltzman constraint" which formally is equivalent to the Tsallis q-average constraint on the random variable X. We derive the Tsallis prior function and study its scaling asymptotic behavior. When the entropic index q tends to 1, m_{q}(x) tends to 1 for all values of x as this should be.
The cornerstones of Boltzmann-Gibbs and nonextensive statistical mechanics respectively are the entropies $S_{BG} \equiv -k \sum_{i=1}^W p_i \ln p_i $ and $S_{q}\equiv k (1-\sum_{i=1}^Wp_i^{q})/(q-1) (q\in{\mathbb R} ; S_1=S_{BG})$. Through them we revisit the concept of additivity, and illustrate the (not always clearly perceived) fact that (thermodynamical) extensivity has a well defined sense {\it only} if we specify the composition law that is being assumed for the subsystems (say $A$ and $B$). If the composition law is {\it not} explicitly indicated, it is {\it tacitly} assumed that $A$ a…
The form invariance of the statement of the maximum entropy principle and the metric structure in quantum density matrix theory, when generalized to nonextensive situations, is shown here to determine the structure of the nonextensive entropies. This limits the range of the nonextensivity parameter to so as to preserve the concavity of the entropies. The Tsallis entropy is thereby found to be appropriately renormalized.
We present a stability analysis of the classical ideal gas in a new theory of nonextensive statistics and use the theory to understand the phenomena of negative specific heat in some self-gravitating systems. The stability analysis is made on the basis of the second variation of Tsallis entropy. It is shown that the system is thermodynamically unstable if the nonextensive parameter is q>5/3, which is exactly equivalent to the condition of appearance of the negative specific heat.
In this note it is shown that in his ``Comments'', Tsallis did not point out any flaws in the main criticism of my paper, namely, that the nonextensive q-entropy formalism fails to satisfy the zeroth law of thermodynamics. Details are also given of a rigorous thermodynamic proof which demonstrates that, contrary to Tsallis's assertion, the application of a nonextensive formalism to black-body radiation does not lead to the well known $T^4$ Stefan-Boltzman law.
In this paper a composed A+B magnetic system, with spins J_A=2 and J_B=3/2, is considered within the mean-field approximation, in the framework of Tsallis nonextensive statistics. Our motivation is twofold: (1) to approach the existing experimental data of manganese oxides (manganites), where Mn^{3+} and Mn^{4+} form two magnetic sublattices, and (2) to investigate the structure of nonextensive density matrices of composed systems. By imposing that thermodynamic quantities, such as the magnetization of sublattices A and B, must be invariant weather the calculation is taken over the total Hilbe…
Starting from the basic prescriptions of the Tsallis' nonextensive thermostatistics, i.e. generalized entropy and normalized q-expectation values, we study the relativistic nonextensive thermodynamics and derive a Boltzmann transport equation that implies the validity of the H-theorem where a local nonextensive four-entropy density is considered. Macroscopic thermodynamic functions and the equation of state for a perfect gas are derived at the equilibrium.
We show that extensive thermostatistics based on Renyi entropy and Kolmogorov-Nagumo averages can be expressed in terms of Tsallis non- extensive thermostatistics. We use this correspondence to generalize thermostatistics to a large class of Kolmogorov-Nagumo means and suitably adapted definitions of entropy.
We have calculated the Tsallis entropy and Fisher information matrix (entropy) of spatially-correlated nonextensive systems, by using an analytic non-Gaussian distribution obtained by the maximum entropy method. Effects of the correlated variability on the Fisher information matrix are shown to be different from those on the Tsallis entropy. The Fisher information is increased (decreased) by a positive (negative) correlation, whereas the Tsallis entropy is decreased with increasing an absolute magnitude of the correlation independently of its sign. This fact arises from the difference in their…
We study the similarities and differences between different models concerning subdiffusion. More particularly, we calculate first passage time (FPT) distributions for subdiffusion, derived from Greens' functions of nonlinear equations obtained from Sharma-Mittal's, Tsallis's and Gauss's nonadditive entropies. Then we compare these with FPT distributions calculated from a fractional model using a subdiffusion equation with a fractional time derivative. All of Greens' functions give us exactly the same standard relation $<(Δx)^2> =2 D_αt^α$ which characterizes subdiffusion ($0<α<1$), but general…
In this lecture we briefly review the definition, consequences and applications of an entropy, $S_q$, which generalizes the usual Boltzmann-Gibbs entropy $S_{BG}$ ($S_1=S_{BG}$), basis of the usual statistical mechanics, well known to be applicable whenever ergodicity is satisfied at the microscopic dynamical level. Such entropy $S_q$ is based on the notion of $q$-exponential and presents properties not shared by other available alternative generalizations of $S_{BG}$. The thermodynamics proposed in this way is generically {\it nonextensive} in a sense that will be qualified. The present frame…
The Tsallis entropy and Fisher information entropy (matrix) are very important quantities expressing information measures in nonextensive systems. Stationary and dynamical properties of the information entropies have been investigated in the $N$-unit coupled Langevin model subjected to additive and multiplicative white noise, which is one of typical nonextensive systems. We have made detailed, analytical and numerical study on the dependence of the stationary-state entropies on additive and multiplicative noise, external inputs, couplings and number of constitutive elements ($N$). By solving t…
Engle's ARCH algorithm is a generator of stochastic time series for financial returns (and similar quantities) characterized by a time-dependent variance. It involves a memory parameter $b$ ($b=0$ corresponds to {\it no memory}), and the noise is currently chosen to be Gaussian. We assume here a generalized noise, namely $q_n$-Gaussian, characterized by an index $q_{n} \in {\cal R}$ ($q_{n}=1$ recovers the Gaussian case, and $q_n>1$ corresponds to tailed distributions). We then match the second and fourth momenta of the ARCH return distribution with those associated with the $q$-Gaussian distr…
The nonextensive statistical ensembles are revisited for the complex systems with long-range interactions and long-range correlations. An approximation, the value of nonextensive parameter (1-q) is assumed to be very tiny, is adopted for the limit of large particle number for most normal systems. In this case, Tsallis entropy can be expanded as a function of energy and particle number fluctuation, and thus the power-law forms of the generalized Gibbs distribution and grand canonical distribution can be derived. These new distribution functions can be applied to derive the free energy and grand…
We investigate the limiting cases of Tsallis statistics. The viewpoint adopted is not the standard information-theoretic one, where one derives the distribution from a given measure of information. Instead the mechanical approach recently proposed in [M. Campisi, G.B. Bagci, Phys. Lett. A (2006), doi:10.1016/j.physleta.2006.09.081], is adopted, where the distribution is given and one looks for the associated physical entropy. We show that, not only the canonical ensemble is recovered in the limit of $q$ tending to one, as one expects, but also the microcanonical ensemble is recovered in the li…
Boltzmann-Gibbs statistical mechanics is based on the entropy $S_{BG}=-k \sum_{i=1}^W p_i \ln p_i$. It enables a successful thermal approach of ubiquitous systems, such as those involving short-range interactions, markovian processes, and, generally speaking, those systems whose dynamical occupancy of phase space tends to be ergodic. For systems whose microscopic dynamics is more complex, it is natural to expect that the dynamical occupancy of phase space will have a less trivial structure, for example a (multi)fractal or hierarchical geometry. The question naturally arises whether it is possi…
Nonextensive aspects of the degree distribution in Watts-Strogatz (WS) small-world networks, $P_{SW}(k)$, have been discussed in terms of a generalized Gaussian (referred to as {\it $Q$-Gaussian}) which is derived by the three approaches: the maximum-entropy method (MEM), stochastic differential equation (SDE), and hidden-variable distribution (HVD). In MEM, the degree distribution $P_Q(k)$ in complex networks has been obtained from $Q$-Gaussian by maximizing the nonextensive information entropy with constraints on averages of $k$ and $k^2$ in addition to the normalization condition. In SDE, $…
In this paper we extend our recent results [Physica A340 (2004)110] on q-nonextensive statistics with non-Tsallis entropies. In particular, we combine an axiomatics of Renyi with the q-deformed version of Khinchin axioms to obtain the entropy which accounts both for systems with embedded self-similarity and q-nonextensivity. We find that this entropy can be uniquely solved in terms of a one-parameter family of information measures. The corresponding entropy maximizer is expressible via a special function known under the name of the Lambert W-function. We analyze the corresponding "high" and "l…
The generalized zeroth law of thermodynamics indicates that the physical temperature in nonextensive statistical mechanics is different from the inverse of the Lagrange multiplier, beta. This leads to modifications of some of thermodynamic relations for nonextensive systems. Here, taking the first law of thermodynamics and the Legendre transform structure as the basic premises, it is found that Clausius definition of the thermodynamic entropy has to be appropriately modified, and accordingly the thermodynamic relations proposed by Tsallis, Mendes and Plastino [Physica A 261 (1998) 534] are als…
The homogeneous entropy for continuous systems in nonextensive statistics reads $S^{H}_{q}=k_B\,{(1 - (K \int dΓρ^{1/q}(Γ))^{q})}/({1-q})$, where $Γ$ is the phase space variable. Optimization of $S^{H}_{q}$ combined with normalization and energy constraints gives an implicit expression of the distribution function $ρ(Γ)$ which can be computed explicitly for the ideal gas. From this result, we compute properties such as the energy fluctuations and the specific heat. Similar results are also presented using the formulation based on the Tsallis entropy. From the analysis, we discuss the validity …
We have shown that the weak-coupling limit superconductors are well described by $ q \sim 1 $, where $ q $ is a real parameter which characterizes the degree of nonextensivity of Tsallis' entropy. Nevertheless, small deviations with respect to q=1 provide better agreement when compared with experimental results. We have also shown that the generalized BCS theory with $ q \neq 1 $ exhibit power-law behavior of several measurable macroscopic functions in the low-temperature regime. These power-law properties are found in many high-Tc oxides superconductors and motivated us to extend Tsallis' ent…
This is a study of the information evolution of complex systems by geometrical consideration. We look at chaotic systems evolving in fractal phase space. The entropy change in time due to the fractal geometry is assimilated to the information growth through the scale refinement. Due to the incompleteness of the state number counting at any scale on fractal support, the incomplete normalization $\sum_ip_i^q=1$ is applied throughout the paper, where $q$ is the fractal dimension divided by the dimension of the smooth Euclidean space in which the fractal structure of the phase space is embedded. I…
