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C Tsallis

Publications and source records attributed to C Tsallis.

At least 19 recordsLinked to original sources
PubMed2004

Comment on "Critique of q-entropy for thermal statistics".

It was recently argued [M. Nauenberg, Phys. Rev. E 67, 036114 (2003)] that the theory sometimes referred to as nonextensive statistical mechanics has no physical basis, for a considerable variety of reasons, including the impossibility of measuring the temperature out of the Boltzmann-Gibbs (BG) theory. We comment here on virtually all the physically and mathematically relevant issues, and point out what we consider to be severe inadvertences contained in that paper. In particular, we factually argue, through computer simulations, the validity of the zeroth principle of thermodynamics, and of the basic rules of thermometry for nonextensive systems. This fact further supports the possible connection with the thermodynamics of nonextensive statistical mechanics, which is already known to be consistent with the first, second, and third principles. All the foundational steps (e.g., the uniqueness of the entropy and the stationary state distribution) have already been established for nonextensive thermostatistics on similar grounds than those long known for BG statistics, the former corresponding to power laws (expected for long-range interactions when size N diverges before time t), and the latter correspond to the BG exponential law (expected for long-range interactions when N diverges after t, as well as for short-range interactions in any diverging order for N and t). We conclude that the invalidating arguments made by Nauenberg by no means apply.

Comment↗
PubMed2004

Nonextensivity of the cyclic lattice Lotka-Volterra model.

We numerically show that the lattice Lotka-Volterra model, when realized on a square lattice support, gives rise to a finite production, per unit time, of the nonextensive entropy S(q)=(1- summation operator (i)p(q)(i))/(q-1) (S(1)=- summation operator (i)p(i) ln p(i)). This finiteness only occurs for q=0.5 for the d=2 growth mode (growing droplet), and for q=0 for the d=1 one (growing stripe). This strong evidence of nonextensivity is consistent with the spontaneous emergence of local domains of identical particles with fractal boundaries and competing interactions. Such direct evidence is, to our knowledge, exhibited for the first time for a many-body system which, at the mean field level, is conservative.

Journal Article↗
PubMed2003

Crossover in diffusion equation: anomalous and normal behaviors.

Ubiquitous phenomena exist in nature where, as time goes on, a crossover is observed between different diffusion regimes (e.g., anomalous diffusion at early times which becomes normal diffusion at long times, or the other way around). In order to focus on such situations we have analyzed particular relevant cases of the generalized Fokker-Planck equation integral dgamma(')tau(gamma('))[ partial differential (gamma('))rho(x,t)]/ partial differential t(gamma('))= integral dmu(')dnu'D(mu('),nu('))[ partial differential (mu('))[rho(x,t)](nu('))]/ partial differential x(mu(')), where tau(gamma(')) and D(mu('),nu(')) are kernels to be chosen; the choice tau(gamma('))=delta(gamma(')-1) and D(mu('),nu('))=delta(mu(')-2)delta(nu(')-1) recovers the normal diffusion equation. We discuss in detail the following cases: (i) a mixture of the porous medium equation, which is connected with nonextensive statistical mechanics, with the normal diffusion equation; (ii) a mixture of the fractional time derivative and normal diffusion equations; (iii) a mixture of the fractional space derivative, which is related with Lévy flights, and normal diffusion equations. In all three cases a crossover is obtained between anomalous and normal diffusions. In cases (i) and (iii), the less diffusive regime occurs for short times, while at long times the more diffusive regime emerges. The opposite occurs in case (ii). The present results could be easily extended to more complex situations (e.g., crossover between two, or even more, different anomalous regimes), and are expected to be useful in the analysis of phenomena where nonlinear and fractional diffusion equations play an important role. Such appears to be the case for isolated long-ranged interaction Hamiltonians, which along time can exhibit a crossover from a longstanding metastable anomalous state to the usual Boltzmann-Gibbs equilibrium one. Another illustration of such crossover occurs in active intracellular transport.

Journal Article↗
PubMed2001

Non-Gaussian equilibrium in a long-range Hamiltonian system.

We study the dynamics of a system of N classical spins with infinite-range interaction. We show that, if the thermodynamic limit is taken before the infinite-time limit, the system does not relax to the Boltzmann-Gibbs equilibrium, but exhibits different equilibrium properties, characterized by stable non-Gaussian velocity distributions, Lévy walks, and dynamical correlation in phase space.

Journal Article↗
PubMed2001

Specific heat of multifractal energy spectra.

Motivated by the self-similar character of energy spectra demonstrated for quasicrystals, we investigate the case of multifractal energy spectra, and compute the specific heat associated with simple archetypal forms of multifractal sets as generated by iterated maps. We considered the logistic map and the circle map at their threshold to chaos. Both examples show nontrivial structures associated with the scaling properties of their respective chaotic attractors. The specific heat displays generically log-periodic oscillations around a value that characterizes a single exponent, the "fractal dimension," of the distribution of energy levels close to the minimum value set to 0. It is shown that when the fractal dimension and the frequency of log oscillations of the density of states are large, the amplitude of the resulting log oscillation in the specific heat becomes much smaller than the log-periodic oscillation measured on the density of states.

Journal Article↗
PubMed2000

Anomalous diffusion with absorption: exact time-dependent solutions

Recently, analytical solutions of a nonlinear Fokker-Planck equation describing anomalous diffusion with an external linear force were found using a nonextensive thermostatistical Ansatz. We have extended these solutions to the case when an homogeneous absorption process is also present. Some peculiar aspects of the interrelation between the deterministic force, the nonlinear diffusion, and the absorption process are discussed.

Journal Article↗
PubMed2000

Anomalous diffusion associated with nonlinear fractional derivative fokker-planck-like equation: exact time-dependent solutions

We consider the d=1 nonlinear Fokker-Planck-like equation with fractional derivatives ( partial differential/ partial differentialt)P(x,t)=D( partial differential(gamma)/ partial differentialx(gamma))[P(x,t)](nu). Exact time-dependent solutions are found for nu=(2-gamma)/(1+gamma)(-infinity<gamma</=2). By considering the long-distance asymptotic behavior of these solutions, a connection is established, namely, q=(gamma+3)/(gamma+1)(0<gamma</=2), with the solutions optimizing the nonextensive entropy characterized by index q. Interestingly enough, this relation coincides with the one already known for Levy-like superdiffusion (i.e., nu=1 and 0<gamma</=2). Finally, for (gamma,nu)=(2,0) we obtain q=5/3, which differs from the value q=2 corresponding to the gamma=2 solutions available in the literature (nu<1 porous medium equation), thus exhibiting nonuniform convergence.

Journal Article↗
PubMed1999

Nonextensive foundation of Lévy distributions.

A deep connection between the ubiquity of Lévy distributions in nature and the nonextensive thermal statistics introduced a decade ago has been established recently [Tsallis et al., Phys. Rev. Lett. 75, 3589 (1995)], by using unnormalized q-expectation values. It has just been argued on physical grounds that normalized q-expectation values should be used instead. We revisit, within this more appropriate scheme, the Lévy problem and verify that the relevant analytic results become sensibly simplified, whereas the basic physics remains unchanged.

Journal Article↗
PubMed1994

Optimal Hebbian learning rules and the role of asymmetry.

We study the storage properties associated with generalized Hebbian learning rules which present four free parameters that allow for asymmetry. We also introduce two extra parameters in the post-synaptic potentials in order to further improve the critical capacity. Using signal-to-noise analysis, as well as computer simulations on an analog network, we discuss the performance of the rules for arbitrarily biased patterns and find that the critical storage capacity alpha c becomes maximal for a particular symmetric rule (alpha c diverges in the sparse coding limit). Departures from symmetry decrease alpha c but can increase the robustness of the model.

Artificial Intelligence↗