arXiv subjects

Constantino Tsallis

Publications and source records attributed to Constantino Tsallis.

189 records · Page 11Linked to original sources

arXiv1999

Critical Temperature and Nonextensivity in Long-range Interacting Lennard-Jones-like Fluids

Molecular dynamic simulations for systems with $D=2,3$ Lennard-Jones-like interactions are studied. In the model, we assume that, at long distances, the two-body attractive potential decays as $r^{-α}$. Thermodynamic extensivity (nonextensivity) is observed for $α> D$ ($0\leq α\leq D$). Particular attention is payed to the liquid-gas critical point located, in the temperature-pressure plane, at ($T_c,P_c$). ($T_c,P_c$) are, in the $N\to \infty$ limit ($N\equiv$ number of molecules), {\em finite} for $α> D$ and {\em diverge} for $α\leq D$ (as $(α- D)^{-1}$ for $α/D \to 1 + 0$). However, the variables $T_c^* \equiv T_c/N^*$ and $P_c^* \equiv P_c/N^*$ with $N^* \equiv [N^{1-α/D} -1]/[1-α/D]$ remain {\em finite for all} $α$. Thus, the extensive and nonextensive regions become unified, as recently conjectured. These results should be useful for discussing gravitation and some special fluids

cond-mat↗

arXiv1999

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 non extensive 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.

cond-mat.stat-mech↗

arXiv1999

Functions of linear operators: Parameter differentiation

We derive a useful expression for the matrix elements $[\frac{\partial f[A(t)]}{\partial t}]_{i j}$ of the derivative of a function $f[A(t)]$ of a diagonalizable linear operator $A(t)$ with respect to the parameter $t$. The function $f[A(t)]$ is supposed to be an operator acting on the same space as the operator $A(t)$. We use the basis which diagonalizes A(t), i.e., $A_{i j}=λ_i δ_{i j}$, and obtain $[\frac{\partial f[A(t)]}{\partial t}]_{i j}=[\frac{\partial A}{\partial t}]_ {i j}\frac{f(λ_j) - f(λ_i)} {λ_j - λ_i}$. In addition to this, we show that further elaboration on the (not necessarily simple) integral expressions given by Wilcox 1967 (who basically considered $f[A(t)]$ of the exponential type) and generalized by Rajagopal 1998 (who extended Wilcox results by considering $f[A(t)]$ of the $q$-exponential type where $\exp_q(x) \equiv [1+(1-q)x]^{1/(1-q)}$ with $q \in {\cal {R}}$; hence, $\exp_1 (x)=\exp(x))$ yields this same expression. Some of the lemmas first established by the above authors are easily recovered.

cond-mat.stat-mech↗

arXiv1999

Are citations of scientific papers a case of nonextensivity ?

The distribution $N(x)$ of citations of scientific papers has recently been illustrated (on ISI and PRE data sets) and analyzed by Redner [Eur. Phys. J. B {\bf 4}, 131 (1998)]. To fit the data, a stretched exponential ($N(x) \propto \exp{-(x/x_0)^β}$) has been used with only partial success. The success is not complete because the data exhibit, for large citation count $x$, a power law (roughly $N(x) \propto x^{-3}$ for the ISI data), which, clearly, the stretched exponential does not reproduce. This fact is then attributed to a possibly different nature of rarely cited and largely cited papers. We show here that, within a nonextensive thermostatistical formalism, the same data can be quite satisfactorily fitted with a single curve (namely, $N(x) \propto 1/[1+(q-1) λx]^{q/{q-1}}$ for the available values of $x$. This is consistent with the connection recently established by Denisov [Phys. Lett. A {\bf 235}, 447 (1997)] between this nonextensive formalism and the Zipf-Mandelbrot law. What the present analysis ultimately suggests is that, in contrast to Redner's conclusion, the phenomenon might essentially be one and the same along the entire range of the citation number $x$.

cond-mat.stat-mech↗

arXiv1999

Generalization of the Lie-Trotter Product Formula for q-Exponential Operators

The Lie-Trotter formula $e^{\hat{A}+\hat{B}} = \lim_{N\to \infty} (e^{\hat{A}/N} e^{\hat{B}/N})^N$ is of great utility in a variety of quantum problems ranging from the theory of path integrals and Monte Carlo methods in theoretical chemistry, to many-body and thermostatistical calculations. We generalize it for the q-exponential function $e_q (x) = [1+ (1-q) x]^{(1/(1-q))}$ (with $e_1(x)=e^x$), and prove $e_q(\hat{A}+\hat{B}+(1-q) [\hat{A}\hat{B}+\hat{B}\hat{A}] /2) = \lim_{N\to \infty} {[e_{1-(1-q)N}(\hat{A}/N)] [e_{1-(1-q)N}(\hat{B}/N)]}^N$. This extended formula is expected to be similarly useful in the nonextensive situations

cond-mat.stat-mech↗

arXiv1998

Circular-like Maps: Sensitivity to the Initial Conditions, Multifractality and Nonextensivity

We generalize herein the usual circular map by considering inflexions of arbitrary power $z$, and verify that the scaling law which has been recently proposed [Lyra and Tsallis, Phys.Rev.Lett. 80 (1998) 53] holds for a large range of $z$. Since, for this family of maps, the Hausdorff dimension $d_f$ equals unity for all $z$ values in contrast with the nonextensivity parameter $q$ which does depend on $z$, it becomes clear that $d_f$ plays no major role in the sensitivity to the initial conditions.

cond-mat.stat-mech↗

arXiv1996

Sensitivity to Initial Conditions and Nonextensivity in Biological Evolution

We consider biological evolution as described within the Bak and Sneppen 1993 model. We exhibit, at the self-organized critical state, a power-law sensitivity to the initial conditions, calculate the associated exponent, and relate it to the recently introduced nonextensive thermostatistics. The scenario which here emerges without tuning strongly reminds that of the tuned onset of chaos in say logistic-like onedimensional maps. We also calculate the dynamical exponent z.

cond-mat.stat-mech↗

arXiv1995

Anomalous diffusion in the presence of external forces: exact time-dependent solutions and entropy

The optimization of the usual entropy $S_1[p]=-\int du p(u) ln p(u)$ under appropriate constraints is closely related to the Gaussian form of the exact time-dependent solution of the Fokker-Planck equation describing an important class of normal diffusions. We show here that the optimization of the generalized entropic form $S_q[p]=\{1- \int du [p(u)]^q\}/(q-1)$ (with $q=1+μ-ν\in {\bf \cal{R}}$) is closely related to the calculation of the exact time-dependent solutions of a generalized, nonlinear, Fokker Planck equation, namely $\frac{\partial}{\partial t}p^μ= -\frac{\partial}{\partial x}[F(x)p^μ]+D \frac{\partial^2} {\partial x^2}p^ν$, associated with anomalous diffusion in the presence of the external force $F(x)=k_1-k_2x$. Consequently, paradigmatic types of normal ($q=1$) and anomalous ($q \neq 1$) diffusions occurring in a great variety of physical situations become unified in a single picture.

cond-mat↗

arXiv1995

Generalized Simulated Annealing

We propose a new stochastic algorithm (generalized simulated annealing) for computationally finding the global minimum of a given (not necessarily convex) energy/cost function defined in a continuous D-dimensional space. This algorithm recovers, as particular cases, the so called classical ("Boltzmann machine") and fast ("Cauchy machine") simulated annealings, and can be quicker than both. Key-words: simulated annealing; nonconvex optimization; gradient descent; generalized statistical mechanics.

cond-mat↗