We provide evidence that cumulative distributions of absolute normalized returns for the 100 Amer... more We provide evidence that cumulative distributions of absolute normalized returns for the 100 American companies with the highest market capitalization, uncover a critical behavior for different time scales ∆t. Such cumulative distributions, in accordance with a variety of complex-and financial-systems, can be modeled by the cumulative distribution functions of q-Gaussians, the distribution function that, in the context of nonextensive statistical mechanics, maximizes a non-Boltzmannian entropy. These q-Gaussians are characterized by two parameters, namely (q, β), that are uniquely defined by ∆t. From these dependencies, we find a monotonic relationship between q and β, which can be seen as evidence of criticality. We numerically determine the various exponents which characterize this criticality.
We numerically study the two-dimensional, area preserving, web map. When the map is governed by e... more We numerically study the two-dimensional, area preserving, web map. When the map is governed by ergodic behavior, it is, as expected, correctly described by Boltzmann-Gibbs statistics, based on the additive entropic functional S BG [p(x)] = −k dx p(x) ln p(x). In contrast, possible ergodicity breakdown and transitory sticky dynamical behavior drag the map into the realm of generalized qstatistics, based on the nonadditive entropic functional S q [p(x)] = k 1− dx [p(x)] q q−1
Journal of Statistical Mechanics: Theory and Experiment, 2017
The standard map, paradigmatic conservative system in the (x, p) phase space, has been recently s... more The standard map, paradigmatic conservative system in the (x, p) phase space, has been recently shown [Tirnakli and Borges (2016)] to exhibit interesting statistical behaviors directly related to the value of the standard map external parameter K. A detailed numerical description is achieved in the present paper. More precisely, for large values of K (e.g., K = 10), the Lyapunov exponents are neatly positive over virtually the entire phase space, and, consistently with Boltzmann-Gibbs (BG) statistics, we verify q ent = q sen = q stat = q rel = 1, where q ent is the q-index for which the nonadditive entropy S q ≡ k 1− W i=1 p q i q−1 (with S 1 = S BG ≡ −k W i=1 p i ln p i) grows linearly with time before achieving its W-dependent saturation value; q sen characterizes the time increase of the sensitivity ξ to the initial conditions, i.e., ξ ∼ e λq sen t qsen (λ qsen > 0), where e z q ≡ [1 + (1 − q)z] 1/(1−q) ; q stat is the index associated with the q stat-Gaussian distribution of the time average of successive iterations of the x-coordinate; finally, q rel characterizes the q rel-exponential relaxation with time of the entropy S qent towards its saturation value. In remarkable contrast, for small values of K (e.g., K = 0.2), the Lyapunov exponents are virtually zero over the entire phase space, and, consistently with q-statistics, we verify q ent = q sen = 0, q stat ≃ 1.935, and q rel ≃ 1.4. The situation corresponding to intermediate values of K, where both stable orbits and a chaotic sea are present, is discussed as well. The present results transparently illustrate when BG behavior or q-statistical behavior are observed.
We study a symmetric generalization p (N) k (η, α) of the binomial distribution recently introduc... more We study a symmetric generalization p (N) k (η, α) of the binomial distribution recently introduced by Bergeron et al, where η ∈ [0, 1] denotes the win probability, and α is a positive parameter. This generalization is based on q-exponential generating functions (e z q gen ≡ [1+(1−q gen)z] 1/(1−q gen) ; e z 1 = e z) where q gen = 1 + 1/α. The numerical calculation of the probability distribution function of the number of wins k, related to the number of realizations N , strongly approaches a discrete q disc-Gaussian distribution, for win-loss equiprobability (i.e., η = 1/2) and all values of α. Asymptotic N → ∞ distribution is in fact a q att-Gaussian e −βz 2 q att , where q att = 1 − 2/(α − 2) and β = (2α − 4). The behavior of the scaled quantity k/N γ is discussed as well. For γ < 1, a large-deviation-like property showing a q ldl-exponential decay is found, where q ldl = 1 + 1/(ηα). For η = 1/2, q ldl and q att are related through 1/(q ldl − 1) + 1/(q att − 1) = 1, ∀α. For γ = 1, the law of large numbers is violated, and we consistently study the large-deviations with respect to the probability of the N → ∞ limit distribution, yielding a power law, although not exactly a q LD-exponential decay. All q-statistical parameters which emerge are univocally defined by (η, α). Finally we discuss the analytical connection with the Pólya urn problem.
In a recent paper ͓Phys. Rev. Lett. 86, 2038 ͑2001͔͒ a simple fluid with a particular density-dep... more In a recent paper ͓Phys. Rev. Lett. 86, 2038 ͑2001͔͒ a simple fluid with a particular density-dependent pair potential was shown to exhibit, together with the vapor-liquid transition, a liquid-liquid phase separation and it was evidenced that, in order to adequately define the correct boundaries of stability, a simulation procedure based on the use of local densities had to be devised. It was found that for certain thermodynamic states the potential drives the system toward a phase separation that is otherwise frustrated by the change in the interactions induced by density fluctuations. Therefore, when integral equations or global density simulations are used, the critical points estimated from the thermodynamics are not associated with divergent correlations and vice versa. Here, we will explore in depth this fluid and introduce a detailed account of the proposed local density simulation technique. The results presented bear general significance for density-dependent potentials, like those of liquid metals or charge-stabilized colloids.
The modified weighted density approximation is investigated in order to determine the thermodynam... more The modified weighted density approximation is investigated in order to determine the thermodynamics of the ordered CsCl and NaCl structures. Our results indicate that one of the essential assumptions of the theory, namely, the uniform fluid limit, is not well defined for mixtures. ͓S1063-651X͑98͒11810-X͔
A large-deviation connection of statistical mechanics is provided by N independent binary variabl... more A large-deviation connection of statistical mechanics is provided by N independent binary variables, the (N → ∞) limit yielding Gaussian distributions. The probability of n = N/2 out of N throws is governed by e −Nr , r related to the entropy. Large deviations for a strong correlated model characterized by indices (Q , γ) are studied, the (N → ∞) limit yielding Q-Gaussians (Q → 1 recovers a Gaussian). Its large deviations are governed by e −Nrq q (∝ 1/N 1/(q−1) , q > 1), q = (Q − 1)/(γ [3 − Q ]) + 1. This illustration opens the door towards a large-deviation foundation of nonextensive statistical mechanics.
Evidence of a liquid-liquid equilibrium in simple fluids has recently been exposed for a densityd... more Evidence of a liquid-liquid equilibrium in simple fluids has recently been exposed for a densitydependent pair potential in the framework of a van der Waals theory. Here this double criticality is investigated by means of computer simulation, a perturbation theory, and integral equation theory. It is found that the critical point estimated from the integral equation thermodynamics is not associated with divergent correlations. To cope with these features, a special simulation procedure, based on the definition of local densities, is devised. Monte Carlo calculations confirm the existence of two critical points, in agreement with the predictions of perturbation theory.
Physica A: Statistical Mechanics and its Applications, 2007
We numerically study two conservative two-dimensional maps, namely the baker map (whose Lyapunov ... more We numerically study two conservative two-dimensional maps, namely the baker map (whose Lyapunov exponent is known to be positive), and a typical one (exhibiting a vanishing Lyapunov exponent) chosen from the generalized shift family of maps introduced by C. Moore [Phys Rev Lett 64, 2354 (1990)] in the context of undecidability. We calculated the time evolution of the entropy S q ≡ 1− W i=1 p q i q−1 (S 1 = S BG ≡ − W i=1 p i ln p i), and exhibited the dramatic effect introduced by numerical precision. Indeed, in spite of being area-preserving maps, they present, well after the initially concentrated ensemble has spread virtually all over the phase space, unexpected pseudo-attractors (fixed-point like for the baker map, and more complex structures for the Moore map). These pseudo-attractors, and the apparent time (partial) reversibility they provoke, gradually disappear for increasingly large precision. In the case of the Moore map, they are related to zero Lebesgue-measure effects associated with the frontiers existing in the definition of the map. In addition to the above, and consistently with the results by V. Latora and M. Baranger [Phys. Rev. Lett. 82, 520 (1999)], we find that the rate of the far-from-equilibrium entropy production of baker map, numerically coincides with the standard Kolmogorov-Sinai entropy of this strongly chaotic system.
The low-density phase diagrams of charge-stabilized colloidal suspensions of the Derjaguin–Landau... more The low-density phase diagrams of charge-stabilized colloidal suspensions of the Derjaguin–Landau–Verwey–Overbeek theory with an approximate effective one-component Hamiltonian given by the volume term and effective pair interactions, and of the classical theory (without including the volume term), are obtained from the hypernetted-chain integral equation at low colloidal charges. In the salt-free case both phase diagrams exhibit a vapour–liquid transition with
International Journal of Bifurcation and Chaos, 2012
We study chaotic orbits of conservative low-dimensional maps and present numerical results showin... more We study chaotic orbits of conservative low-dimensional maps and present numerical results showing that the probability density functions (pdfs) of the sum of N iterates in the large N limit exhibit very interesting time-evolving statistics. In some cases where the chaotic layers are thin and the (positive) maximal Lyapunov exponent is small, long-lasting quasi-stationary states (QSS) are found, whose pdfs appear to converge to q-Gaussians associated with nonextensive statistical mechanics. More generally, however, as N increases, the pdfs describe a sequence of QSS that pass from a q-Gaussian to an exponential shape and ultimately tend to a true Gaussian, as orbits diffuse to larger chaotic domains and the phase space dynamics becomes more uniformly ergodic.
... «Mira, Guiomar me dijo algo así podría objetarlo un cura, ¡pero un científico no!». ... Es ... more ... «Mira, Guiomar me dijo algo así podría objetarlo un cura, ¡pero un científico no!». ... Es verdad que las horas de escalas en el piano o de dibujar 13 José Antonio Molina, Teoría de la inteligencia creadora, Anagrama, Barcelona 1993. 16 Page 18. ...
... Dr. Carlos Fernández Tejero Dr. Enrique Lomba Garc´ıa Page 3. ... directa colaboradora Sánche... more ... Dr. Carlos Fernández Tejero Dr. Enrique Lomba Garc´ıa Page 3. ... directa colaboradora SánchezBalmaseda. ... La disponibilidad del Dr. Fernández Tejero para asumir una nueva tarea de di-rección, después de mi interrupción durante algunos anos de toda vinculación con la ...
We introduce a new universality class of one-dimensional unimodal dissipative maps. The new famil... more We introduce a new universality class of one-dimensional unimodal dissipative maps. The new family, from now on referred to as the (z 1 , z 2)-logarithmic map, corresponds to a generalization of the z-logistic map. The Feigenbaum-like constants of these maps are determined. It has been recently shown that the probability density of sums of iterates at the edge of chaos of the z-logistic map is numerically consistent with a q-Gaussian, the distribution which, under appropriate constraints, optimizes the nonadditive entropy S q. We focus here on the presently generalized maps to check whether they constitute a new universality class with regard to q-Gaussian attractor distributions. We also study the generalized q-entropy production per unit time on the new unimodal dissipative maps, both for strong and weak chaotic cases. The q-sensitivity indices are obtained as well. Our results are, like those for the z-logistic maps, numerically compatible with the q-generalization of a Pesin-like identity for ensemble averages.
We provide evidence that cumulative distributions of absolute normalized returns for the 100 Amer... more We provide evidence that cumulative distributions of absolute normalized returns for the 100 American companies with the highest market capitalization, uncover a critical behavior for different time scales ∆t. Such cumulative distributions, in accordance with a variety of complex-and financial-systems, can be modeled by the cumulative distribution functions of q-Gaussians, the distribution function that, in the context of nonextensive statistical mechanics, maximizes a non-Boltzmannian entropy. These q-Gaussians are characterized by two parameters, namely (q, β), that are uniquely defined by ∆t. From these dependencies, we find a monotonic relationship between q and β, which can be seen as evidence of criticality. We numerically determine the various exponents which characterize this criticality.
We numerically study the two-dimensional, area preserving, web map. When the map is governed by e... more We numerically study the two-dimensional, area preserving, web map. When the map is governed by ergodic behavior, it is, as expected, correctly described by Boltzmann-Gibbs statistics, based on the additive entropic functional S BG [p(x)] = −k dx p(x) ln p(x). In contrast, possible ergodicity breakdown and transitory sticky dynamical behavior drag the map into the realm of generalized qstatistics, based on the nonadditive entropic functional S q [p(x)] = k 1− dx [p(x)] q q−1
Journal of Statistical Mechanics: Theory and Experiment, 2017
The standard map, paradigmatic conservative system in the (x, p) phase space, has been recently s... more The standard map, paradigmatic conservative system in the (x, p) phase space, has been recently shown [Tirnakli and Borges (2016)] to exhibit interesting statistical behaviors directly related to the value of the standard map external parameter K. A detailed numerical description is achieved in the present paper. More precisely, for large values of K (e.g., K = 10), the Lyapunov exponents are neatly positive over virtually the entire phase space, and, consistently with Boltzmann-Gibbs (BG) statistics, we verify q ent = q sen = q stat = q rel = 1, where q ent is the q-index for which the nonadditive entropy S q ≡ k 1− W i=1 p q i q−1 (with S 1 = S BG ≡ −k W i=1 p i ln p i) grows linearly with time before achieving its W-dependent saturation value; q sen characterizes the time increase of the sensitivity ξ to the initial conditions, i.e., ξ ∼ e λq sen t qsen (λ qsen > 0), where e z q ≡ [1 + (1 − q)z] 1/(1−q) ; q stat is the index associated with the q stat-Gaussian distribution of the time average of successive iterations of the x-coordinate; finally, q rel characterizes the q rel-exponential relaxation with time of the entropy S qent towards its saturation value. In remarkable contrast, for small values of K (e.g., K = 0.2), the Lyapunov exponents are virtually zero over the entire phase space, and, consistently with q-statistics, we verify q ent = q sen = 0, q stat ≃ 1.935, and q rel ≃ 1.4. The situation corresponding to intermediate values of K, where both stable orbits and a chaotic sea are present, is discussed as well. The present results transparently illustrate when BG behavior or q-statistical behavior are observed.
We study a symmetric generalization p (N) k (η, α) of the binomial distribution recently introduc... more We study a symmetric generalization p (N) k (η, α) of the binomial distribution recently introduced by Bergeron et al, where η ∈ [0, 1] denotes the win probability, and α is a positive parameter. This generalization is based on q-exponential generating functions (e z q gen ≡ [1+(1−q gen)z] 1/(1−q gen) ; e z 1 = e z) where q gen = 1 + 1/α. The numerical calculation of the probability distribution function of the number of wins k, related to the number of realizations N , strongly approaches a discrete q disc-Gaussian distribution, for win-loss equiprobability (i.e., η = 1/2) and all values of α. Asymptotic N → ∞ distribution is in fact a q att-Gaussian e −βz 2 q att , where q att = 1 − 2/(α − 2) and β = (2α − 4). The behavior of the scaled quantity k/N γ is discussed as well. For γ < 1, a large-deviation-like property showing a q ldl-exponential decay is found, where q ldl = 1 + 1/(ηα). For η = 1/2, q ldl and q att are related through 1/(q ldl − 1) + 1/(q att − 1) = 1, ∀α. For γ = 1, the law of large numbers is violated, and we consistently study the large-deviations with respect to the probability of the N → ∞ limit distribution, yielding a power law, although not exactly a q LD-exponential decay. All q-statistical parameters which emerge are univocally defined by (η, α). Finally we discuss the analytical connection with the Pólya urn problem.
In a recent paper ͓Phys. Rev. Lett. 86, 2038 ͑2001͔͒ a simple fluid with a particular density-dep... more In a recent paper ͓Phys. Rev. Lett. 86, 2038 ͑2001͔͒ a simple fluid with a particular density-dependent pair potential was shown to exhibit, together with the vapor-liquid transition, a liquid-liquid phase separation and it was evidenced that, in order to adequately define the correct boundaries of stability, a simulation procedure based on the use of local densities had to be devised. It was found that for certain thermodynamic states the potential drives the system toward a phase separation that is otherwise frustrated by the change in the interactions induced by density fluctuations. Therefore, when integral equations or global density simulations are used, the critical points estimated from the thermodynamics are not associated with divergent correlations and vice versa. Here, we will explore in depth this fluid and introduce a detailed account of the proposed local density simulation technique. The results presented bear general significance for density-dependent potentials, like those of liquid metals or charge-stabilized colloids.
The modified weighted density approximation is investigated in order to determine the thermodynam... more The modified weighted density approximation is investigated in order to determine the thermodynamics of the ordered CsCl and NaCl structures. Our results indicate that one of the essential assumptions of the theory, namely, the uniform fluid limit, is not well defined for mixtures. ͓S1063-651X͑98͒11810-X͔
A large-deviation connection of statistical mechanics is provided by N independent binary variabl... more A large-deviation connection of statistical mechanics is provided by N independent binary variables, the (N → ∞) limit yielding Gaussian distributions. The probability of n = N/2 out of N throws is governed by e −Nr , r related to the entropy. Large deviations for a strong correlated model characterized by indices (Q , γ) are studied, the (N → ∞) limit yielding Q-Gaussians (Q → 1 recovers a Gaussian). Its large deviations are governed by e −Nrq q (∝ 1/N 1/(q−1) , q > 1), q = (Q − 1)/(γ [3 − Q ]) + 1. This illustration opens the door towards a large-deviation foundation of nonextensive statistical mechanics.
Evidence of a liquid-liquid equilibrium in simple fluids has recently been exposed for a densityd... more Evidence of a liquid-liquid equilibrium in simple fluids has recently been exposed for a densitydependent pair potential in the framework of a van der Waals theory. Here this double criticality is investigated by means of computer simulation, a perturbation theory, and integral equation theory. It is found that the critical point estimated from the integral equation thermodynamics is not associated with divergent correlations. To cope with these features, a special simulation procedure, based on the definition of local densities, is devised. Monte Carlo calculations confirm the existence of two critical points, in agreement with the predictions of perturbation theory.
Physica A: Statistical Mechanics and its Applications, 2007
We numerically study two conservative two-dimensional maps, namely the baker map (whose Lyapunov ... more We numerically study two conservative two-dimensional maps, namely the baker map (whose Lyapunov exponent is known to be positive), and a typical one (exhibiting a vanishing Lyapunov exponent) chosen from the generalized shift family of maps introduced by C. Moore [Phys Rev Lett 64, 2354 (1990)] in the context of undecidability. We calculated the time evolution of the entropy S q ≡ 1− W i=1 p q i q−1 (S 1 = S BG ≡ − W i=1 p i ln p i), and exhibited the dramatic effect introduced by numerical precision. Indeed, in spite of being area-preserving maps, they present, well after the initially concentrated ensemble has spread virtually all over the phase space, unexpected pseudo-attractors (fixed-point like for the baker map, and more complex structures for the Moore map). These pseudo-attractors, and the apparent time (partial) reversibility they provoke, gradually disappear for increasingly large precision. In the case of the Moore map, they are related to zero Lebesgue-measure effects associated with the frontiers existing in the definition of the map. In addition to the above, and consistently with the results by V. Latora and M. Baranger [Phys. Rev. Lett. 82, 520 (1999)], we find that the rate of the far-from-equilibrium entropy production of baker map, numerically coincides with the standard Kolmogorov-Sinai entropy of this strongly chaotic system.
The low-density phase diagrams of charge-stabilized colloidal suspensions of the Derjaguin–Landau... more The low-density phase diagrams of charge-stabilized colloidal suspensions of the Derjaguin–Landau–Verwey–Overbeek theory with an approximate effective one-component Hamiltonian given by the volume term and effective pair interactions, and of the classical theory (without including the volume term), are obtained from the hypernetted-chain integral equation at low colloidal charges. In the salt-free case both phase diagrams exhibit a vapour–liquid transition with
International Journal of Bifurcation and Chaos, 2012
We study chaotic orbits of conservative low-dimensional maps and present numerical results showin... more We study chaotic orbits of conservative low-dimensional maps and present numerical results showing that the probability density functions (pdfs) of the sum of N iterates in the large N limit exhibit very interesting time-evolving statistics. In some cases where the chaotic layers are thin and the (positive) maximal Lyapunov exponent is small, long-lasting quasi-stationary states (QSS) are found, whose pdfs appear to converge to q-Gaussians associated with nonextensive statistical mechanics. More generally, however, as N increases, the pdfs describe a sequence of QSS that pass from a q-Gaussian to an exponential shape and ultimately tend to a true Gaussian, as orbits diffuse to larger chaotic domains and the phase space dynamics becomes more uniformly ergodic.
... «Mira, Guiomar me dijo algo así podría objetarlo un cura, ¡pero un científico no!». ... Es ... more ... «Mira, Guiomar me dijo algo así podría objetarlo un cura, ¡pero un científico no!». ... Es verdad que las horas de escalas en el piano o de dibujar 13 José Antonio Molina, Teoría de la inteligencia creadora, Anagrama, Barcelona 1993. 16 Page 18. ...
... Dr. Carlos Fernández Tejero Dr. Enrique Lomba Garc´ıa Page 3. ... directa colaboradora Sánche... more ... Dr. Carlos Fernández Tejero Dr. Enrique Lomba Garc´ıa Page 3. ... directa colaboradora SánchezBalmaseda. ... La disponibilidad del Dr. Fernández Tejero para asumir una nueva tarea de di-rección, después de mi interrupción durante algunos anos de toda vinculación con la ...
We introduce a new universality class of one-dimensional unimodal dissipative maps. The new famil... more We introduce a new universality class of one-dimensional unimodal dissipative maps. The new family, from now on referred to as the (z 1 , z 2)-logarithmic map, corresponds to a generalization of the z-logistic map. The Feigenbaum-like constants of these maps are determined. It has been recently shown that the probability density of sums of iterates at the edge of chaos of the z-logistic map is numerically consistent with a q-Gaussian, the distribution which, under appropriate constraints, optimizes the nonadditive entropy S q. We focus here on the presently generalized maps to check whether they constitute a new universality class with regard to q-Gaussian attractor distributions. We also study the generalized q-entropy production per unit time on the new unimodal dissipative maps, both for strong and weak chaotic cases. The q-sensitivity indices are obtained as well. Our results are, like those for the z-logistic maps, numerically compatible with the q-generalization of a Pesin-like identity for ensemble averages.
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Papers by Guiomar Ruiz