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In the case of fully chaotic systems the distribution of the Poincare'recurrence times is an exponential whose decay rate is the Kolmogorov-Sinai(KS) entropy.We address the discussion of the same problem, the connection between dynamics and thermodynamics,in the case of sporadic randomness,using the Manneville map as a prototype of this class of processes. We explore the possibility of relating the distribution of Poincare' recurrence times to `thermodynamics',in the sense of the KS entropy,also in the case of an inverse power law. This is the dynamic property that Zaslavsly [Phys.Today(8), 39(1999)] finds to be responsible for a striking deviation from ordinary statistical mechanics under the form of Maxwell's Demon effect. We show that this way of estabi- lishing a connection between thermodynamics and dynamics is valid only in the case of strong chaos. In the case of sporadic randomness, resulting at long times in the Levy diffusion processes,the sensitivity to initial conditions is initially an inverse pow erlaw,but it becomes exponential in the long-time scale, whereas the distribution of Poincare times keeps its inverse power law forever. We show that a nonextensive thermodynamics would imply the Maxwell's Demon effect to be determined by memory and thus to be temporary,in conflict with the dynamic approach to Levy statistics. The adoption of heuristic arguments indicates that this effect,is possible, as a form of genuine equilibrium,after completion of the process of memory erasure., Comment: 13 pages,two-column,latex-revtex,2 figures(2 files .eps)

We study the influence of a dissipation process on diffusion dynamics triggered by fluctuations with long-range correlations. We make the assumption that the perturbation process involved is of the same kind as those recently studied numerically and theoretically, with a good agreement between theory and numerical treatment. As a result of this assumption the equilibrium distribution departs from the ordinary canonical distribution. The distribution tails are truncated, the distribution border is signaled by sharp peaks, and, in the weak dissipation limit, the central distribution body becomes identical to a truncated L\'evy distribution.

We study the entropy time evolution of a quantum mechanical model, which is frequently used as a prototype for Anderson's localization. Recently Latora and Baranger [V. Latora, M. Baranger, Phys. Rev.Lett. 82, 520(1999)] found that there exist three entropy regimes, a transient regime of passage from dynamics to thermodynamics, a linear in time regime of entropy increase, namely a thermodynamic regime of Kolmogorov kind, and a saturation regime. We use the non-extensive entropic indicator recently advocated by Tsallis [ C. Tsallis, J. Stat. Phys. 52, 479 (1988)] with a mobile entropic index q, and we find that with the adoption of the ``magic'' value q = Q = 1/2 the Kolmogorov regime becomes more extended and more distinct than with the traditional entropic index q = 1. We adopt a two-site model to explain these properties by means of an analytical treatment and we argue that Q =1/2 might be a typical signature of the occurrence of Anderson's localization., 13 pages, 8 figures submitted to Phys. Rev. E

We adapt the Kolmogorov-Sinai entropy to the non-extensive perspective recently advocated by Tsallis. The resulting expression is an average on the invariant distribution, which should be used to detect the genuine entropic index Q. We argue that the condition Q >1 is time dependent.

We address the problem of relating thermodynamics to mechanics in the case of microscopic dynamics without a finite time scale. The solution is obtained by expressing the Tsallis entropic index q as a function of the Levy index alpha, and using dynamical rather than probabilistic arguments., 4 pages, new revised version resubmitted to Phys. Rev. Lett

We generalize the method of Van Hove so as to deal with the case of non-ordinary statistical mechanics, that being phenomena with no time-scale separation. We show that in the case of ordinary statistical mechanics, even if the adoption of the Van Hove method imposes randomness upon Hamiltonian dynamics, the resulting statistical process is described using normal calculus techniques. On the other hand, in the case where there is no time-scale separation, this generalized version of Van Hove's method not only imposes randomness upon the microscopic dynamics, but it also transmits randomness to the macroscopic level. As a result, the correct description of macroscopic dynamics has to be expressed in terms of the fractional calculus., 20 pages, 1 figure

We address the issue of the quantum-classical correspondence in chaotic systems using, as recently done by Zurek [e-print quant-ph/9802054], the solar system as a whole as a case study: this author shows that the classicality of the planetary motion is ensured by the environment-induced decoherence. We show that equivalent results are provided by the theories of spontaneous fluctuations and that these latter theories, in some cases, result in a still faster process of decoherence. We show that, as an additional benefit, the assumption of spontaneous fluctuation makes it possible to genuinely derive thermodynamics from mechanics, namely, without implicitly assuming thermodynamics., 9 pages, two tables included, RevTex. Concluding part of Sec. IV revised and shortened

We investigate the tunneling process between two symmetric stable islands of a forced pendulum Hamiltonian in the weak chaos regime. We show that when the tunneling doublet is quantized over a classical non-linear resonance the tunneling rate strongly deviates from the semiclassical prediction. This mechanism is responsible for the irregular dependence of the tunneling rate on the system parameters. The weak-chaos condition allows us to make a theoretical prediction which agrees very well with the numerical results. This opens up a possible avenue to a general theory on the dependence of quantum tunneling on classical chaos., 4 pages, 2 figures

Herein we study the effects on Anderson localization of correlations in the energy distribution of the sites of a tight-binding Hamiltonian. The lattice correlations are introduced by means of classical maps generating anomalous diffusion, that have recently been found to account for the correlated disorder of ‘‘biological’’ lattices. We show that the enhancement of localization length takes place on a much wider band of energies than in the case of the random-dimer model if the random walk on the site energies of the tight-binding Hamiltonian is determined by the joint action of short- and long-range correlations. @S0163-1829~96!08641-9#

We derive the diffusion process generated by a correlated dichotomous fluctuating variable y starting from a Liouville-like equation by means of a projection procedure. This approach makes it possible to derive all statistical properties of the diffusion process from the correlation function of the dichotomous fluctuating variable F y(t). Of special interest is that the distribution of the times of sojourn in the two states of the fluctuating process is proportional to d 2 F y(t)/dt 2 . Furthermore, in the special case where F y(t) has an inverse power law, with the index b ranging from 0 to 1, thus making it nonintegrable, we show analytically that the

This article discusses the linear response of Hamiltonian chaotic systems as a function of the number of degrees of freedom.

We argue that the condition of local thermal equilibrium realized several years ago by Rich and Visscher [Phys. Rev. B 11, 2164 (1975)] through a process of mathematical convergence can be obtained dynamically by adopting the prescription of a recent paper [M. Bianucci, R. Mannella, B. J. West, and P. Grigolini, Phys. Rev. E 51, 3002 (1995)]. This should contribute to shedding light on the still unsolved problem of the microscopic derivation of the heat Fourier law. (c) 1995 The American Physical Society

We study the effect of weak environmental fluctuations on a deterministic map dynamics, which, in the unperturbed case, is characterized by anomalous diffusion. We show, with the help of numerical calculations, that there is a crossover time 1/\ensuremath{\epsilon}, at which the waiting time distributions change from an inverse power-law distribution into an exponential behavior. We prove theoretically that the diffusion coefficient of the long-time process is proportional to 1/${\mathrm{\ensuremath{\epsilon}}}^{\mathrm{\ensuremath{\alpha}}}$, with \ensuremath{\alpha} positive or negative, according to whether we consider the superdiffusive or the subdiffusive case. With very weak environmental fluctuations the diffusion coefficient of the former case becomes anomalously large and that of the latter case anomalously small. The theoretical predictions are confirmed by the numerical results.

Erratum Erratum to ‘‘A new measure of network efficiency’’ [Physica A 391 (4) (2012) 1894–1899] Nicholas W. Hollingshad , Malgorzata Turalska, Paolo Allegrini, Bruce J. West, Paolo Grigolini Center for Nonlinear Science, University of North Texas, P.O. Box 311427, Denton, TX 76203-1427, USA Istituto di Fisiologia Clinica (IFC-CNR), Via Moruzzi 1, 56124 Pisa, Italy Information Science Directorate, US Army Research Office, Durham, NC 27709, USA

We study the response of a classical Hamiltonian system to a weak perturbation in the regime where the dynamics is mixing, with the purpose of critically examining both the foundation of the Kubo linear response theory (LRT) and van Kampen's well known objections to LRT [Phys. Norv. 5, 279 (1971)]. Although the exactness of LRT for short times is not surprising, we prove that for the class of model studied here the LRT must also become accurate in the limit of long times, even for macroscopically large external perturbations. Hence, if the LRT breaks down, the breakdown occurs in the region of intermediate times. We also show that, for a given system, if any macroscopic linear response exists, it must coincide with Kubo LRT; thus if a generic system responds nonlinearly to an external perturbation, this nonlinear response is observable only in an intermediate-time range. Numerical calculations carried out on some model systems with only a few degrees of freedom support these arguments.

L\'evy statistics are derived from a dynamical system, which can be either Hamiltonian or not, using a master equation approach. We compare these predictions to the random walk approach recently developed by Zumofen and Klafter for both the nonstationary [Phys. Rev. E 47, 851 (1993)] and stationary [Physica A 196, 102 (1993)] case. We study the unperturbed dynamics of the system analytically and numerically and evaluate the time evolution of the second moment of the probability distribution. We also study the response of the dynamical system undergoing anomalous diffusion to an external perturbation and show that if the slow regression to equilibrium of the variable ``velocity'' is triggered by the perturbation, the process of diffusion of the ``space'' variable takes place under nonstationary conditions and a conductivity steadily increasing with time is generated in the early part of the response process. In the regime of extremely long times the conductivity becomes constant with a value, though, that does not correspond to the prescriptions of the ordinary Green-Kubo treatments.

We focus our attention on a fractal model recently proposed by Liebovitch to account for the lack of a time scale in ion channel kinetics. We establish a connection between the dwell-time distributions and the correlation time of the ion channel signal, thereby making it possible to derive analytical predictions on the diffusion properties of a random walk constructed from the sum of the current fluctuations of ion channels. With the help of a numerical simulation of the Liebovitch model, it is shown that the Hurst analysis can provide a reliable determination of the standard (or anomalous) diffusion properties. On the basis of results of computer simulation we argue that by applying the Hurst analysis to the experimental distribution of closed times it is possible, in principle, to establish whether the Liebovitch model is valid.

We study the effect that the state-vector reduction of Ghirardi, Rimini, and Weber (GRW) [Phys. Rev. D 34, 470 (1986)] has on the quantum fluctuation-dissipation process. A conventional fluctuation-dissipation process stems from the interaction between a system of interest and a macroscopic number of microscopic variables, thereby suggesting the possible occurrence of an enhancement of the GRW wave-function collapse. We carry out an exact treatment of the joint action of conventional quantum dissipation and GRW wave-function collapse in the special case where the system of interest is a harmonic oscillator. We prove that the GRW dynamics does not lead to any significant departure from the predictions of ordinary quantum mechanics, if the region of astronomically large times is disregarded, where the energy of the system diverges as an effect of a slow but steady increase caused by the GRW collapses.

We apply the theory developed in [Marco Bianucci. (2004) Ordinary chemical reaction process induced by a unidimensional map. Physical Review E 70] to two deterministic nonlinear systems with few degrees of freedom (mappings). The first mapping considered is known to exactly satisfy the prescriptions laid down in I to obtain the Fokker-Planck equation associated to Brownian motion. For the second mapping, due to its more complex form, it is not possible to prove analytically that it too satisfies the prescriptions of I; however, we show numerically that this fact is plausible, at least in the chaotic regime. For both cases we show that indeed Brownian motion for the variable of interest w arises. We conclude arguing that the theory developped in I is generally applicable to systems for which the "thermal bath" is in a fully chaotic state. Read More: http://www.worldscientific.com/doi/abs/10.1142/S0217979294000580

We study the dynamics of a lazy random walker who is inactive for extended times and tries to make up for her laziness with very large jumps. She remains in a condition of rest for a time τ derived from a waiting-time distribution , with μW 1. The most convenient choice to make up for the random walker laziness would be to select μξ 2 would produce Levy flights with scaling δ = 1/(μξ − 1) and consequently super-diffusion. According to the Sparre Andersen theorem, the distribution density of the first times to go from xA to xB > xA has the inverse power law form with μFPT = μSA = 1.5. We find the surprising result that there exists a region of the phase space (μξ, μW) with μW μSA and the lazy walker compensates for her laziness. There also exists an extended region breaking the Sparre Andersen theorem, where the lazy runner cannot compensate for her laziness. We make conjectures concerning the possible relevance of this mathematical prediction, supported by numerical calculations, for the problem of animal foraging.

We consider the spin-boson model as a measurement device, with the 1/2-spin system acting as a pointer and the influence of the environment simulated by means of the interaction between the 1/2-spin system and a dissipative oscillator. In accordance with the recent finding of Bonilla and Guinea [L. L. Bonilla and F. Guinea, Phys. Rev. A 45, 7718 (1992)] the assumption that the oscillator is classical results, under specific conditions, in the occurrence of a wave-function collapse. However, our theoretical analysis shows that the wave-function collapse is provoked by the action exerted by the 1/2-spin system on the oscillator (reaction field) and that the reaction field is as intense as the quantum-mechanical fluctuation neglected by the semiclassical approximation

We show that in classically chaotic systems the quantum uncertainty, in spite of being assumed to be extremely small so as to make the classical approximation possible, increases very quickly thereby making these classical systems strongly quantum. It is argued that these systems also exhibit an irreversible-like behavior, albeit the phase correlations among different regions of the wavefunction make them reversible. From a statistical point of view it is difficult to distinguish the quantum behavior of these systems from the predictions of the classical approximation, unless the resulting transport process does not essentially rest either on tunneling or on localization processes (or on both). The dinstinctively quantum mechanical features of the transport process might be destroyed by the interaction between the system and environment. This possible role of environmental fluctuations, dealt within a fully quantum formalism, is discussed.

Kenkre and Wu1 have studied the nonadiabatic effects on the nonlinear dimer, assuming a finite relaxation time for the vibrations.

This paper addresses the problem of deriving the probability distribution density of a diffusion process generated by a nonergodic dichotomous fluctuation using the Liouville equation (density method). The velocity of the diffusing particles fluctuates from the value of 1 to the value of −1, and back, with the distribution density of time durations τ of the two states proportional to 1/τμ in the asymptotic time limit. The adopted density method allows us to establish an exact analytical expression for the probability distribution density of the diffusion process generated by these fluctuations. Contrary to intuitive expectations, the central part of the diffusion distribution density is not left empty when moving from μ>2 (ergodic condition) to μ μcr, the monomodal distribution density with a minimum at the origin is turned into a bimodal one, with a central bump whose intensity increases for μ→2. The ...

Using the Wigner method, it is shown that a classical-like equation of motion for a quasiprobability distribution ${\mathrm{\ensuremath{\rho}}}_{\mathit{W}}$ can be built up, \ensuremath{\partial}${\mathrm{\ensuremath{\rho}}}_{\mathit{W}}$/\ensuremath{\partial}t=(L${\mathrm{^}}_{\mathrm{cl}}$+L${\mathrm{^}}_{\mathrm{QGD}}$)${\mathrm{\ensuremath{\rho}}}_{\mathit{W}}$, which is rigorously equivalent to the quantum von Neumann--Liouville equation. The operator L${\mathrm{^}}_{\mathrm{cl}}$ is equivalent to integrating classical trajectories, which are then averaged over an initial distribution, broadened so as to fulfill the requirements of the quantum uncertainty principle. It is shown that this operator produces semiclassical chaos and is responsible for quantum irreversibility and the fast growth of quantum uncertainty. Carrying out explicit calculations for a spin-boson Hamiltonian, the joint action of L${\mathrm{^}}_{\mathrm{cl}}$ and L${\mathrm{^}}_{\mathrm{QGD}}$ is illustrated. It is shown that the latter operator L${\mathrm{^}}_{\mathrm{QGD}}$ (where QGD stands for quantum generating diffusion), makes the 1/2-spin system ``remember'' its quantum nature, and competes with the irreversibility induced by the former operator. Some ambiguous aspects of ``irreversibility'' and ``growth of quantum fluctuations'' as indicators of semiclassical chaos are discussed.

The goal of this thesis is to contribute to the ambitious program of the foundation of developing statistical physics using chaos. We build a deterministic model of Brownian motion and provide a microscpoic derivation of the Fokker-Planck equation. Since the Brownian motion of a particle is the result of the competing processes of diffusion and dissipation, we create a model where both diffusion and dissipation originate from the same deterministic mechanism - the deterministic interaction of that particle with its environment. We show that standard diffusion which is the basis of the Fokker-Planck equation rests on the Central Limit Theorem, and, consequently, on the possibility of deriving it from a deterministic process with a quickly decaying correlation function. The sensitive dependence on initial conditions, one of the defining properties of chaos insures this rapid decay. We carefully address the problem of deriving dissipation from the interaction of a particle with a fully deterministic nonlinear bath, that we term the booster. We show that the solution of this problem essentially rests on the linear response of a booster to an external perturbation. This raises a long-standing problem concerned with Kubo's Linear Response Theory and the strong criticism against it by van Kampen. Kubo's theory is based on a perturbation treatment of the Liouville equation, which, in turn, is expected to be totally equivalent to a first-order perturbation treatment of single trajectories. Since the boosters are chaotic, and chaos is essential to generate diffusion, the single trajectories are highly unstable and do not respond linearly to weak external perturbation. We adopt chaotic maps as boosters of a Brownian particle, and therefore address the problem of the response of a chaotic booster to an external perturbation. We notice that a fully chaotic map is characterized by an invariant measure which is a continuous function of the control parameters of the map. Consequently if the external perturbation is made to act on a control parameter of the map, we show that the booster distribution undergoes slight modifications as an effect of the weak external perturbation, thereby leading to a linear response of the mean value of the perturbed variable of the booster. This approach to linear response completely bypasses the criticism of van Kampen. The joint use of these two phenomena, diffusion and friction stemming from the interaction of the Brownian particle with the same booster, makes the microscopic derivation of a Fokker-Planck equation and Brownian motion, possible.

We discuss the problem of deriving Levy diffusion, in the form of a Levy walk, from a density approach, namely using a Liouville equation. We address this problem for a case that has already been discussed using the method of the continuous time random walk, and consequently an approach based on trajectory dynamics rather than density time evolution. We show that the use of the Liouville equation requires the knowledge of the correlation functions of the fluctuation that generates the Levy diffusion. We benefit from the results of earlier work proving that the correlation function is not factorized as in the Poisson case. We show that the Liouville equation generates a long-time diffusion whose probability distribution density keeps a memory of the detailed form of the fluctuation correlation function, and not only of its long-time inverse power law structure. Although the main result of this paper rests on an approximate expression for higher-order correlation functions, it becomes exact in the long-time limit. Thus, we argue that it explains the failure to derive Levy diffusion from the Liouville equation, thereby supporting the claim that there exists a conflict between trajectory and density approaches in this case.

The response of a system with ON–OFF intermittency to an external harmonic perturbation is discussed. ON–OFF intermittency is described by means of a sequence of random events, i.e., the transitions from the ON to the OFF state and vice versa. The unperturbed waiting times (WTs) between two events are assumed to satisfy a renewal condition, i.e., the WTs are statistically independent random variables. The response of a renewal model with non-Poisson ON–OFF intermittency, associated with non-exponential WT distribution, is analyzed by looking at the changes induced in the WT statistical distribution by the harmonic perturbation. The scaling properties are also studied by means of diffusion entropy analysis. It is found that, in the range of fast and relatively strong perturbation, the non-Poisson system displays a Poisson-like behavior in both WT distribution and scaling. In particular, the histogram of perturbed WTs becomes a sequence of equally spaced peaks, with intensity decaying exponentially in time. Further, the diffusion entropy detects an ordinary scaling (related to normal diffusion) instead of the expected unperturbed anomalous scaling related to the inverse power-law decay. Thus, an analysis based on the WT histogram and/or on scaling methods has to be considered with some care when dealing with perturbed intermittent systems.

The dynamical properties of a semiclassical model of spin relaxation are carefully studied with the help of computer simulation. It is shown that, as an effect of the reaction field, the relaxation time as a function of temperature ranges from an Arrhenius-like behavior at weak temperatures to a linear power-law behavior at high temperatures. It is also shown that a fairly accurate theoretical prediction of the relaxation time can be obtained in both the low- and high-temperature limit. In the latter regime a simple factorization prescription, although leading to small deviations from the exact equilibrium value for the spin components, is shown to provide the most accurate predictions.

We study the effects of the reaction field on the spin relaxation. We find that they are not only responsible for a nonlinear friction process, as recently pointed out by several authors, but, in the case of sufficiently strong coupling, they also result in a bistable precession process. This implies that at weak temperatures the rate of the spin relaxation is expected to exhibit an Arrhenius-like dependence on temperature.

We have obtained results for the activation energy in bistable overdamped systems driven by colored noise. Our results cover the region of small diffusion coefficients and small to large correlation times. A scaling relation between the escape rate from one attractor to the other and the diffusion coefficient allows us to obtain results for the activation energy in the limit of vanishingly small diffusion coefficients. Numerical results for both the activation energy and the mean first-passage time are compared with theoretical predictions.

Article replying to a comment by Massimo Falcioni and Angelo Vulpiani. The authors discuss research on the linear response theory (LRT).

As stated in the text, Figs. 3 and 4 (transmission versus probe laser detuning) were taken at an energy of 10 nJ in a 0.5-mm-diam beam. We should have noted that when the probe energy is increased to 150 n3 (— 10" W/cm ) the absorption cross section at the transmission maximum increases by a factor of approximately 2. At this energy density, with or without the coupling laser present, transmitted probe energy is no longer linear with incident probe energy.

Herein we establish a relation between quantum irreversibility and the chaotic semi-classical solutions for a spin-boson Hamiltonian system. We obtain quantum averages by numerically integrating the appropriate Liouville-Von Neumann equations of motion and find these averages to be less erratic than the corresponding chaotic semi-classical trajectories. However, the quantum averages are shown to be dissipative as measured by the entropy of the spin subsystem and to suppress the phenomenon of "revivals".

The perturbative approximation theory recently developed in order to study the decay of non-markovian dissipative systems in the presence of high-intensity electromagnetic fields, is here shown to be especially suitable for applications to the theory of molecular systems undergoing both chemical relaxation and irradiation by external electromagnetic fields. In fact, such an approach allows irradiation processes to be studied without disregarding the non-markovian nature of chemical relaxation, which seems to be the most interesting new feature of the unimolecular reaction rate theory recently developed by Nordholm. Moreover, the paper shows how the theory by Nordhom can be improved in some respects, by defining, for example, the dissipative part of its effective hamiltonian in terms of a rigorous quantum mechanical formalism.

Correctly renormalized drift coefficients can be straightforwardly derived using the linear version of the generalized Langevin equation and linear reaction potentials (parabolas or inverted parabolas). Recent investigation via computer simulation of molecular dynamics and numerical solution of stochastic differential equations, shows that interesting cases exist where the nonlinear nature of the interaction between reacting system and ‘‘bath’’ and that of the reaction potential must be taken into account. Then it is shown that a Smoluchowski diffusion equation with correctly renormalized drift coefficients can be obtained by adopting a local linearization assumption, which, nevertheless allows the reaction coordinate to ‘‘feel’’ the influence of different transport properties in different regions of the reaction potential. Under the special condition where the system–bath interaction is assumed to be linear, this Smoluchowski equation is shown to coincide with that recently proposed by Okuyama and Oxtoby...