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Absorption and Emission in the Non-Poissonian Case
This article discusses absorption and emission in the Non-Poissonian Case. Abstract: This Letter addresses the challenging problems posed to the Kubo-Anderson (KA) theory by the discovery of intermittent resonant fluorescence with a nonexponential distribution of waiting times. We show how to extend the KA theory from aged to aging systems, aging for a very extended time period or even forever, being a crucial consequence of non-Poisson statistics. digital.library.unt.edu/ark:/67531/metadc67641/
Aging and Rejuvenation with Fractional Derivatives
This article discusses aging rejuvenation with fractional derivatives. Abstract: We discuss a dynamic procedure that makes a fractional derivatives emerge in the time asymptotic limit of non-Poisson processes. We find that two-state fluctuations, with an inverse power-law distribution of waiting times, finite first moment, and divergent second moment, namely, with the power index μ in the interval 2<μ<3, yield a generalized master equation equivalent to the sum of an ordinary Markov contribution and a fractional derivative term. We show that the order of the fractional derivative depends on the age of the process under study. If the system is infinitely old, the order of the fractional derivative, o, is given by o=3-μ. A brand new system is characterized by the degree o=μ-2. If the system is prepared at time -tₐ<0 and the observation begins at time t=0, we derive the following scenario. For times 0<t«tₐ the system is satisfactorily described by the fractional derivative with o=3-μ. Upon time increase the system undergoes a rejuvenation process that in the time limit t⪢tₐ yields o=μ-2. The intermediate time regime is probably incompatible with a picture based on fractional derivatives, or, at least, with a mono-order fractional derivative. digital.library.unt.edu/ark:/67531/metadc67638/
Anomalous diffusion and ballistic peaks: A quantum perspective
This article discusses anomalous diffusion and ballistic peaks. Abstract: The quantum kicked rotor and the classical kicked rotor are both shown to have truncated Lévy distributions in momentum space, when the classical phase space has accelerator modes embedded in a chaotic sea. The survival probability for classical particles at the interface of an accelerator mode and the chaotic sea has an inverse power-law structure, whereas that for quantum particles has a periodically modulated inverse power law, with the period of oscillation being dependent on Planck's constant. These logarithmic oscillations are a renormalization group property that disappears as ħ → 0 in agreement with the correspondence principle. digital.library.unt.edu/ark:/67531/metadc75417/
Anomalous diffusion and environment-induced quantum decoherence
This article discusses anomalous diffusion and environment-induced quantum decoherence. Abstract: We study the anomalous diffusion resulting from the standard map in the so-called accelerating state, and we observe that it is determined by unusually large times of sojourn of the classical trajectories in the fractal region at the border between the chaotic sea and the acceleration island. The quantum-mechanical breakdown of this property implies a coherence among so slightly different values of momentum as to become much more robust against environment fluctuations than the quantum localization corresponding to normal diffusion. digital.library.unt.edu/ark:/67531/metadc139477/
Anomalous diffusion associated with nonlinear fractional derivative Fokker-Planck-like equation: Exact time-dependent solutions
This article discusses anomalous diffusion associated with nonlinear fractional derivative Fokker-Planck-like equation. Abstract: We consider the d=1 nonlinear Fokker-Planck-like equation with fractional derivatives (∂/∂t)P(x,t) = D(∂ƴ/∂xƴ)[P(x,t]v. Exact time-dependent solutions are found for v = (2 - y)/(1 + y)(-∞ < y ⩽ 2). By considering the long-distance asymptotic behavior of these solutions, a connection is established, namely, q = (y + 3)/(Y + 1)(0 < y ⩽ 2), with the solutions optimizing the nonextensive entropy characterized by index q. Interestingly enough, this relation coincides with the only already known for Lévy-like superdiffusion (i.e., v = 1 and 0 < y ⩽ 2). Finally, for (y,v) = (2,0) the authors obtain q=5/3, which differs from the value q = 2 corresponding to the y = 2 solutions available in the literature (v < 1 porous medium equation), thus exhibiting nonuniform convergence. digital.library.unt.edu/ark:/67531/metadc77162/
Beyond the Death of Linear Response: 1/f Optimal Information Transport
This article discusses linear response and 1/f optimal information transport. Article: Nonergodic renewal processes have recently been shown by several authors to be insensitive to periodic perturbations, thereby apparently sanctioning the death of linear response, a building block of nonequilibrium statistical physics. The authors show that it is possible to go beyond the "death of linear response" and establish a permanent correlation between an external stimulus and the response of a complex network generating nonergodic renewal processes, by taking as stimulus a similar nonergodic process. The ideal condition of 1/f noise corresponds to a singularity that is expected to be relevant in several experimental conditions. digital.library.unt.edu/ark:/67531/metadc40407/
Bianucci, Mannella, and Grigolini Reply
This article is a reply to a comment by Massimo Falcioni and Angelo Vulpiani. In a previous letter, the authors have discussed the linear response theory (LRT) and shown that the breakdown of this theory occurring at intermediate times, observed in an earlier paper [2] as well as in [1], disappears upon an increase of the number of degrees of freedom. In a comment to [1] Falcioni and Vulpiani [3] claim that this breakdown is rather a consequence of the lack of mixing: according to them, regardless of the number of degrees of freedom, mixing is the key ingredient behind the LRT. digital.library.unt.edu/ark:/67531/metadc77166/
Brain, Music, and Non-Poisson Renewal Processes
This article discusses brain, music, and non-Poisson renewal processes. Abstract: In this paper we show that both music composition and brain function, as revealed by the electroencephalogram (EEG) analysis, are renewal non-Poisson processes living in the nonergodic dominion. To reach this important conclusion the authors process the data with the minimum spanning tree method, so as to detect significant events, thereby building a sequence of times, which is the time series to analyze. The the authors show that in both cases, EEG and music composition, these significant events are the signature of a non-Poisson renewal process. This conclusion is reached using a technique of statistical analysis recently developed by the authors' group, the aging experiment (AE). First, the authors find that in both cases the distances between two consecutive events are described by nonexponential histograms, thereby proving the non-Poisson nature of these processes. The corresponding survival probabilities ψ(t) are well fitted by stretched exponentials [ψ(t) ∝ exp (-(yt)a), with 0.5<a<1.] The second step rests on the adoption of AE, which shows that these are renewal processes. The authors show that the stretched exponential, due to its renewal character, is the emerging tip of an iceberg, whose underwater part has slow tails with an inverse power law structure with power index μ=1+ơ. Adopting the AE procedure, the authors find that both EEG and music composition yield μ<2. On the basis of the recently discovered complexity driving signal P with μp⩽μs, the authors conclude that the results of their analysis may explain the influence of music on the human brain. digital.library.unt.edu/ark:/67531/metadc40398/
Canonical and noncanonical equilibrium distribution
This article discusses canonical and noncanonical equilibrium distribution. Abstract: We address the problem of the dynamical foundation of noncanonical equilibrium. We consider, as a source of divergence from ordinary statistical mechanics, the breakdown of the condition of time scale separation between microscopic and macroscopic dynamics. We show that this breakdown has the effect of producing a significant deviation from the canonical prescription. We also show that, while the canonical equilibrium can be reached with no apparent dependence on dynamics, the specific form of noncanonical equilibrium is, in fact, determined by dynamics. We consider the special case where the thermal reservoir driving the system of interest to equilibrium is a generator of intermittent fluctuations. We assess the form of the noncanonical equilibrium reached by the system in this case. Using both theoretical and numerical arguments we demonstrate that Lévy statistics are the best description of the dynamics and that the Lévy distribution is the correct basin of attraction. We show that the correct path to noncanonical equilibrium by means of strictly thermodynamic arguments has not yet been found, and that further research has to be done to establish a connection between dynamics and thermodynamics. digital.library.unt.edu/ark:/67531/metadc77164/
Chaos and thermal conductivity
This article discusses chaos and thermal conductivity. Abstract: 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 fo the microscopic derivation of the heat Fourier law. digital.library.unt.edu/ark:/67531/metadc139502/
Complexity and Synchronization
This article discusses complexity and synchronization. Abstract: We study a fully connected network (cluster) of interacting two-state units as a model of cooperative decision making. Each unit in isolation generates a Poisson process with rate g. We show that when the number of nodes is finite, the decision-making process becomes intermittent. The decision-time distribution density is characterized by inverse power-law behavior with index μ=1.5 and is exponentially truncated. We find that the condition of perfect consensus is recovered by means of a fat tail that becomes more and more extended with increasing numbers of nodes N. The intermittent dynamics of the global variable are described by the motion of a particle in a double well potential. The particle spends a portion of the total time τs at the top of the potential barrier. Using theoretical and numerical arguments it is proved that τs ∝ (1/g)1n(const X N). The second portion of its time, τk, is spent by the particle at the bottom of the potential well and it is given by τk=(1/g)exp(const X N). We show that the time τk is responsible for the Kramers fat tail. This generates a stronger ergodicity breakdown than that generated by the inverse power law without truncation. The authors establish that the condition of partial consensus can be transmitted from one cluster to another provided that both networks are in a cooperative condition. No significant information transmission is possible if one of the two networks is not yet self-organized. We find that partitioning a large network into a set of smaller interacting clusters has the effect of converting the fat Kramers tail into an inverse power law with μ=1.5. digital.library.unt.edu/ark:/67531/metadc40410/
Compression and Diffusion: A Joint Approach to Detect Complexity
This article discusses a joint approach to detect complexity. Abstract: The adoption of the Kolmogorov-Sinai (KS) entropy is becoming a popular research tool among physicists, especially when applied to a dynamical system fitting the conditions of validity of the Pesin theorem. The study of time series that are a manifestation of system dynamics whose rules are either unknown or too complex for a mathematical treatment, is still a challenge since the KS entropy is not computable, in general, in that case. Here the authors present a plan of action based on the joint action of two procedures, both related to the KS entropy, but compatible with computer implementation through fast and efficient programs. The former procedure, called Compression Algorithm Sensitive To Regularity (CASToRe), establishes the amount of order by the numerical evaluation of algorithmic compressibility. The latter, called Complex Analysis of Sequences via Scaling AND Randomness Assessment (CASSANDRA), establishes the complexity degree through the numerical evaluation of the strength of an anomalous effect. This is the departure, of the diffusion process generated by the observed fluctuations, from ordinary Brownian motion. The CASSANDRA algorithm shares with CASToRe a connection with the Kolmogorov complexity. This makes both algorithms especially suitable to study the transition from dynamics to thermodynamics, and the case of non-stationary time series as well. The benefit of the joint action of these two methods is proven by the analysis of artificial sequences with the same main properties as the real time series to which the joint use of these two methods will be applied in future research work. digital.library.unt.edu/ark:/67531/metadc139462/
Cooperation in neural systems: Bridging complexity and periodicity
This article discusses cooperation in neural systems. Abstract: Inverse power law distributions are generally interpreted as a manifestation of complexity, and waiting time distributions with power index μ < 2 reflect the occurrence of ergodicity-breaking renewal events. In this paper we show how to combine these properties with the apparently foreign clocklike nature of biological processes. We use a two-dimensional regular network of leaky integrate-and-fire neurons, each of which is linked to its four nearest neighbors, to show that both complexity and periodicity are generated by locality breakdown: Links of increasing strength have the effect of turning local interactions into long-range interactions, thereby generating time complexity followed by time periodicity. Increasing the density of neuron firings reduces the influence of periodicity, thus creating a cooperation-induced renewal condition that is distinctly non-Poissonian. digital.library.unt.edu/ark:/67531/metadc132986/
Cooperation-induced topological complexity: a promising road to fault tolerance and Hebbian learning
This article discusses cooperation-induced topological complexity. Abstract: According to an increasing number of researchers intelligence emerges from criticality as a consequence of locality breakdown and long-range correlation, well known properties of phase transition processes. The authors study a model of interacting units, as an idealization of real cooperative systems such as the brain or a flock of birds, for the purpose of discussing the emergence of long-range correlation from the coupling of any unit with its nearest neighbors. The authors focus on the critical condition that has been recently shown to maximize information transport and the authors study the topological structure of the network of dynamically linked nodes. Although the topology of this network depends on the arbitrary choice of correlation threshold, namely the correlation intensity selected to establish a link between two nodes; the numerical calculations of this paper afford some important indications on the dynamically induced topology. The first important property is the emergence of a perception length as large as the flock size, thanks to some nodes with a large number of links, thus playing the leadership role. All the units are equivalent and leadership moves in time from one to another set of nodes, thereby insuring fault tolerance. Then the authors focus on the correlation threshold generating a scale-free topology with power index v ≈ 1 and the authors find that if this topological structure is selected to establish consensus through the linked nodes, the control parameter necessary to generate criticality is close to the critical value corresponding to the all-to-all coupling condition. The authors find that criticality in this case generates also a third state, corresponding to a total lack of consensus. However, the authors make a numerical analysis of the dynamically induced network, and the authors find that it consists of two almost independent structures, each of which is equivalent to a network in the all-to-all coupling condition. This observation confirms that cooperation makes the system evolve toward favoring consensus topological structures. The authors argue that these results are compatible with both Hebbian learning and fault tolerance. digital.library.unt.edu/ark:/67531/metadc132972/
Correlation Function and Generalized Master Equation of Arbitrary Age
This article discusses correlation function and generalized master equation of arbitrary age. Abstract: We study a two-state statistical process with a non-Poisson distribution of sojourn times. In accordance with earlier work, we find that this process is characterized by aging and we study three different ways to define the correlation function of arbitrary age of the corresponding dichotomous fluctuation. These three methods yield exact expressions, thus coinciding with the recent result by Godrèche and Luck [J. Stat. Phys. 104, 489 (2001)]. Actually, non-Poisson statistics yields infinite memory at the probability level, thereby breaking any form of Markovian approximation, including the one adopted herein, to find an approximated analytical formula. For this reason, we check the accuracy of this approximated formula by comparing it with the numerical treatment of the second of the three exact expressions. We find that, although not exact, a simple analytical expression for the correlation function of arbitrary age is very accurate. We establish a connection between the correlation function and a generalized master equation of the same age. Thus this formalism, related to models used in glassy materials, allows us to illustrate an approach to the statistical treatment of blinking quantum dots, bypassing the limitations fo the conventional Liouville treatment. digital.library.unt.edu/ark:/67531/metadc40401/
Criticality and Transmission of Information in a Swarm of Cooperative Units
This article discusses criticality and transmission of information in a swarm of cooperative units. Abstract: We show that the intelligence of a swarm of cooperative units (birds) emerges at criticality, as an effect of the joint action of frequent organizational collapses and of spatial correlation as extended as the flock size. The organizational collapses make the birds become independent of one another, thereby allowing the flock to follow the direction of the lookout birds. Long-range correlation violates the principle of locality, making the lookout birds transmit information on either danger or resources with a time delay determined by the time distance between two consecutive collapses. digital.library.unt.edu/ark:/67531/metadc40392/
Diffusion Entropy and Waiting Time Statistics of Hard-X-Ray Solar Flares
This article discusses diffusion entropy and waiting time statistics of hard-x-ray solar flares. Abstract: We show at work a technique of scaling detection based on evaluating the Shannon entropy of the diffusion process obtained by converting the time series under study into trajectories. This method, called diffusion entropy, affords information that cannot be derived from the direct evaluation of waiting times. We apply this method to the analysis of the distribution of time distance τ between two nearest-neighbor solar flares. This traditional part of the analysis is based on the direct evaluation of the distribution function ψ(τ), or of the probability ψ(τ), that no time distance smaller than a given τ is found. We adopt the paradigm of the inverse power-law behavior, and the authors focus on the determination of the inverse power index μ, without ruling out different asymptotic properties that might be revealed, at larger scales, with the help of richer statistics. We then use the DE method, with three different walking rules, and the authors focus on the regime of transition to scaling. This regime of transition and the value of the scaling parameter itself, δ, depends on the walking rule adopted, a property of interest to shed light on the slow process of transition from dynamics to thermodynamics often occurring under anomalous statistical conditions. With the first two rules the transition regime occurs through-out a large time interval, and the information contained in the time series is transmitted, to a great extent, to it, as well as to the scaling regime. By using the third rule, on the contrary, the same information is essentially conveyed to the scaling regime, which, in fact, emerges very quickly after a fast transition process. We show that the DE method not only causes to emerge the long-range correlation with a given μ<3, and so a basin of attraction different from the ordinary Gaussian one, but it also reveals the presence of memory effects induced by the time dependence of the solar flare rate. When this memory is annihilated by shuffling, the scaling parameter δ is shown to fit the theoretically expected function of μ. All this leads us to the compelling conclusion that μ=2.138±0.01. digital.library.unt.edu/ark:/67531/metadc67629/
Dynamic Approach to the Thermodynamics of Superdiffusion
This article discusses dynamic approach to the thermodynamics of superdiffusion. Abstract: 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 Lévy index α, and using dynamic rather than probabilistic arguments. digital.library.unt.edu/ark:/67531/metadc77167/
Dynamical approach to Lévy processes
This article discusses a dynamical approach to Lévy processes.Abstract: 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 Φy(t). Of special interest is that the distribution of the times of sojourn in the two states of the fluctuating process is proportional to d²Φy(t)/dt². Furthermore, in the special case where Φy(t) has an inverse power law, with the index β ranging from 0 to 1, thus making it nonintegrable, the authors show analytically that the statistics of the diffusing variable approximate in the long-time limit the α-stable Lévy distributions. The departure of the diffusion process of dynamical origin from the ideal condition of the Lévy statistics is established by means of a simple analytical expression. We note, first of all, that the characteristic function of a genuine Lévy process should be an exponential in time. We evaluate the correction to this exponential and show it to be expressed by a harmonic time oscillation modulated by the correlation function Φy(t). Since the characteristic function can be given a spectroscopic significance, we also discuss the relevance of the results within this context. digital.library.unt.edu/ark:/67531/metadc139498/
Dynamical model for DNA sequences
This article discusses a dynamical model for DNA sequences. Abstract: We address the problem of DNA sequences, developing a "dynamical" method based on the assumption that the statistical properties of DNA paths are determined by the joint action of two processes, one deterministic with long-range correlations and the other random and δ-function correlated. The generator of the deterministic evolution is a nonlinear map belonging to a class of maps recently tailored to mimic the processes of weak chaos responsible for the birth of anomalous diffusion. It is assumed that the deterministic process corresponds to unknown biological rules that determine the DNA path, whereas the noise mimics the influence of an infinite-dimensional environment on the biological process under study. We prove that the resulting diffusion process, if the effect of the random process is determined by the joint action of the deterministic and the random process, the correlation effects of the "deterministic dynamics" are canceled on the short-range scale, but show up in the long-range one. We denote their prescription to generate statistical sequences as the copying mistake map (CMM). We carry out their analysis of several DNA sequences and their CMM realizations with a variety of techniques and the authors especially focus on a method of regression to equilibrium, which they call the Onsager analysis. With these techniques the authors establish the statistical equivalence of the real DNA sequences with their CMM realizations. We show that long-range correlations are present in exons as well as in introns, but are difficult to detect, since the exon "dynamics" is shown to be determined by the entanglement of three distinct and independent CMM's. digital.library.unt.edu/ark:/67531/metadc139499/
Dynamical Origin of Memory and Renewal
This article discusses a dynamical origin of memory and renewal. Abstract: We show that the dynamic approach to fractional Brownian motion (FBM) establishes a link between a non-Poisson renewal process with abrupt jumps resetting to zero the system's memory and correlated dynamic processes, whose individual trajectories keep a nonvanishing memory of their past time evolution. It is well known that the recrossings of the origin by an ordinary one-dimensional diffusion trajectory generates a Lévy (and thus renewal) process of index θ=1/2. We prove with theoretical and numerical arguments that this is the special case of a more general condition, insofar as the recrossings produced by the dynamic FBM generates a Lévy process with 0<θ<1. This result is extended to produce a satisfactory model for the fluorescent signal of blinking quantum dots. digital.library.unt.edu/ark:/67531/metadc40399/
The Dynamics of EEG Entropy
This article discusses the dynamics of EEG entropy. Abstract: EEG time series are analyzed using the diffusion entropy method. The resulting EEG entropy manifests short-time scaling, asymptotic saturation and an attenuated alpha-rhythm modulation. These properties are faithfully modeled by a phenomenological Langevin equation interpreted within a neural network context. digital.library.unt.edu/ark:/67531/metadc132967/
Dynamics of Electroencephalogram Entropy and Pitfalls of Scaling Detection
This article discusses dynamics of electroencephalogram entropy and pitfalls of scaling detection. Abstract: In recent studies a number of research groups have determined that human electroencephalograms (EEG) have scaling properties. In particular, a crossover between two regions with different scaling exponents has been reported. Herein the authors study the time evolution of diffusion entropy to elucidate the scaling of EGG time series. For a cohort of 20 awake healthy volunteers with closed eyes, the authors find that the diffusion entropy of EEG increments (obtained from EEG waveforms by differencing) exhibits three features: short-time growth, an alpha wave related oscillation whose amplitude gradually decays in time, and asymptotic saturation which is achieved after approximately 1 s. This analysis suggests a linear, stochastic Ornstein-Uhlenbeck Langevin equation with a quasiperiodic forcing (whose frequency and/or amplitude may vary in time) as the model for the underlying dynamics. This model captures the salient properties of EEG dynamics. In particular, both the experimental and simulated EEG time series exhibit short-time scaling which is broken by a strong periodic component, such as alpha waves. The saturation of EEG diffusion entropy precludes the existence of asymptotic scaling. We find that the crossover between two scaling regions seen in detrended fluctuation analysis (DFA) of EEG increments does not originate from the underlying dynamics but is merely an artifact of the algorithm. This artifact is rooted in the failure of the "trend plus signal" paradigm of DFA. digital.library.unt.edu/ark:/67531/metadc40408/
Event-Driven Power-Law Relaxation in Weak Turbulence
This article discusses event-driven power-law relaxation in weak turbulence. Abstract: We characterize the spectral properties of weak turbulence in a liquid crystal sample driven by an external electric field, as a function of the applied voltage, and we find a 1/f noise spectrum S(f) ∝ 1/fn within the whole range 0< ɳ <2. We theoretically explore the hypothesis that the system complexity is driven by non-Poisson events resetting the system through creation and annihilation of coherent structures, retaining no memory of previous history (crucial events). The authors study the time asymptotic regime by means of the density ψ(τ) of the time distances between two crucial events, yielding ɳ = 3 - μ, where μ is defined through the long-time form ψ(τ) ∝ 1/τµ, with 1 < µ < 3. The system regression to equilibrium after an abrupt voltage change experimentally confirms the theory, proving violations of the ordinary linear response theory for both ɳ > 1 and ɳ < 1. digital.library.unt.edu/ark:/67531/metadc40395/
Experimental Quenching of Harmonic Stimuli: Universality of Linear Response Theory
This article discusses experimental quenching of harmonic stimuli. Abstract: We show that liquid crystals in the weak turbulence electroconvective regime respond to harmonic perturbations with oscillations whose intensity decay with an inverse power law of time. We use the results of this experiment to prove that this effect is the manifestation of a form of linear response theory (LRT) valid in the out-of-equilibrium case, as well as at thermodynamic equilibrium where it reduces to the ordinary LRT. We argue that this theory is a universal property, which is not confined to physical processes such as turbulent or excitable media, and that it holds true in all possible conditions, and for all possible systems, including a complex networks, thereby establishing a bridge between statistical physics and all the fields of research in complexity. digital.library.unt.edu/ark:/67531/metadc40394/
A fluctuating environment as a source of periodic modulation
This article discusses a fluctuating environment as a source of periodic modulation. Abstract: We study the intermittent fluorescence of a single molecule, jumping from the "light on" to the "light off" state, as a Poisson process modulated by a fluctuating environment. We show that the quasi-periodic and quasi-deterministic environmental fluctuations make the distribution of the times of sojourn in the "light off" state depart from the exponential form, and that their succession in time mirrors environmental dynamics. As an illustration, the authors discuss some recent experimental results, where the environmental fluctuations depend on enzymatic activity. digital.library.unt.edu/ark:/67531/metadc132981/
Fluctuation-dissipation process without a time scale
This article discusses fluctuation-dissipation process without a time scale. Abstract: 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 Lévy distribution. digital.library.unt.edu/ark:/67531/metadc77161/
Fluctuation-Dissipation Theorem for Event-Dominated Processes
This article discusses fluctuation-dissipation theorem for event-dominated processes. Abstract: We study a system whose dynamics are driven by non-Poisson, renewal, and nonergodic events. We show that external perturbations influencing the times at which these events occur violate the standard fluctuation-dissipation prescription due to renewal aging. The fluctuation-dissipation relation of this Letter is shown to be the linear response limit of an exact expression that has been recently proposed to account for the luminescence decay in a Gibbs ensemble of semiconductor nanocrystals, with intermittent fluorescence. digital.library.unt.edu/ark:/67531/metadc40397/
Fractional Brownian motion as a nonstationary process: An alternative paradigm for DNA sequences
This article discusses fractional Brownian motion as a nonstationary process. Abstract: The long-range correlations in DNA sequences are currently interpreted as an example of stationary fractional Brownian motion (FBM). First the authors show that the dynamics of a dichotomous stationary process with long-range correlations such as that used to model DNA sequences should correspond to Lévy statistics and not to FBM. To explain why, in spite of this, the statistical analysis of the data seems to be compatible with FBM, the authors notice that an initial Gaussian condition, generated by a process foreign to the mechanism establishing the long-range correlations and consequently implying a departure from the stationary condition is maintained approximately unchanged for very long times. This is so because due to the nature itself of the long-range correlation process, it takes virtually an infinite time for the system to reach the genuine stationary state. Then the authors discuss a possible generator of initial Gaussian conditions, based on a folding mechanism of the nucleic acid in the cell nucleus. The model adopted is compatible with the known biological and physical constraints, namely, it is shown to be consistent with the information of current biological literature on folding as well as with the statistical analyses of DNA sequences. digital.library.unt.edu/ark:/67531/metadc75416/
Fractional calculus as a macroscopic manifestation of randomness
This article discusses fractional calculus as a macroscopic manifestation of randomness. Abstract: We generalize the method of Van Hove [Physica (Amsterdam) 21, 517 (1955)] so as to deal with the case of nonordinary 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. digital.library.unt.edu/ark:/67531/metadc77121/
From power law intermittence to macroscopic coherent regime
This article offers discussions from power law intermittence to macroscopic coherent regime. Abstract: We address the problem of establishing which is the proper form of quantum master equation generating a survival probability identical to that corresponding to the nonergodic sequence of "light on" and "light off" fluorescence fluctuations in blinking quantum dots. We adopt a theoretical perspective based on the assumption that the abrupt transitions from the light on to light off state are the results of many collisions between system and environment, properly described by the Linkblad equation, and that between two consecutive collisions the system dynamics are frozen. This generates a quantum master equation belonging to the recently proposed class of generalized Lindblad equations, with a time convoluted structure, involving in the specific case of this paper both the unitary and the nonunitary contribution of the Lindlad equation. This is the property that under the low-frequency condition makes the new class of generalized Lindblad equation generates the required survival probability. We make the conjecture that this equation corresponds to the cooperative dynamics of many units that, in isolation, are described by the ordinary Lindblad equation. When the time scale of the unitary term of the Lindblad equation is shorter than the dephasing time, the cooperation generates a surprisingly extended macroscopic coherence. digital.library.unt.edu/ark:/67531/metadc132992/
From self-organized to extended criticality
This article includes discussions from self-organized to extended criticality. Abstract: We address the issue of criticality that is attracting the attention of an increasing number of neurophysiologists. Our main purpose is to establish the specific nature of some dynamical processes that although physically different, are usually termed as "critical," and we focus on those characterized by the cooperative interaction of many units. We notice that the term "criticality" has been adopted to denote both noise-induced phase transitions and Self-Organized Criticality (SOC) with no clear connection with the traditional phase transitions, namely the transformation of a thermodynamic system from one state of matter to another. We notice the recent attractive proposal of extended criticality advocated by Bailly and Longo, which is realized through a wide set of critical points rather than emerging as a singularity from a unique value of the control parameter. We study a set of cooperatively firing neurons and we show that for an extended set of interaction couplings the system exhibits a form of temporal complexity similar to that emerging at criticality from ordinary phase transitions. This extended criticality regime is characterized by three main properties: (i) In the ideal limiting case of infinitely large time period, temporal complexity corresponds to Mittag-Leffler complexity; (ii) For large values of the interaction coupling the periodic nature of the process becomes a predominant while maintaining to some extent, in the intermediate time asymptotic region, the signature of complexity; (iii) Focusing their attention on firing neuron avalanches, We find two of the popular SOC properties, namely the power indexes 2 and 1.5 respectively for time length and for the intensity of the avalanches. We derive the main conclusion that SOC emerges from extended criticality, thereby explaining the experimental observation of Plenz and Beggs: avalanches occur in time with surprisingly regularity, in apparent conflict with the temporal complexity of physical critical points. digital.library.unt.edu/ark:/67531/metadc132990/
Generalized Master Equation Via Aging Continuous-Time Random Walks
This article discusses generalized master equation via aging continuous-time random walks. Abstract: We discuss the problem of the equivalence between continuous-time random walk (CTRW) and generalized master equation (GME). The walker, making instantaneous jumps from one site of the lattice to another, resides in each site for extended times. The sojourn times have a distribution density ψ(t) that is assumed to be an inverse power law with the power index μ. We assume that the Onsager principle is fulfilled, and we use this assumption to establish a complete equivalence between GME and the Montroll-Weiss CTRW.We prove that this equivalence is confined to the case where ψ(t) is an exponential. We argue that is so because the Montroll-Weiss CTRW, as recently proved by Barkai [E. Barkai, Phys. Rev. Lett. 90, 104101 (2003)], is nonstationary, thereby implying aging, while the Onsager principle is valid only in the case of fully aged systems. The case of a Poisson distribution of sojourn times is the only one with no aging associated to it, and consequently with no need to establish special initial conditions to fulfill the Onsager principle. We consider the case of a dichotomous fluctuation, and we prove that the Onsager principle is fulfilled for any form of regression to equilibrium provided that the stationary condition holds true. We set the stationary condition on both the CTRW and the GME, thereby creating a condition of total equivalence, regardless of the nature of the waiting-time distribution. As a consequence of this procedure the authors create a GME that is a bona fide master equation, in spite of being non-Markov. We note that the memory kernel of the GME affords information on the interaction between system of interest and its bath. The Poisson case yields a bath with infinitely fast fluctuations. We argue that departing from the Poisson form has the effect of creating a condition of infinite memory and that these results might be useful to shed light on the problem of how to unravel non-Markov quantum master equations. digital.library.unt.edu/ark:/67531/metadc67635/
Lévy diffusion as an effect of sporadic randomness
This article discusses Lévy diffusion as an effect of sporadic randomness. Abstract: The Lévy diffusion processes are a form of nonordinary statistical mechanics resting, however, on the conventional Markov property. As a consequence of this, their dynamic derivation is possible provided that (i) a source of randomness is present in the corresponding microscopic dynamics and (ii) the consequent process of memory erasure is properly taken into account by the theoretical treatment. digital.library.unt.edu/ark:/67531/metadc77160/
Lévy Scaling: The Diffusion Entropy Analysis Applied to DNA Sequences
This article discusses Lévy scaling and the diffusion entropy analysis applied to DNA sequences. Abstract: We address the problem of the statistical analysis of a time series generated by complex dynamics with the diffusion entropy analysis (DEA) [N. Scafetta, P. Hamilton, and P. Grigolini, Fractals 9, 193 (2001)]. This method is based on the evaluation of the Shannon entropy of the diffusion process generated by the time series imagined as a physical source of fluctuations, rather than on the measurement of the variance of this diffusion process, as done with the traditional methods. We compare the DEA to the traditional methods of scaling detection and prove that the DEA is the only method that always yields the correct scaling value, if the scaling condition applies. Furthermore, DEA detects the real scaling of a time series without requiring any form of detrending. We show that the joint use of DEA and variance method allows to assess whether a time series is characterized by Lévy or Gauss statistics. We apply the DEA to the study of DNA sequences and prove that their large-time scales are characterized by Lévy statistics, regardless of whether they are coding or noncoding sequences. We show that the DEA is a reliable technique and, at the same time, we use it to confirm the validity of the dynamic approach to the DNA sequences, proposed in earlier work. digital.library.unt.edu/ark:/67531/metadc67631/
Linear response at criticality
This article discusses a linear response to criticality. Abstract: We study a set of cooperatively interacting units at criticality, and we prove with analytical and numerical arguments that they generate the same renewal non-Poisson intermittency as that produced by blinking quantum dots, thereby giving a stronger support to the results of earlier investigation. By analyzing how this out-of-equilibrium system responds to harmonic perturbations, we find that the response can be described only using only a new form of linear response theory that accounts for aging and the nonergodic behavior of the underlying process. We connect the undamped response of the system at criticality to the decaying response predicted by the recently established nonergodic fluctuation-dissipation theorem for dichotomous processes using information about the second moment of the fluctuations. We demonstrate that over a wide range of perturbation frequencies the response of the cooperative system is greatest when at criticality. digital.library.unt.edu/ark:/67531/metadc132985/
Linear Response of Hamiltonian Chaotic Systems as a Function of the Number of Degrees of Freedom
This article discusses the linear response of Hamiltonian chaotic systems as a function of the number of degrees of freedom. Abstract: Using numerical simulations we show that the response to weak perturbations of a variable of Hamiltonian chaotic systems depend on the number of degrees of freedom: When this is small (≈2) the response is not linear, in agreement with the well known objections to the Kubo linear response theory, while, for a larger number of degrees of freedom, the response becomes linear. This is due to the fact that increasing the number of degrees of freedom the shape of the distribution function, projected onto the subspace of the variable of interest, becomes fairly "regular." digital.library.unt.edu/ark:/67531/metadc139479/
Linear Response to Perturbation of Nonexponential Renewal Processes
This article discusses the linear response to perturbation of nonexponential renewal processes. Abstract: We study the linear response of a two-state stochastic process, obeying the renewal condition, by means of a stochastic rate equation equivalent to a master equation with infinite memory. We show that the condition of perennial aging makes the response to coherent perturbation vanish in the long-time limit. digital.library.unt.edu/ark:/67531/metadc67626/
Long- and Short-Time Analysis of Heartbeat Sequences: Correlation with Mortality Risk in Congestive Heart Failure Patients
This paper discusses long- and short-time analysis of heartbeat sequences and the correlation with mortality risk in congestive heart failure patients. Abstract: We analyze RR heartbeat sequences with a dynamic model that satisfactorily reproduces both the long- and the short-time statistical properties of heart beating. These properties are expressed quantitatively by means of two significant parameters, the scaling δ concerning the asymptotic effects of long-range correlation, and the quantity 1 - π establishing the amount of uncorrelated fluctuations. We find a correlation between the position in the phase space (δ,π) of patients with congestive heart failure and their mortality risk. digital.library.unt.edu/ark:/67531/metadc67633/
The Markov approximation revisited: inconsistency of the standard quantum Brownian motion model
In this article, the authors revisit the Markov approximation necessary to derive ordinary Brownian motion from a model widely adopted in literature for this specific purpose. The authors show that this leads to internal inconsistencies, thereby implying that further search for a more satisfactory model is required. digital.library.unt.edu/ark:/67531/metadc139465/
Mega et al. Reply
This article is a reply to a comment by A. Helmstetter and D. Sornette about the article 'Power-Law Time Distribution of Large Earthquakes' from 2003. digital.library.unt.edu/ark:/67531/metadc67640/
Memory Beyond Memory in Heart Beating, a Sign of a Healthy Physiological Condition
In this article, the authors describe two types of memory and illustrate each using artificial and actual heartbeat data sets. The first type of memory, yielding anomalous diffusion, implies the inverse power-law nature of the waiting time distribution and the second the correlation among distinct times, and consequently also the occurrence of many pseudoevents, namely, not genuinely random events. Using the method of diffusion entropy analysis, the authors establish the scaling that would be determined by the real events alone. The authors prove that the heart beating of healthy patients reveals the existence of many more pseudoevents than in the patients with congestive heart failure. digital.library.unt.edu/ark:/67531/metadc67628/
Memory Effects in Fractional Brownian Motion with Hurst Exponent H<1/3
In this article, the authors study the regression to the origin of a walker driven by dynamically generated fractional Brownian motion (FBM) and the authors prove that when the FBM scaling, i.e., the Hurst exponent H<1/3, the emerging inverse power law is characterized by a power index that is a compelling signature of the infinitely extended memory of the system. Strong memory effects leads to the relation H=θ/2 between the Hurst exponent and the persistent exponent θ, which is different from the widely used relation H=1 - θ. The latter is valid for 1/3<H<1 and is known to be compatible with the renewal assumption. digital.library.unt.edu/ark:/67531/metadc40405/
Networking of psychophysics, psychology, and neurophysiology
This article focuses on dynamic networking and dynamic networks in complex research on psychophysics, psychology, and neurophysiology. It states that new ways were suggested by dynamic networking and dynamic networks to transfer information utilizing the long-distance communication through local cooperative interaction. It says that the fluctuations in brain and social dynamics reveal the emergence of complex behavior when analyzed with advanced methods of fractal statistical analysis. digital.library.unt.edu/ark:/67531/metadc132991/
Noise-induced transition from anomalous to ordinary diffusion: The crossover time as a function of noise intensity
In this article, the authors study the interplay between a deterministic process of weak chaos, responsible for the anomalous diffusion of a variable x, and a white noise of intensity ≡. The deterministic process of anomalous diffusion results from the correlated fluctuations of a statistical variable ξ between two distinct values +1 and -1, each of them characterized by the same waiting time distribution ψ(t), given by ψ(t)≃ t(-μ) with 2 < μ < 3, in the long-time limit. The authors prove that under the influence of a weak white noise of intensity ≡, the process of anomalous diffusion becomes normal at a time t(c) given by t(c) ~ 1/≡(β)(μ). Here β(μ) is a function of μ which depends on the dynamical generator of the waiting-time distribution ψ(t). The authors derive an explicit expression for β(μ) in the case of two dynamical systems, a one-dimensional superdiffusive map and the standard map in the accelerating state. The theoretical prediction is supported by numerical calculations. digital.library.unt.edu/ark:/67531/metadc139501/
Non-Gaussian statistics of anomalous diffusion: The DNA sequences of prokaryotes
This article discusses non-Gaussian statistics of anomalous diffusion. The authors adopt a non-Gaussian indicator to measure the deviation from Gaussian statistics of a diffusion process generated by dichotomous fluctuations with infinite memory. The authors also make analytical predictions on the transient behavior of the non-Gaussian indicator as well as on its stationary value. The authors then apply this non-Gaussian analysis to the DNA sequences of prokaryotes adopting a theoretical model where the "DNA dynamics" are assumed to be determined by the statistical superposition of two independent generators of fluctuations: a generator of fluctuations with no correlation and a generator of fluctuations with infinite correlation "time". The authors study also the influence that the finite length of the observed sequences has on the short-range fluctuation and sequence truncation. Nevertheless, under proper conditions, fulfilled by all the DNA sequences of prokaryotes that have been examined, a non-Gaussian signature remains to signal the correlated nature of the driving process. digital.library.unt.edu/ark:/67531/metadc75418/
Non-Markovian Nonstationary Completely Positive Open-Quantum-System Dynamics
This article discusses non-Markovian nonstationary completely positive open-quantum-system dynamics. By modeling the interaction of a system with an environment through a renewal approach, the authors demonstrate that completely positive non-Markovian dynamics may develop some unexplored nonstandard statistical properties. The renewal approach is defined by a set of disruptive events, consisting in the action of a completely positive superoperator over the system density matrix. The random time intervals between events are described by an arbitrary waiting-time distribution. The authors show that, in contrast to the Markovian case, if one performs a system preparation (measurement) at an arbitrary time, the subsequent evolution of the density-matrix evolution is modified. The nonstationary character refers to the absence of an asymptotic master equation even when the preparation is performed at arbitrary long times. In spite this property, the authors demonstrate that operator expectation values and operators correlations have the same dynamical structure, establishing the validity of a nonstationary quantum regression hypothesis. The nonstationary property of the dynamics is also analyzed through the response of the system to an external weak perturbation. digital.library.unt.edu/ark:/67531/metadc40393/
Non-Poisson Dichotomous Noise: Higher-Order Correlation Functions and Aging
In this article, the authors study a two-state symmetric noise, with a given waiting time distribution ψ(τ), and focus their attention on the connection between the four-time and two-time correlation functions. The transition of ψ(τ) from the exponential to the nonexponential condition yields the breakdown of the usual factorization condition of high-order correlation functions, as well as the birth of aging effects. The authors discuss the subtle connections between these two properties and establish the condition that the Liouville-like approach has to satisfy in order to produce a correct description of the resulting diffusion process. digital.library.unt.edu/ark:/67531/metadc40403/
Non-Poisson distribution of the time distances between two consecutive clusters of earthquakes
This article discusses non-Poisson distribution of the time distances between two consecutive clusters of earthquakes. With the help of the Diffusion Entropy technique the authors show the non-Poisson statistics of the distances between consecutive Omori's swarms of earthquakes. The authors give an analytical proof of the numerical results of an earlier paper [Mega et al., Phys. Rev. Lett. 90 (2003) 188501]. digital.library.unt.edu/ark:/67531/metadc132976/
Power-Law Time Distribution of Large Earthquakes
In this article, the authors study the statistical properties of time distribution of seismicity in California by means of a new method of analysis, the diffusion entropy. The authors find that the distribution of time intervals between a large earthquake (the main shock of a given seismic sequence) and the next one does not obey Poisson statistics, as assumed by the current models. The authors prove that this distribution is an inverse power law with an exponent μ = 2.06 ± 0.01. The authors propose the long-range model, reproducing the main properties of the diffusion entropy and describing the seismic triggering mechanisms induced by large earthquakes. digital.library.unt.edu/ark:/67531/metadc67639/
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