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We consider two different proposals to generate a time series with the same non-Poisson distribution of waiting times, to which we refer to as renewal and modulation. We show that, in spite of the apparent statistical equivalence, the two time series generate different physical effects. Renewal generates aging and anomalous scaling, while modulation yields no aging and either ordinary or anomalous diffusion, according to the prescription used for its generation. We argue, in fact, that the physical realization of modulation involves critical events, responsible for scaling. In conclusion, modulation rather than ruling out the action of critical events, sets the challenge for their identification.
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We study time series produced by the blinking quantum dots, by means of an aging experiment, and we examine the results of this experiment in the light of two distinct approaches to complexity, renewal and slow modulation. We find that the renewal approach fits the result of the aging experiment, while the slow modulation perspective does not. We make also an attempt at establishing the existence of an intermediate condition.
Superstatistics [C. Beck and E.G.D. Cohen, Physica A 322, 267 (2003)] is a formalism aimed at describing statistical properties of a generic extensive quantity E in complex out-of-equilibrium systems in terms of a superposition of equilibrium canonical distributions weighted by a function P(beta) of the intensive thermodynamic quantity beta conjugate to E. It is commonly assumed that P(beta) is determined by the spatiotemporal dynamics of the system under consideration. In this work we show by examples that, in some cases fulfilling all the conditions for the superstatistics formalism to be applicable, P(beta) is actually affected also by the way the measurement of E is performed, and thus is not an intrinsic property of the system.
Generalized superstatistics, i.e., a statistics of superstatistics, is proposed. A generalized superstatistical system comprises a set of superstatistical subsystems and represents a generalized hyperensemble. There exists a random control parameter that determines both the density of energy states and the distribution of the intensive parameter for each superstatistical subsystem, thereby forming the third, upper level of dynamics. Generalized superstatistics can be used for nonstationary nonequilibrium systems. The system in which a supercritical multitype age-dependent branching process takes place is an example of a nonstationary generalized superstatistical system. The theory is applied to pair production in a neutron star magnetosphere.
We discuss the situations under which Brownian yet non-Gaussian (BnG) diffusion can be observed in the model of a particles motion in a random landscape of diffusion coefficients slowly varying in space. Our conclusion is that such behavior is extremely unlikely in the situations when the particles, introduced into the system at random at $t=0$, are observed from the preparation of the system on. However, it indeed may arise in the case when the diffusion (as described in Ito interpretation) is observed under equilibrated conditions. This paradigmatic situation can be translated into the model of the diffusion coefficient fluctuating in time along a trajectory, i.e. into a kind of the diffusing diffusivity model.
We investigate the statistics of the first detected passage time of a quantum walk. The postulates of quantum theory, in particular the collapse of the wave function upon measurement, reveal an intimate connection between the wave function of a process free of measurements, i.e. the solution of the Schrodinger equation, and the statistics of first detection events on a site. For stroboscopic measurements a quantum renewal equation yields basic properties of quantum walks. For example, for a tight binding model on a ring we discover critical sampling times, diverging quantities such as the mean time for first detection, and an optimal detection rate. For a quantum walk on an infinite line the probability of first detection decays like $(mbox{time})^{-3}$ with a superimposed oscillation, critical behavior for a specific choice of sampling time, and vanishing amplitude when the sampling time approaches zero due to the quantum Zeno effect.
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