The Origin of the Solar Flare Waiting-Time Distribution

Strong Scattering of ~keV Pickup Ions in the Local Interstellar Magnetic Field Draped Around Our Heliosphere: Implications for the IBEX Ribbon's Source and IMAP

The leading hypothesis for the origin of the Interstellar Boundary Explorer (IBEX) "ribbon" of enhanced energetic neutral atoms (ENAs) from the outer heliosphere is the secondary ENA mechanism, whereby neutralized solar wind ions escape the heliosphere and, after several charge-exchange processes, may propagate back toward Earth primarily in directions perpendicular to the local interstellar magnetic field (ISMF). However, the physical processes governing the parent protons outside of the heliopause are still unconstrained. In this study, we compute the "spatial retention" model proposed by Schwadron & McComas (2013) in a 3D simulated heliosphere. In their model, pickup ions outside the heliopause that originate from the neutral solar wind are spatially-retained in a region of space via strong pitch angle scattering before becoming ENAs. We find that the ribbon's intensity and shape can vary greatly depending on the pitch angle scattering rate both inside and outside the spatial retention region, potentially contributing to the globally distributed flux. The draping of the ISMF around the heliopause creates an asymmetry in the average distance to the ribbon's source as well as an asymmetry in the ribbon's shape, i.e., radial cross section of ENA flux through the circular ribbon. The spatial retention model adds an additional asymmetry to the ribbon's shape due to the enhancement of ions in the retention region close to the heliopause. Finally, we demonstrate how the ribbon's structure observed at 1 au is affected by different instrument capabilities, and how the Interstellar Mapping and Acceleration Probe (IMAP) may observe the ribbon.

Can a time varying external drive give rise to apparent criticality in neural systems?

The finding of power law scaling in neural recordings lends support to the hypothesis of critical brain dynamics. However, power laws are not unique to critical systems and can arise from alternative mechanisms. Here, we investigate whether a common time-varying external drive to a set of Poisson units can give rise to neuronal avalanches and exhibit apparent criticality. To this end, we analytically derive the avalanche size and duration distributions, as well as additional measures, first for homogeneous Poisson activity, and then for slowly varying inhomogeneous Poisson activity. We show that homogeneous Poisson activity cannot give rise to power law distributions. Inhomogeneous activity can also not generate perfect power laws, but it can exhibit approximate power laws with cutoffs that are comparable to those typically observed in experiments. The mechanism of generating apparent criticality by time-varying external fields, forces or input may generalize to many other systems like dynamics of swarms, diseases or extinction cascades. Here, we illustrate the analytically derived effects for spike recordings in vivo and discuss approaches to distinguish true from apparent criticality. Ultimately, this requires causal interventions, which allow separating internal system properties from externally imposed ones.

Short-ranged memory model with preferential growth

In this work we introduce a variant of the Yule-Simon model for preferential growth by incorporating a finite kernel to model the effects of bounded memory. We characterize the properties of the model combining analytical arguments with extensive numerical simulations. In particular, we analyze the lifetime and popularity distributions by mapping the model dynamics to corresponding Markov chains and branching processes, respectively. These distributions follow power laws with well-defined exponents that are within the range of the empirical data reported in ecologies. Interestingly, by varying the innovation rate, this simple out-of-equilibrium model exhibits many of the characteristics of a continuous phase transition and, around the critical point, it generates time series with power-law popularity, lifetime and interevent time distributions, and nontrivial temporal correlations, such as a bursty dynamics in analogy with the activity of solar flares. Our results suggest that an appropriate balance between innovation and oblivion rates could provide an explanatory framework for many of the properties commonly observed in many complex systems.

Stochastic stability of some state-dependent growth-collapse processes

In this paper we consider a discrete-time process which grows according to a random walk with nonnegative increments between crash times at which it collapses to 0. We assume that the probability of crashing depends on the level of the process. We study the stochastic stability of this growth-collapse process. Special emphasis is given to the case in which the probability of crashing tends to 0 as the level of the process increases. In particular, we show that the process may exhibit long-range dependence and that the crash sizes may have a power law distribution.

Stochastic stability of some state-dependent growth-collapse processes

In this paper we consider a discrete-time process which grows according to a random walk with nonnegative increments between crash times at which it collapses to 0. We assume that the probability of crashing depends on the level of the process. We study the stochastic stability of this growth-collapse process. Special emphasis is given to the case in which the probability of crashing tends to 0 as the level of the process increases. In particular, we show that the process may exhibit long-range dependence and that the crash sizes may have a power law distribution.

Modelling the influence of photospheric turbulence on solar flare statistics

Solar flares stem from the reconnection of twisted magnetic field lines in the solar photosphere. The energy and waiting time distributions of these events follow complex patterns that have been carefully considered in the past and that bear some resemblance with earthquakes and stockmarkets. Here we explore in detail the tangling motion of interacting flux tubes anchored in the plasma and the energy ejections resulting when they recombine. The mechanism for energy accumulation and release in the flow is reminiscent of self-organized criticality. From this model, we suggest the origin for two important and widely studied properties of solar flare statistics, including the time-energy correlations. We first propose that the scale-free energy distribution of solar flares is largely due to the twist exerted by the vorticity of the turbulent photosphere. Second, the long-range temporal and time-energy correlations appear to arise from the tube-tube interactions. The agreement with satellite measurements is encouraging.

Intertime jump statistics of state-dependent Poisson processes

A method to obtain the probability distribution of the interarrival times of jump occurrences in systems driven by state-dependent Poisson noise is proposed. Such a method uses the survivor function obtained by a modified version of the master equation associated to the stochastic process under analysis. A model for the timing of human activities shows the capability of state-dependent Poisson noise to generate power-law distributions. The application of the method to a model for neuron dynamics and to a hydrological model accounting for land-atmosphere interaction elucidates the origin of characteristic recurrence intervals and possible persistence in state-dependent Poisson models.

Intensity thresholds and the statistics of the temporal occurrence of solar flares

Introducing thresholds to analyze time series of emission from the Sun enables a new and simple definition of solar flare events and their interoccurrence times. Rescaling time by the rate of events, the waiting and quiet time distributions both conform to scaling functions that are independent of the intensity threshold over a wide range. The scaling functions are well-described by a two-parameter function, with parameters that depend on the phase of the solar cycle. For flares identified according to the current, standard definition, similar behavior is found.

Interoccurrence times in the Bak-Tang-Wiesenfeld sandpile model: a comparison with the observed statistics of solar flares

A sequence of bursts observed in an intermittent time series may be caused by a single avalanche, even though these bursts appear as distinct events when noise and/or instrument resolution impose a detection threshold. In the Bak-Tang-Wiesenfeld sandpile, the statistics of quiet times between bursts switches from Poissonian to scale invariant on raising the threshold for detecting instantaneous activity, since each zero-threshold avalanche breaks into a hierarchy of correlated bursts. Calibrating the model with the time resolution of GOES data, qualitative agreement with the interoccurrence time statistics of solar flares at different intensity thresholds is found.

Point process model of 1/f noise vs a sum of Lorentzians

We present a simple point process model of 1/f(beta) noise, covering different values of the exponent beta . The signal of the model consists of pulses or events. The interpulse, interevent, interarrival, recurrence, or waiting times of the signal are described by the general Langevin equation with the multiplicative noise and stochastically diffuse in some interval resulting in a power-law distribution. Our model is free from the requirement of a wide distribution of relaxation times and from the power-law forms of the pulses. It contains only one relaxation rate and yields 1/f(beta) spectra in a wide range of frequencies. We obtain explicit expressions for the power spectra and present numerical illustrations of the model. Further we analyze the relation of the point process model of 1/f noise with the Bernamont-Surdin-McWhorter model, representing the signals as a sum of the uncorrelated components. We show that the point process model is complementary to the model based on the sum of signals with a wide-range distribution of the relaxation times. In contrast to the Gaussian distribution of the signal intensity of the sum of the uncorrelated components, the point process exhibits asymptotically a power-law distribution of the signal intensity. The developed multiplicative point process model of 1/f(beta)noise may be used for modeling and analysis of stochastic processes in different systems with the power-law distribution of the intensity of pulsing signals.

Radiative intermittent events during Fermi's stochastic acceleration

We investigate the dynamics of a realization of Fermi's relativistic acceleration mechanism that is a charged test particle oscillating between two reflecting plates that move stochastically. By allowing the charge to radiate energy during each collision, we find that the main features of the system are (1) due to the radiation drag the energy gained by the particle is bounded, and (2) the radiated energy represents a typical realization of an on-off intermittent process, due to numerous continuous encounters with a very small emission, interrupted by short and intense bursts of radiation. This intermittent radiative process exhibits non-Gaussian statistical features.

Temporal dynamics of recurrent airway symptoms and cellular random walk

Asthma is a complex chronic inflammatory disease of the small airways that has dramatically increased in prevalence in industrialized countries during the last decades. Risk factors for adult asthma have been related to the complex array of gene-environment interactions and exposure of the immune system to allergens in early childhood. In genetically predisposed subjects, continuous exposure to environmental agents such as allergens or infections can lead to recurrent airway symptoms characterized by recurrent episodes of airway inflammation and bronchoconstriction with clinical symptoms of cough, dyspnea, or wheezing. In this study, we report that the longterm temporal dynamics of recurrent airway symptoms in a population of unselected infants display a complex intermittent pattern and that the distribution of interepisode intervals follows a power law. We interpret the data by using a model of the dynamics of attack episodes in which an attack is triggered by an avalanche of airway constrictions. We map the dynamics of this model to the known problem of a random walk in the presence of an absorbing boundary in which the walker corresponds to the fluctuations in contractile state of airway smooth muscle cells. These findings may provide new insight into the mechanisms of otherwise unexplained symptom episodes.

Solar flares as cascades of reconnecting magnetic loops

A model for the solar coronal magnetic field is proposed where multiple directed loops evolve in space and time. Loops injected at small scales are anchored by footpoints of opposite polarity moving randomly on a surface. Nearby footpoints of the same polarity aggregate, and loops can reconnect when they collide. This may trigger a cascade of further reconnection, representing a solar flare. Numerical simulations show that a power law distribution of flare energies emerges, associated with a scale-free network of loops, indicating self-organized criticality.

Diffusion entropy and waiting time statistics of hard-x-ray solar flares

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 tau between two nearest-neighbor solar flares. This traditional part of the analysis is based on the direct evaluation of the distribution function psi(tau), or of the probability Psi(tau), that no time distance smaller than a given tau is found. We adopt the paradigm of the inverse power-law behavior, and we focus on the determination of the inverse power index mu, 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 we focus on the regime of transition to scaling. This regime of transition and the value of the scaling parameter itself, delta, 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 throughout 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 mu < 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 delta is shown to fit the theoretically expected function of mu. All this leads us to the compelling conclusion that mu = 2.138+/-0.01.