Log-Normal Waiting Time Widths Characterize Dynamics

Many astronomical phenomena, including Fast Radio Bursts and Soft Gamma Repeaters, consist of brief, separated, seemingly aperiodic events. The intervals between these events vary randomly, but there are epochs of greater activity, with shorter mean intervals, and of lesser activity, with longer mean intervals. This variability can be quantified by a single dimensionless parameter, the width of a log-normal fit to the distribution of waiting times between events. If the distribution of event strengths is a power law, as is often the case, this parameter is independent of the detection threshold and is a robust measure of the intrinsic variability of the waiting times and of the underlying dynamics.


Introduction
Many episodic natural phenomena originate in complex and imperfectly understood physical processes e.g., repeating Fast Radio Bursts (FRB) (Zhang 2023) and Soft Gamma Repeaters (SGR) (Hurley 2011).In some, such as FRB, the responsible physical processes and their environment are not known.In others we may know which physical processes are responsible (magnetic reconnection in SGR), but lack sufficient understanding to calculate them quantitatively.Burst activity may be described by the distribution of intervals (waiting times) between detected bursts.This reflects two distinct properties of the underlying process: the correlation (or lack thereof) between consecutive bursts (short-term memory) and slower variations in the mean level of activity (long-term memory).When the level of activity varies, the width of the distribution increases; SGR 1806−20 is an example (Table 1).
The distributions of waiting times in repeating FRB (Katz 2019;Aggarwal et al. 2022;Hewitt et al. 2022;Li et al. 2021;Niu et al. 2022;Zhang et al. 2022) and in SGR burst storms (Hurley et al. 1994;Younes et al. 2020) have been well fit by log-normal functions.
This paper describes the application of log-normal fits to waiting time distributions, and suggests their dimensionless standard deviation σ as a robust and fundamental quantitative metric.Alternative statistical methods have recently been used to describe FRB waiting times by Du et al. (2023); Wang et al. (2023).Their conclusions about memory in burst recurrence intervals ("short waiting times tend to be followed by short ones, and long by long") are qualitatively consistent with the picture of varying levels of activity presented here.

Bursting Phenomena
Bursting phenomena may be described by their distribution of strengths (energy, flux, fluence, electromagnetic field) and by their temporal distribution.Distributions of strengths are power laws (Younes et al. 2020;Zhang et al. 2022) when there is no characteristic value of strength (earthquake magnitude, flare or burst energy, etc.) or scale over a wide range (Kolmogorov 1941a,b;Katz 1986).Any deviation from a power law would define a characteristic strength, the value of strength where its distribution deviates from a straight line on a log-log plot, contradicting the assumption that there is no characteristic strength.
In some FRB a break (a change in the exponent of the fitted energy distribution f (E)) is observed (Zhang et al. 2022), defining a characteristic value of the burst energy E or fluence where the exponent changes, and some SGR have shown extreme outliers, also not described by a power law distribution (Katz 2021); power laws are widespread but not universal.As a result, many but not all log-normal widths are independent of detection sensitivity.
Most episodically outbursting phenomena display periods of greater and lesser activity with shorter and longer mean waiting times, respectively.If the activity level is uniform the process is, by definition, Poissonian and the width of the waiting time distribution σ = 0.723, while if there are periods of greater and lesser activity there are excesses of short and long waiting times, leading to a broader distribution of waiting times.The width σ of this distribution thus measures the variability of the activity.In one limit outbursts are regularly periodic and σ = 0, while in the other limit brief periods of frequent activity are separated by long periods of quietude and σ ≫ 1; shot noise, with random events occurring at a constant mean rate, is intermediate.

Log-Normal Fits
Log-normal fitting functions are widely used because with only three parameters, a midpoint, a maximum and a width, they provide good fits to a broad range of single-peaked distributions, wide as well as narrow (Johnson, Kotz & Balakrishnan 1994;Limpert, Stahel & Abbt 2001).These fits may be entirely phenomenological, without basis in a causal model.Most uses of log-normal fits describe the distribution of some parameter of the individual events, but here we are concerned with the waiting times between them.
The logarithm of the product of a series of multiplications of a random variable is the sum of the logarithms of the variables, and executes a random walk.If there are many factors in the product the central limit theorem applies to the sum of the logarithms, and the result approaches a log-normal distribution (Shockley 1957;Montroll & Shlesinger 1982).The rate of convergence depends on the distributions of these logarithms, and for pathological distributions the sum may not converge at all.A log-normal distribution of a variable is therefore a natural consequence if it is the result of a series of independent stochastic multiplicative steps.The observed σ of a waiting time distribution would be ∝ N −1/2 , where N is the number of independent steps, in series, weighted by their frequency: if there is a necessary step that occurs at a low rate, the observed waiting time distribution would reflect the distribution of that infrequent step, while a step that occurs at a high rate would have little effect on σ.A log-normal distribution will be a good fit if there are several required steps, each with a comparable rate, making the central limit theorem applicable to the sum of their logarithms.
Power-law distributions are not fit by log-normal functions because they do not have a peak.All power-law distributions of data must have at least one cutoff, either a threshold for detection or an intrinsic characteristic scale, in order that the total number of events and the total energy (or an analogous quantity) be finite.The distribution peaks at the cutoff because it marks the transition between an increasing function that would diverge if not cut off and a decreasing function.Abruptly cut off power laws may be fit by log-normal functions, although not closely, and the present method may be applied.
A log-normally distributed probability density function of N measurements of the variable x, normalized to unity, is where ln x 0 is the mean logarithm and σ the standard deviation of the logarithms (Evans, Hastings & Peacock 2000;Johnson, Kotz & Balakrishnan 1994).Applying this to a distribution of waiting times ∆t between events, the distribution of their logarithms, normalized to N events, is where ln ∆t 0 is the peak of the distribution of logarithms and σ is its dimensionless standard deviation.If there are N values of ∆t in the data the normalization factor A = N/( √ 2πσ).
We are interested in σ as a parameter that describes the underlying physical processes.
For data with the empirical distribution f (ln ∆t), ln ∆t 0 is taken to be the mean The standard deviation 4. Simple Models

Exact Periodicity
One limiting case of a waiting time distribution is that of periodic pulses, all of which are bright enough to be detected; all waiting times are then the same.This is often a excellent approximation for radio pulsars, whose period derivatives 10 −21 Ṗ 10 −9 .Then the fractional variation in inter-pulse intervals over a span T of observations (generally, very intermittent rather than continuous) is T Ṗ /P and the fitted σ ∼ T Ṗ /P .For observed pulsars σ is in the range 10 −14 -0.03 (Manchester, et al. 2005), with the largest values for young, rapidly slowing, pulsars (like the Crab) observed for decades, and the smallest values for a recycled (low-field) pulsar briefly observed.For a steadily slowing but nearly strictly periodic phenomenon like pulsar emission, σ is not very meaningful because of its dependence on T .

Shot Noise
The definition of shot noise (Poissonian statistics) with mean rate ν is that the probability per unit time of an event after a waiting time ∆t from the immediately preceding event decays exponentially at the rate ν: Performing the integrals from 0 to ∞ in Eq. 3 ln (ν∆t) 0 = −0.577, and Eq. 4 shorter waiting times may be attributable to substructure within individual bursts rather than to the intervals between distinct bursts, and are not considered further.

Log-Normal Fit Examples
Log-normal widths have been fitted to the waiting time distributions of many astronomical object with episodic activity.This section collects the best-characterized examples.The object must have discrete, cleanly separated, events such as FRB or SGR outbursts, and enough events must have been observed ( 100) to determine σ with smalll uncertainty.Solar flares display memory on many time scales (Aschwanden & Johnson 2021) but are beyond the scope of this study.
Table 1 shows the values of σ of waiting time distributions of shot noise and of several astronomical datasets.These include very active FRB, burst storms from SGR 1806−20 and SGR 1935+2154 (associated with a Galactic FRB) and microglitches of the Vela pulsar.substructure within a single event that may be erroneously classified as distinct outbursts.

The values of σ in the
The criteria for exclusion are necessarily subjective, but plausible variations change σ by only a few hundredths.For SGR 1806-20 σ was calculated by the cited references.The intervals range over several orders of magnitude-it has intense burst storms separated by years of quiescence, explaining the large value of σ.Other σ were calculated from the cited published data.

Discussion
This paper compares the widths of log-normal fits to interval (lag) times of several Vela PSR µ-glitches 57 0.79 ± 0.07 Hurley et al. (1994) Vela PSR δ ν 57 0.33 ± 0.03 Hurley et al. (1994) Table 1: Values of σ obtained for log-normal fits to several datasets.N is the number of events in each dataset.The uncertainties of σ are their standard errors based on the number of data (where FRB intervals are evidently double-peaked the peak at longer intervals is used).Hurley et al. (1994) gave values of σ but the other σ (and that for Vela PSR δ ν) were measured from the published figures.The FRB (see Table 6 of Hu & Huang (2022) for a review), SGR and flare star data refer to the intervals (waiting times) between bursts or flares.For FRB the data are averages over several observing epochs by the same telescope.
The Vela PSR intervals are the waiting times between its microglitches and δ ν the changes in its spindown rate following these glitches.The flare star data are averages over targets authors provided uncertainty estimates, those are used.When they are not available, uncertainties have been estimated from the published figures, as indicated in the Table caption.The data and their analyses are described in the papers cited here.
Inspection of these figures (the reader is referred to the cited original papers) indicates that the log-normal functions fit the data well, but it is not possible to say with confidence whether deviations from the log-normal are entirely explained by the random statistics of comparatively small samples.Even if this is not the case, the log-normals explain most of the distribution of waiting times, and are a valid characterization, the the dominant terms in a moment expansion about the mean.It is not claimed that the log-normal functions describe the data exactly (aside from the statistics of small samples) but that, considered as a leading term in an expansion, they contain fundamental information about the underlying physical processes.
The virtue of σ as a measure of the statistics of a bursting source is its independence of the observing sensitivity if the distribution of event strengths is a power law.This is simply demonstrated: If σ depended on a detection threshold T , then the function σ(T ) would define the characteristic signal strength equal to T where σ is some specific value, such as 0.723 (its value for shot noise).That would be inconsistent with a power law distribution of signal strength, a straight line on a log-log plot of number of events vs. signal strength, with no characteristic value (Kolmogorov 1941a,b;Katz 1986).Most of the phenomena under consideration (astronomical fast radio bursts, soft gamma repeaters and pulsar glitches) do not have obvious characteristic strengths, at least over a range several orders of magnitude wide, but there have been exceptions (Zhang et al. 2022).
Very small (≪ 1) values of σ of a waiting time distribution indicate periodicity.Values < 0.723, the value for shot noise, indicate a memory effect, like that of a noisy relaxation oscillator, that does not produce periodicity but instead has a characteristic repetition time scale, with shorter or longer waiting times less likely than for shot noise.This is often described as quasi-periodicity, and is shown in Table 1 for the post-microglitch changes of the spin-down rate of the Vela pulsar.These δ ν have a preferred scale, although the timing of the microglitches is consistent with shot noise.True periodicity is described by σ → 0.
Although this limit cannot be achieved in a finite dataset, phenomena that are called periodic, such as binary orbits or pulsar rotation, have nonzero but very small σ ≪ 1. show an extreme example for FRB 20201124A, that resumed activity after an extended quiet period (Niu 2022).A different toy model is discussed in Sec.4.3, in which intervals may be either long or short, but there is no memory bunching long intervals together, nor short intervals together.The fact that σ of SGR1935+2154, the only SGR associated with a FRB, is much less than that of SGR 1806−20 indicates that these are fundamentally different objects; unfortunately, insufficient data are available from other SGR to explore this further.
Table 1 shows that for the repeating FRB for which sufficient waiting time data are available, σ has approximately the same value, indicating common underlying analogous to the Solar cycle, but this may be attributable to the limited temporal extent of the data.
The universality classes of critical phenomena (Pelissetto & Vicari 2002) suggest a path to qualitative categorization.Phenomena of different microphysical origin are sorted into universality classes on the basis of their critical exponents.Detailed knowledge of their microphysics, such as intermolecular potentials in liquid-gas critical points or the Hamiltonian of a ferromagnet, is not required to identify these classes and to establish fundamentally common dynamics among phenomena involving different physical processes.
This paper suggests that the width of a log-normal distribution is analogous to the critical exponents that characterize universality classes in renormalization group theory.
Values of σ > 0.723 indicate varying rates of activity (shot noise has the most random possible statistics if the mean or statistically expected rate of activity is unchanging).Most of the data series shown in Table1have σ larger than 0.723, indicating changing levels of activity and long term memory in the mechanism.In contrast, the smaller value of σ for the Vela δ ν indicates a characteristic scale of the phenomenon (because these data are not time intervals, it doesn't indicate periodicity of quasi-periodicity).Much larger values of σ, as found for SGR 1806−20, may indicate periods of much greater ("burst storms") and of lesser activity; long intervals are likely to be followed and preceded by long intervals, and short intervals by short intervals.The fact that SGR and FRB show periods of greater and lesser activity has long been known;Zhou et al. (2022)