Non-Gaussian distributions in extended dynamical systems

Abstract

We propose a novel mechanism for the origin of non-Gaussian tails in the probability distribution functions (PDFs) of local variables in nonlinear, diffusive, dynamical systems including passive scalars advected by chaotic velocity fields. Intermittent fluctuations on appropriate time scales in the amplitude of the (chaotic) noise can lead to exponential tails. We provide numerical evidence for such behavior in deterministic, discrete-time passive scalar models. Different possibilities for PDFs are also outlined.

Earthquakes in a model of seismic zone with embedded pre‐existing faults

“…Note that the exponent values for cases A and B are in agreement with the results obtained for the seismic zone models (without pre-existing faults) driven by a uniformly increasing shear stress, studied earlier by Bhagavatula et al [1994] and and Chen et al [1991] respectively. This suggests that the presence of a simple linear fault structure and the choice of driving do not alter scaling behaviors.…”

“…The value of • obtained for the model with case A is consistent with the seismological data gathered over the past one hundred years [Pacheco et al, 1993] as discussed in our earlier paper [Bhagavatula et al, 1994], suggesting that case A is more relevant than case B for the description of e_arthquakes in seismic zones. Also, note that beyond E the distribution for case A has a hump.…”

Passive scalar conditional statistics in a model of random advection

“…1 A series of experimental studies on Rayleigh-Bénard convection further reveal that the temperature fluctuation itself can also be non-Gaussian when the Rayleigh number is high enough. 2 Such a discovery has motivated several studies [3][4][5][6][7][8][9] to understand the statistics of a randomly advected passive scalar, which is a theoretically more tractable problem.…”

Thermal vortex dynamics in a two-dimensional condensate

“…This equation is of the same form as that for a passive scalar advected to background fluid flow V(r, t) [24,25]. It is well established-particularly in the best-studied case of incompressible fluid flow ∇ · V(r, t) = 0-that in contrast to pure diffusion, an advected passive scalar may display non-Gaussian (for example, exponential) tails in the probability density function (PDF) c(r, t).…”

“…It is well established-particularly in the best-studied case of incompressible fluid flow ∇ • V(r, t) = 0-that in contrast to pure diffusion, an advected passive scalar may display non-Gaussian (for example, exponential) tails in the probability density function (PDF) c(r, t). These tails arise due to nontrivial spacetime correlations in the advecting velocity field [24,25]. Interestingly, neither diffusion nor advection when acting alone can produce non-Gaussian tails in the PDF's.…”