210 research outputs found
Phenomenology for the decay of energy-containing eddies in homogeneous MHD turbulence
We evaluate a number of simple, oneâpoint phenomenological models for the decay of energyâcontaining eddies in magnetohydrodynamic(MHD) and hydrodynamicturbulence. The MHDmodels include effects of cross helicity and AlfvĂ©nic couplings associated with a constant mean magnetic field, based on physical effects wellâdescribed in the literature. The analytic structure of three separate MHDmodels is discussed. The single hydrodynamic model and several MHDmodels are compared against results from spectralâmethod simulations. The hydrodynamic model phenomenology has been previously verified against experiments in wind tunnels, and certain experimentally determined parameters in the model are satisfactorily reproduced by the present simulation. This agreement supports the suitability of our numerical calculations for examining MHDturbulence, where practical difficulties make it more difficult to study physical examples. When the tripleâdecorrelation time and effects of spectral anisotropy are properly taken into account, particular MHDmodels give decay rates that remain correct to within a factor of 2 for several energyâhalving times. A simple model of this type is likely to be useful in a number of applications in space physics, astrophysics, and laboratory plasma physics where the approximate effects of turbulence need to be included
The Effect of Coherent Structures on Stochastic Acceleration in MHD Turbulence
We investigate the influence of coherent structures on particle acceleration
in the strongly turbulent solar corona. By randomizing the Fourier phases of a
pseudo-spectral simulation of isotropic MHD turbulence (Re ), and
tracing collisionless test protons in both the exact-MHD and phase-randomized
fields, it is found that the phase correlations enhance the acceleration
efficiency during the first adiabatic stage of the acceleration process. The
underlying physical mechanism is identified as the dynamical MHD alignment of
the magnetic field with the electric current, which favours parallel
(resistive) electric fields responsible for initial injection. Conversely, the
alignment of the magnetic field with the bulk velocity weakens the acceleration
by convective electric fields - \bfu \times \bfb at a non-adiabatic stage of
the acceleration process. We point out that non-physical parallel electric
fields in random-phase turbulence proxies lead to artificial acceleration, and
that the dynamical MHD alignment can be taken into account on the level of the
joint two-point function of the magnetic and electric fields, and is therefore
amenable to Fokker-Planck descriptions of stochastic acceleration.Comment: accepted for publication in Ap
Analysis of unbounded operators and random motion
We study infinite weighted graphs with view to \textquotedblleft limits at
infinity,\textquotedblright or boundaries at infinity. Examples of such
weighted graphs arise in infinite (in practice, that means \textquotedblleft
very\textquotedblright large) networks of resistors, or in statistical
mechanics models for classical or quantum systems. But more generally our
analysis includes reproducing kernel Hilbert spaces and associated operators on
them. If is some infinite set of vertices or nodes, in applications the
essential ingredient going into the definition is a reproducing kernel Hilbert
space; it measures the differences of functions on evaluated on pairs of
points in . And the Hilbert norm-squared in will represent
a suitable measure of energy. Associated unbounded operators will define a
notion or dissipation, it can be a graph Laplacian, or a more abstract
unbounded Hermitian operator defined from the reproducing kernel Hilbert space
under study. We prove that there are two closed subspaces in reproducing kernel
Hilbert space which measure quantitative notions of limits at
infinity in , one generalizes finite-energy harmonic functions in
, and the other a deficiency index of a natural operator in
associated directly with the diffusion. We establish these
results in the abstract, and we offer examples and applications. Our results
are related to, but different from, potential theoretic notions of
\textquotedblleft boundaries\textquotedblright in more standard random walk
models. Comparisons are made.Comment: 38 pages, 4 tables, 3 figure
Linear representations of probabilistic transformations induced by context transitions
By using straightforward frequency arguments we classify transformations of
probabilities which can be generated by transition from one preparation
procedure (context) to another. There are three classes of transformations
corresponding to statistical deviations of different magnitudes: (a)
trigonometric; (b) hyperbolic; (c) hyper-trigonometric. It is shown that not
only quantum preparation procedures can have trigonometric probabilistic
behaviour. We propose generalizations of -linear space probabilistic
calculus to describe non quantum (trigonometric and hyperbolic) probabilistic
transformations. We also analyse superposition principle in this framework.Comment: Added a physical discussion and new reference
Big Entropy Fluctuations in Statistical Equilibrium: The Macroscopic Kinetics
Large entropy fluctuations in an equilibrium steady state of classical
mechanics were studied in extensive numerical experiments on a simple
2--freedom strongly chaotic Hamiltonian model described by the modified Arnold
cat map. The rise and fall of a large separated fluctuation was shown to be
described by the (regular and stable) "macroscopic" kinetics both fast
(ballistic) and slow (diffusive). We abandoned a vague problem of "appropriate"
initial conditions by observing (in a long run)spontaneous birth and death of
arbitrarily big fluctuations for any initial state of our dynamical model.
Statistics of the infinite chain of fluctuations, reminiscent to the Poincar\'e
recurrences, was shown to be Poissonian. A simple empirical relation for the
mean period between the fluctuations (Poincar\'e "cycle") has been found and
confirmed in numerical experiments. A new representation of the entropy via the
variance of only a few trajectories ("particles") is proposed which greatly
facilitates the computation, being at the same time fairly accurate for big
fluctuations. The relation of our results to a long standing debates over
statistical "irreversibility" and the "time arrow" is briefly discussed too.Comment: Latex 2.09, 26 pages, 6 figure
Big Entropy Fluctuations in Nonequilibrium Steady State: A Simple Model with Gauss Heat Bath
Large entropy fluctuations in a nonequilibrium steady state of classical
mechanics were studied in extensive numerical experiments on a simple 2-freedom
model with the so-called Gauss time-reversible thermostat. The local
fluctuations (on a set of fixed trajectory segments) from the average heat
entropy absorbed in thermostat were found to be non-Gaussian. Approximately,
the fluctuations can be discribed by a two-Gaussian distribution with a
crossover independent of the segment length and the number of trajectories
('particles'). The distribution itself does depend on both, approaching the
single standard Gaussian distribution as any of those parameters increases. The
global time-dependent fluctuations turned out to be qualitatively different in
that they have a strict upper bound much less than the average entropy
production. Thus, unlike the equilibrium steady state, the recovery of the
initial low entropy becomes impossible, after a sufficiently long time, even in
the largest fluctuations. However, preliminary numerical experiments and the
theoretical estimates in the special case of the critical dynamics with
superdiffusion suggest the existence of infinitely many Poincar\'e recurrences
to the initial state and beyond. This is a new interesting phenomenon to be
farther studied together with some other open questions. Relation of this
particular example of nonequilibrium steady state to a long-standing persistent
controversy over statistical 'irreversibility', or the notorious 'time arrow',
is also discussed. In conclusion, an unsolved problem of the origin of the
causality 'principle' is touched upon.Comment: 21 pages, 7 figure
Adaptation of Autocatalytic Fluctuations to Diffusive Noise
Evolution of a system of diffusing and proliferating mortal reactants is
analyzed in the presence of randomly moving catalysts. While the continuum
description of the problem predicts reactant extinction as the average growth
rate becomes negative, growth rate fluctuations induced by the discrete nature
of the agents are shown to allow for an active phase, where reactants
proliferate as their spatial configuration adapts to the fluctuations of the
catalysts density. The model is explored by employing field theoretical
techniques, numerical simulations and strong coupling analysis. For d<=2, the
system is shown to exhibits an active phase at any growth rate, while for d>2 a
kinetic phase transition is predicted. The applicability of this model as a
prototype for a host of phenomena which exhibit self organization is discussed.Comment: 6 pages 6 figur
Non-Hermitian Localization and Population Biology
The time evolution of spatial fluctuations in inhomogeneous d-dimensional
biological systems is analyzed. A single species continuous growth model, in
which the population disperses via diffusion and convection is considered.
Time-independent environmental heterogeneities, such as a random distribution
of nutrients or sunlight are modeled by quenched disorder in the growth rate.
Linearization of this model of population dynamics shows that the fastest
growing localized state dominates in a time proportional to a power of the
logarithm of the system size. Using an analogy with a Schrodinger equation
subject to a constant imaginary vector potential, we propose a delocalization
transition for the steady state of the nonlinear problem at a critical
convection threshold separating localized and extended states. In the limit of
high convection velocity, the linearized growth problem in dimensions
exhibits singular scaling behavior described by a (d-1)-dimensional
generalization of the noisy Burgers' equation, with universal singularities in
the density of states associated with disorder averaged eigenvalues near the
band edge in the complex plane. The Burgers mapping leads to unusual transverse
spreading of convecting delocalized populations.Comment: 22 pages, 11 figure
Uniform approximation of Poisson integrals of functions from the class H_omega by de la Vallee Poussin sums
We obtain asymptotic equalities for least upper bounds of deviations in the
uniform metric of de la Vall\'{e}e Poussin sums on the sets
C^{q}_{\beta}H_\omega of Poisson integrals of functions from the class H_\omega
generated by convex upwards moduli of continuity \omega(t) which satisfy the
condition \omega(t)/t\to\infty as t\to 0. As an implication, a solution of the
Kolmogorov-Nikol'skii problem for de la Vall\'{e}e Poussin sums on the sets of
Poisson integrals of functions belonging to Lipschitz classes H^\alpha,
0<\alpha <1, is obtaine
Diffusion on networked systems is a question of time or structure
Network science investigates the architecture of complex systems to understand their functional and dynamical properties. Structural patterns such as communities shape diffusive processes on networks. However, these results hold under the strong assumption that networks are static entities where temporal aspects can be neglected. Here we propose a generalized formalism for linear dynamics on complex networks, able to incorporate statistical properties of the timings at which events occur. We show that the diffusion dynamics is affected by the network community structure and by the temporal properties of waiting times between events. We identify the main mechanismânetwork structure, burstiness or fat tails of waiting timesâdetermining the relaxation times of stochastic processes on temporal networks, in the absence of temporalâstructure correlations. We identify situations when fine-scale structure can be discarded from the description of the dynamics or, conversely, when a fully detailed model is required due to temporal heterogeneities
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