Diffusion In Phase Space Research Articles

A stochastic model for the relative motion of high Stokes number particles in isotropic turbulence

AbstractThe probability density function (PDF) kinetic equation describing the relative motion of inertial particle pairs in a turbulent flow requires closure of the phase-space diffusion current. A novel analytical closure for the diffusion current is presented that is applicable to high-inertia particle pairs with Stokes numbers $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}{\mathit{St}}_r \gg 1$. Here ${\mathit{St}}_r$ is a Stokes number based on the time scale $\tau _r$ of eddies whose size scales with pair separation $r$. In the asymptotic limit of ${\mathit{St}}_r \gg 1$, the pair PDF kinetic equation reduces to an equation of Fokker–Planck form. The diffusion tensor characterizing the diffusion current in the Fokker–Planck equation is equal to $1/\tau _v^2$ multiplied by the time integral of the Lagrangian correlation of fluid relative velocities along particle-pair trajectories. Here, $\tau _v$ is the particle viscous relaxation time. Closure of the diffusion tensor is achieved by converting the Lagrangian correlations of fluid relative velocities ‘seen’ by pairs into Eulerian fluid-velocity correlations at pair separations that remain essentially constant during time scales of $O(\tau _r)$; the pair centre of mass, however, is not stationary and responds to eddies with time scales comparable to or smaller than $\tau _v$. For isotropic turbulence, Eulerian fluid-velocity correlations may be expressed as Fourier transforms of the velocity spectrum tensor, enabling us to derive a closed-form expression for the diffusion tensor. A salient feature of this closure is that it has a single, unique form for pair separations spanning the entire spectrum of turbulence scales, unlike previous closures that involve velocity structure functions with different forms for the integral, inertial subrange, and Kolmogorov-scale separations. Using this closure, Langevin equations, which are statistically equivalent to the Fokker–Planck equation, were solved to evolve particle-pair relative velocities and separations in stationary isotropic turbulence. The Langevin equation approach enables the simulation of the full PDF of pair relative motion, instead of only the first few moments of the PDF as is the case in a moments-based approach. Accordingly, PDFs $\varOmega (U|r)$ and $\varOmega (U_r|r)$ are computed and presented for various separations $r$, where the former is the PDF of relative velocity $U$ and the latter is the PDF of the radial component of relative velocity $U_r$, both conditioned upon the separation $r$. Consistent with the direct numerical simulation (DNS) study of Sundaram & Collins (J. Fluid Mech., vol. 335, 1997, pp. 75–109), the Langevin simulations capture the transition of $\varOmega (U|r)$ from being Gaussian at integral-scale separations to an exponential PDF at Kolmogorov-scale separations. The radial distribution functions (RDFs) computed from these simulations also show reasonable quantitative agreement with those from the DNS study of Février, Simonin & Legendre (Proceedings of the Fourth International Conference on Multiphase Flow, New Orleans, 2001).

Anomalous waves as an object of statistical topography: Problem statement

Based on ideas of statistical topography, we analyze the boundary-value problem of the appearance of anomalous large waves (rogue waves) on the sea surface. The boundary condition for the sea surface is regarded as a closed stochastic quasilinear equation in the kinematic approximation. We obtain the stochastic Liouville equation, which underlies the derivation of an equation describing the joint probability density of fields of sea surface displacement and its gradient. We formulate the statistical problem with the stochastic topographic inhomogeneities of the sea bottom taken into account. It describes diffusion in the phase space, and its solution must answer the question whether information about the existence of anomalous large waves is contained in the quasilinear equation under consideration.

Clustering and Uncertainty in Perfect Chaos Systems

The goal of this investigation was to derive strictly new properties of chaotic systems and their mutual relations. The generalized Fokker-Planck equation with a nonstationary diffusion has been derived and used for chaos analysis. An anomalous transport turned out to be natural property of this equation. A nonlinear dispersion of the considered motion allowed us to find a principal consequence: a chaotic system with uniform dynamic properties tends to instable clustering. Small fluctuations of particles density increase by time and form attractors and stochastic islands even if the initial transport properties have uniform distribution. It was shown that an instability of phase trajectories leads to the nonlinear dispersion law and consequently to a space instability. A fixed boundary system was considered, using a standard Fokker-Planck equation. We have derived that such a type of dynamic systems has a discrete diffusive and energy spectra. It was shown that phase space diffusion is the only parameter that defines a dynamic accuracy in this case. The uncertainty relations have been obtained for conjugate phase space variables with account of transport properties. Given results can be used in the area of chaotic systems modelling and turbulence investigation.

A family of dissipative two-dimensional mappings: Chaotic, regular and steady state dynamics investigation

Some dynamical properties for a family of two dimensional mappings controlled by three parameters are considered in this paper. For null dissipation and depending on the other control parameters, diffusion in phase space is observed. A connection with the standard mapping is made to determine the range of control parameters leading to unlimited diffusion. In the dissipative case, for low nonlinearity, we obtain the fixed points and an expression to find any periodic orbit with any period, if it exists. For large nonlinearity we investigate the Lyapunov exponents and characterise the steady state of the system for a long time.

Turbulent heating and acceleration of He++ ions by spectra of Alfvén‐cyclotron waves in the expanding solar wind: 1.5‐D hybrid simulations

Both remote sensing and in situ measurements show that the fast solar wind plasma significantly deviates from thermal equilibrium and is strongly permeated by turbulent electromagnetic waves, which regulate the ion temperature anisotropies and relative drifts. Thus, the ion kinetics is governed by heating and cooling related to absorption and emission of ion‐acoustic and ion‐cyclotron waves, as well as nonresonant pitch angle scattering and diffusion in phase space. Additionally, the solar wind properties are affected by its nonadiabatic expansion as the wind travels away from the Sun. In this study we present results from 1.5‐D hybrid simulations to investigate the effects of a nonlinear turbulent spectrum of Alfvén‐cyclotron waves and the solar wind expansion on the anisotropic heating and differential acceleration of protons and He++ ions. We compare the different heating and acceleration by turbulent Alfvén‐cyclotron wave spectra and by pure monochromatic waves. For the waves and the wave spectra used in our model, we find that the He++ ions are preferentially heated and by the end of the simulations acquire much more than mass‐proportional temperature ratios, Tα/Tp>mα/mp. The differential acceleration between the two species strongly depends on the initial wave amplitude and the related spectral index and is often suppressed by the solar wind expansion. We also find that the expansion leads to perpendicular cooling for both species, and depending on the initial wave spectra, it can either heat or cool the ions in parallel direction. Despite the cooling effect of the expansion in perpendicular direction, the wave‐particle interactions provide an additional heating source, and the perpendicular temperature components remain higher than the adiabatic predictions.

On the origin of the phase–space diffusion limit in (dis)ordered protein aggregation

Derivation of a phase–space diffusion limit (D-L) allows to obtain a useful formula for a characteristic width of the macroion-channeling filter, controlling model (dis)ordered protein aggregations in a non-ideal aqueous solution. The channel’s width is estimated at the order of an inner half-width of the Stern-type double layer circumventing the growing object and depends in turn on an interplay of the local thermal and electrostatic conditions. The interfacial channeling effect manifests at the edge of biomolecular hydration-duration dependent (non)Markovianity of the system. The interface vs. solution aggregation late-time dynamics are discussed in such local (non)isothermal context with the aim to suggest their experimental assessment.

Whistler mode wave growth and propagation in the prenoon magnetosphere

Pitch angle scattering of electrons can limit the stably trapped particle flux in the magnetosphere and precipitate energetic electrons into the ionosphere. Whistler mode waves generated by a temperature anisotropy can mediate this pitch angle scattering over a wide range of radial distances and latitudes, but in order to correctly predict the phase space diffusion, it is important to characterize the whistler mode wave distributions that result from the instability. We use previously published observations of number density, pitch angle anisotropy, and phase space density to model the plasma in the quiet prenoon magnetosphere (defined as periods when AE < 100 nT). We investigate the global propagation and growth of whistler mode waves by studying millions of growing raypaths and demonstrate that the wave distribution at any one location is a superposition of many waves at different points along their trajectories and with different histories. We show that for observed electron plasma properties, very few raypaths undergo magnetospheric reflection; most rays grow and decay within 30 degrees of the magnetic equator. The frequency range of the wave distribution at large L can be adequately described by the solutions of the local dispersion relation, but the range of wave normal angle is different. The wave distribution is asymmetric with respect to the wave normal angle. The numerical results suggest that it is important to determine the variation of magnetospheric parameters as a function of latitude, as well as local time and L‐shell.

Super-diffusion in optical realizations of Anderson localization

We discuss the dynamics of particles in one dimension in potentials that are random in both space and time. The results are applied to recent optics experiments on Anderson localization, in which the transverse spreading of a beam is suppressed by random fluctuations in the refractive index. If the refractive index fluctuates along the direction of the paraxial propagation of the beam, the localization is destroyed. We analyze this broken localization in terms of the spectral decomposition of the potential. When the potential has a discrete spectrum, the spread is controlled by the overlap of Chirikov resonances in phase space. As the number of Fourier components is increased, the resonances merge into a continuum, which is described by a Fokker–Planck equation. We express the diffusion coefficient in terms of the spectral intensity of the potential. For a general class of potentials that are commonly used in optics, the solutions to the Fokker–Planck equation exhibit anomalous diffusion in phase space, implying that when Anderson localization is broken by temporal fluctuations of the potential, the result is transport at a rate similar to a ballistic one or even faster. For a class of potentials which arise in some existing realizations of Anderson localization, atypical behavior is found.

On wave–particle interaction in axisymmetric toroidal systems

A general formalism is developed to describe the interaction of charged particles with electromagnetic waves in terms of coupled finite difference mapping equations that incorporate tokamak topology. The approach is based on considering non-adiabatic changes in the constants of particle motion and it covers a range of wave–particle resonance frequencies, from the precessional to cyclotron frequencies of both passing and trapped ions. The concept of overlapping resonances is used to estimate the threshold for a single plane wave to cause stochastic particle motion. In the stochastic regime, the process is Markovian, and particle diffusion in three-dimensional phase space takes place. Estimations of diffusion coefficients are carried out in the two cases of waves interacting with passing and trapped ions by means of the cyclotron and bounce resonances, respectively, and previously known results are recovered in the proper limits.

Boxed Molecular Dynamics: Decorrelation Time Scales and the Kinetic Master Equation.

A number of methods proposed in the past few years have been aimed at accelerating the sampling of rare events in molecular dynamics simulations. We recently introduced a method called Boxed Molecular Dynamics (BXD) for accelerating the calculation of thermodynamics and kinetics ( J. Phys. Chem. B 2009 , 113 , 16603 - 16611 ). BXD relies upon confining the system in a series of adjacent "boxes" by inverting the projection of the system velocities along the reaction coordinate. The potential of mean force along the reaction coordinate is obtained from the mean first passage times (MFPTs) for exchange between neighboring boxes, simultaneously providing both kinetics and thermodynamics. In this paper, we investigate BXD in the context of its natural relation to a kinetic master equation and show that the BXD first passage times (FPTs) include different time scales-a fast short time decay due to correlated dynamical motion and slower long time decay arising from phase space diffusion. Correcting the FPTs to remove the fast correlated motion yields accurate thermodynamics and master equation kinetics. We also discuss interrelations between BXD and a recently described Markovian milestoning technique and use a simple application to show that, despite each method producing distinct nonstatistical effects on time scales on the order of dynamical decorrelation, both yield similar long-time kinetics.

Potential-Decomposition Strategy in Markov Chain Monte Carlo Sampling Algorithms

We introduce the potential-decomposition strategy (PDS), which can he used in Markov chain Monte Carlo sampling algorithms. PDS can be designed to make particles move in a modified potential that favors diffusion in phase space, then, by rejecting some trial samples, the target distributions can be sampled in an unbiased manner. Furthermore, if the accepted trial samples are insufficient, they can be recycled as initial states to form more unbiased samples. This strategy can greatly improve efficiency when the original potential has multiple metastable states separated by large barriers. We apply PDS to the 2d Ising model and a double-well potential model with a large barrier, demonstrating in these two representative examples that convergence is accelerated by orders of magnitude.

Constraining white dwarf kicks in globular clusters – IV. Retarding core collapse

Abstract Observations of white dwarfs in the globular clusters NGC 6397 and Omega Centauri indicate that these stars may get a velocity kick during their time as giants. If the mass loss while on the asymptotic giant branch is slightly asymmetric, the resulting white dwarf could be born with such a velocity kick. These energetic white dwarfs will impart their excess energy on other stars as they travel through the cluster. A Monte Carlo simulation of the white dwarfs kicks combined with estimate of the phase-space diffusion of the white dwarfs reveals that as the white dwarfs equilibrate, they lose most of their energy in the central region of the cluster. They could possibly augment the effect of binaries, delaying core collapse or increasing the size of globular cluster cores.

THE ATTRACTOR AND THE QUANTUM STATES

The dissipative dynamics anticipated in the proof of 't Hooft's existence theorem — "For any quantum system there exists at least one deterministic model that reproduces all its dynamics after prequantization" — is constructed here explicitly. We propose a generalization of Liouville's classical phase space equation, incorporating dissipation and diffusion, and demonstrate that it describes the emergence of quantum states and their dynamics in the Schrödinger picture. Asymptotically, there is a stable ground state and two decoupled sets of degrees of freedom, which transform into each other under the energy-parity symmetry of Kaplan and Sundrum. They recover the familiar Hilbert space and its dual. Expectations of observables are shown to agree with the Born rule, which is not imposed a priori. This attractor mechanism is applicable in the presence of interactions, to few-body or field theories in particular.

Behaviour of the Fokker–Planck–Boltzmann equation near a Maxwellian

We consider the initial value problem for the Fokker–Planck–Boltzmann equation namely, viewed as the Boltzmann equation with an additional diffusion term in velocity space to describe, for instance, the transport in thermal baths of binary elastic collisional particles. The strong solution for initial data near an absolute Maxwellian is proved to exist globally in time and tends asymptotically in the \({L^\infty_y(L_\xi^1)}\) -norm to another time dependent self-similar Maxwellian in large time. The effect of the diffusion in phase space is investigated. It produces a diffusion process in velocity space and results in a heating process on the macroscopic fluid-dynamic observable, accelerating the convergence of solutions to the equilibrium of a self-similar Maxwellian at a faster time-decay rate than the Boltzmann equation. This phenomena is also observed for homogeneous Fokker–Planck–Boltzmann equations, where the time-decay rate in the \({L_\xi^1}\) -norm to the self-similar Maxwellian is proved to be faster than exponential. Moreover, the Fokker–Planck–Boltzmann equation is shown to converge (under an appropriate scaling) strongly to the Boltzmann equation in the process of the zero diffusion limit.

CONTROL OF CHAOTIC ADVECTION

A method of chaos reduction for Hamiltonian systems is applied to control chaotic advection. By adding a small and simple term to the stream function of the system, the construction of invariant tori has a stabilization effect in the sense that these tori act as barriers to diffusion in phase space and the controlled Hamiltonian system exhibits a more regular behaviour.

A model for nonexponential relaxation and aging in dissipative systems

The nonexponential relaxation and aging inherent to complex dynamics manifested in a wide variety of dissipative systems are analyzed through a model of diffusion in phase space in the presence of a nonconservative force. The action of this force establishes a heat flow which maintains the system away from equilibrium. The inability of the system to find its equilibrium state becomes apparent through the presence of an effective temperature field. This is the temperature of the stationary nonequilibrium state reached by the system satisfying a generalized version of the fluctuation-dissipation theorem. The presence of a nonequilibrium temperature leads to a hierarchy of relaxation times responsible for the aging phenomena and to a relation similar to the Vogel-Fulcher-Tammann law [H. Vogel, Phys. Z. 22, 645 (1921); G. S. Fulcher, J. Am. Ceram. Soc. 8, 339 (1925); 8, 789 (1925); G. Tammann and W. Hesse, Z. Anorg. Allg. Chem. 156, 245 (1926)].

Vlasov Hybrid Simulation—An Efficient and Stable Algorithm for the Numerical Simulation of Collision‐Free Plasma

This paper presents a highly efficient and stable algorithm for the numerical simulation of collision free plasma. The algorithm has been successfully used to numerically model non linear electron cyclotron resonance in VLF band radio waves in space, and has produced good simulations of radio emissions such as ‘dawn chorus’ and ‘triggered VLF emissions’. The algorithm fills the phase box with simulation particles which represent phase space trajectories. Particle trajectories are followed forwards continuously in time without restarts, and these particles are followed until either the simulation ends or the particles exit from the phase box. Liouville’s theorem states that distribution function F is conserved along these trajectories, and so F is known at the phase space points represented by the particle positions. No phase space volume is associated with these particles. At each time step F is truly interpolated, and not distributed from particles to the regular phase space grid, solely for the purpose of calculating plasma charge and current fields. For this operation a cheap low order interpolation is useable. The algorithm is simple, robust and highly efficient. There is NO diffusion in phase space whatsoever, and fine structure is merely undersampled and does not cause any instability problems. Regions of phase space where F=0 are cost free as no particles need be provided. The algorithm readily accomodates situations where phase fluid flows into or out of the phase box. The paper presents the latest simulations of VLF chorus and VLF triggered emissions from the VHS/VLF code, which are in excellent agreement with observations.

QUANTUM GRAVITY PHENOMENOLOGY, LORENTZ INVARIANCE AND DISCRETENESS

Contrary to what is often stated, a fundamental spacetime discreteness need not contradict Lorentz invariance. A causal set's discreteness is in fact locally Lorentz invariant, and we recall the reasons why. For illustration, we introduce a phenomenological model of massive particles propagating in a Minkowski spacetime which arises from an underlying causal set. The particles undergo a Lorentz invariant diffusion in phase space, and we speculate on whether this could have any bearing on the origin of high energy cosmic rays.

Simple model for nonexponential relaxation in complex dynamics

The nonexponential relaxation occurring in complex dynamics manifested in a wide variety of systems is analyzed through a simple model of diffusion in phase space. It is found that the inability of the system to find its equilibrium state in any time scale becomes apparent in an effective temperature field, which leads to a hierarchy of relaxation times responsible for the slow relaxation phenomena.

Singularity in classical and quantum Kepler problem with weak anisotropy

Anisotropic Kepler problem is investigated by perturbation method in both classical and quantum mechanics. In classical mechanics, due to the singularity of the potential, global diffusion in phase space occurs at an arbitrarily small perturbation parameter. In quantum mechanics, the singularity induces a large tunneling rate between quasi degenerated eigenstates, which generically decays as ℏ in the semi-classical limit.