Fokker-Planck Diffusion Equation Research Articles

How to perform the coherent measurement of a curved phase space by continuous isotropic measurement. I. Spin and the Kraus-operator geometry of SL(2,C)

The generalized Q-function of a spin system can be considered the outcome probability distribution of a state subjected to a measurement represented by the spin-coherent-state (SCS) positive-operator-valued measure (POVM). As fundamental as the SCS POVM is to the 2-sphere phase-space representation of spin systems, it has only recently been reported that the SCS POVM can be performed for any spin system by continuous isotropic measurement of the three total spin components [E. Shojaee, C. S. Jackson, C. A. Riofrio, A. Kalev, and I. H. Deutsch, Phys. Rev. Lett. 121, 130404 (2018)]. This article develops the theoretical details of the continuous isotropic measurement and places it within the general context of curved-phase-space correspondences for quantum systems. The analysis is in terms of the Kraus operators that develop over the course of a continuous isotropic measurement. The Kraus operators of any spin j are shown to represent elements of the Lie group SL(2,C)≅Spin(3,C), a complex version of the usual unitary operators that represent elements of SU(2)≅Spin(3,R). Consequently, the associated POVM elements represent points in the symmetric space SU(2)∖SL(2,C), which can be recognized as the 3-hyperboloid. Three equivalent stochastic techniques, (Wiener) path integral, (Fokker-Planck) diffusion equation, and stochastic differential equations, are applied to show that the continuous isotropic POVM quickly limits to the SCS POVM, placing spherical phase space at the boundary of the fundamental Lie group SL(2,C) in an operationally meaningful way. Two basic mathematical tools are used to analyze the evolving Kraus operators, the Maurer-Cartan form, modified for stochastic applications, and the Cartan, decomposition associated with the symmetric pair SU(2) ⊂ SL(2,C). Informed by these tools, the three schochastic techniques are applied directly to the Kraus operators in a representation-independent – and therefore geometric – way (independent of any spectral information about the spin components).The Kraus-operator-centric, geometric treatment applies not just to SU(2) ⊂ SL(2,C), but also to any compact semisimple Lie group and its complexification. The POVM associated with the continuous isotropic measurement of Lie-group generators thus corresponds to a type-IV globally Riemannian symmetric space and limits to the POVM of generalized coherent states. This generalization is the focus of a sequel to this article.

Correlated observations and simulations on the buildup of radiation belt electron fluxes driven by substorm injections and chorus waves

We report the multi-satellite (LANL, GOES-10 and Cluster) observation data of electron flux evolutions and chorus wave excitation in the radiation belt during the geomagnetic storm and substorm from 10 to 14 January, 2002. The seed (50–225 keV) electron flux increased 50 times in five hours during the storm main phase, and the relativistic (>0.6 MeV) electron flux increased about 60 times at night side during the recovery phase. In the meanwhile, the Cluster satellites detected intense chorus waves (the wave power up to ∼10−3 nT2 Hz−1) at MLT≈3 when passing through the outer radiation belt. Using a Gaussian fit to the observed chorus spectra, we calculate the drift-averaged diffusion coefficients and then solve a 2-D Fokker-Planck diffusion equation. We simulate the energetic electron flux evolutions driven by chorus waves in two cases: with and without seed electron injections. We show that the energetic electron flux increases 79 times in three days with injection, comparable to the observation. However, the flux increases only 3 times in three days without injection, far below the observation. The current results suggest that the injected seed electrons and chorus waves play important roles in the buildup of the radiation belt electrons.

Radiation belt electron acceleration induced by gyroresonant interaction with magnetosonic waves

Using Cluster 4 satellite data, we examine activities of fast magnetosonic (MS) waves in the outer radiation belt near the location L=4.2 on 28 May 2005. We adopt a Gaussian distribution to fit the observed power spectral density of MS waves and find the fitting wave strength to be 245 pT. We then calculate the bounce-averaged diffusion coefficients and show that these diffusion coefficients are pronounced within a region of pitch angles about 25°–70°. By solving a 2D Fokker-Planck diffusion equation, we simulate the dynamic evolution of the electron phase space density (PSD), and demonstrate that significant increases in electron PSDs at energies of MeVs occur mainly within the aforementioned pitch-angle range over a time scale of several hours. The current results suggest that the interaction between MS waves and electrons could be an important mechanism of electron acceleration in the radiation belt.

Effect of low energy electron injection on storm-time evolution of radiation belt energetic electrons: three-dimensional modeling

Using the STEERB (storm-time evolution of electron radiation belt) code, we simulate the evolution of radiation belt energetic electrons during geomagnetic storms in the case of low energy electron injection. The STEERB code is used to solve the three-dimensional Fokker–Planck diffusion equation which incorporates wave-particle interaction, Coulomb collisions and radial diffusion. Numerical simulations show that under the short time (∼1 h) injection of low energy (0.1 MeV≤Ek≤0.2 MeV) fluxes of radiation belt energetic electrons can increase during the entire storm period. During the main and recovery phases, such injection efficiently enhances chorus-driven acceleration of radiation belt energetic electrons, allowing fluxes of energetic electrons by a factor of 1–2 orders higher than those in the absence of injection. The current results indicate that substorm-induced electron injection must be incorporated to investigate the evolution of radiation belt energetic electrons.

Responses of geostationary orbit energetic electron fluxes to chorus waves under different geomagnetic conditions

We investigate the flux evolution of geostationary orbit energetic electrons during a strong storm on 24 August 2005 (event A, the storm index Dst -50 nT, average AE=320 nT). Data collected by LANL and GOES-12 satellites show that energetic electron fluxes increase by a factor of 10 during the recovery phase compared to the prestorm level for both events A and B. As the substorm continued, the Cluster C4 satellite observed strong whistler-mode chorus waves (with spectral density approaching 10−5 nT2/Hz). The wave amplitude correlates with the substorm AE index, but is less correlated with the storm Dst index. Using a Gaussian distribution fitting method, we solve the Fokker-Planck diffusion equation governing the wave-particle interaction. Numerical results demonstrate that chorus waves efficiently accelerate ∼1 MeV energetic electrons, particularly at high pitch angles. The calculated acceleration time scale and amplitude are comparable to observations. Our results provide new observational support for chorus-driven acceleration of radiation belt energetic electrons.

Fokker-Planck description for the queue dynamics of large tick stocks

Motivated by empirical data, we develop a statistical description of the queue dynamics for large tick assets based on a two-dimensional Fokker-Planck (diffusion) equation. Our description explicitly includes state dependence, i.e., the fact that the drift and diffusion depend on the volume present on both sides of the spread. "Jump" events, corresponding to sudden changes of the best limit price, must also be included as birth-death terms in the Fokker-Planck equation. All quantities involved in the equation can be calibrated using high-frequency data on the best quotes. One of our central findings is that the dynamical process is approximately scale invariant, i.e., the only relevant variable is the ratio of the current volume in the queue to its average value. While the latter shows intraday seasonalities and strong variability across stocks and time periods, the dynamics of the rescaled volumes is universal. In terms of rescaled volumes, we found that the drift has a complex two-dimensional structure, which is a sum of a gradient contribution and a rotational contribution, both stable across stocks and time. This drift term is entirely responsible for the dynamical correlations between the ask queue and the bid queue.

Review on the Brownian Dynamics Simulation of Bead-Rod-Spring Models Encountered in Computational Rheology

Kinetic theory is a mathematical framework intended to relate directly the most relevant characteristics of the molecular structure to the rheological behavior of the bulk system. In other words, kinetic theory is a micro-to-macro approach for solving the flow of complex fluids that circumvents the use of closure relations and offers a better physical description of the phenomena involved in the flow processes. Cornerstone models in kinetic theory employ beads, rods and springs for mimicking the molecular structure of the complex fluid. The generalized bead-rod-spring chain includes the most basic models in kinetic theory: the freely jointed bead-spring chain and the freely-jointed bead-rod chain. Configuration of simple coarse-grained models can be represented by an equivalent Fokker-Planck (FP) diffusion equation, which describes the evolution of the configuration distribution function in the physical and configurational spaces. FP equation can be a complex mathematical object, given its multidimensionality, and solving it explicitly can become a difficult task. Even more, in some cases, obtaining an equivalent FP equation is not possible given the complexity of the coarse-grained molecular model. Brownian dynamics can be employed as an alternative extensive numerical method for approaching the configuration distribution function of a given kinetic-theory model that avoid obtaining and/or resolving explicitly an equivalent FP equation. The validity of this discrete approach is based on the mathematical equivalence between a continuous diffusion equation and a stochastic differential equation as demonstrated by Itô in the 1940s. This paper presents a review of the fundamental issues in the BD simulation of the linear viscoelastic behavior of bead-rod-spring coarse grained models in dilute solution. In the first part of this work, the BD numerical technique is introduced. An overview of the mathematical framework of the BD and a review of the scope of applications are presented. Subsequently, the links between the rheology of complex fluids, the kinetic theory and the BD technique are established at the light of the stochastic nature of the bead-rod-spring models. Finally, the pertinence of the present state-of-the-art review is explained in terms of the increasing interest for the stochastic micro-to-macro approaches for solving complex fluids problems. In the second part of this paper, a detailed description of the BD algorithm used for simulating a small-amplitude oscillatory deformation test is given. Dynamic properties are employed throughout this work to characterise the linear viscoelastic behavior of bead-rod-spring models in dilute solution. In the third and fourth part of this article, an extensive discussion about the main issues of a BD simulation in linear viscoelasticity of diluted suspensions is tackled at the light of the classical multi-bead-spring chain model and the multi-bead-rod chain model, respectively. Kinematic formulations, integration schemes and expressions to calculate the stress tensor are revised for several classical models: Rouse and Zimm theories in the case of multi-bead-spring chains, and Kramers chain and semi-flexible filaments in the case of multi-bead-rod chains. The implemented BD technique is, on the one hand, validated in front of the analytical or exact numerical solutions known of the equivalent FP equations for those classic kinetic theory models; and, on the other hand, is control-set thanks to the analysis of the main numerical issues involved in a BD simulation. Finally, the review paper is closed by some concluding remarks.

Three‐dimensional radiation belt simulations in terms of adiabatic invariants using a single numerical grid

The dynamics of electrons in the Earth's radiation belts can be described by the Fokker‐Planck equation, which includes radial diffusion, and local energy and pitch angle diffusion. Previously, the Versatile Electron Radiation Belt (VERB) code utilized two grids to solve the Fokker‐Planck equation; one grid, which keeps the first and second adiabatic invariants constant, was used for the computation of the radial diffusion, and the other grid, orthogonal in energy and pitch angle at each fixed radial distance, was used for the computation of energy diffusion, pitch angle diffusion, and mixed energy and pitch angle diffusion. At each time step, the results were interpolated between the two grids. In the current work, we present a new method for numerical solution of the Fokker‐Planck diffusion equation written in terms of adiabatic invariants using a single numerical grid. The one‐grid solution allows us to eliminate the time‐consuming interpolation. The solution in terms of adiabatic invariants implicitly accounts for conservation of phase space density during adiabatic magnetic field fluctuations. The Fokker‐Planck equation written in terms of adiabatic invariants is easier to solve with a realistic magnetic field configuration, compared to the Fokker‐Planck equation written in terms of energy and pitch angle. To validate the approach, we compare a 2‐D simulation at fixed L* in terms of the adiabatic invariants (μ, K) with a simulation on a grid orthogonal in energy and pitch angle. Both simulations produce essentially the same result. Accurate simulations on the (μ, K) grid require resolution of 601 × 601 points, while only 101 × 101 grid points are sufficient for performing an accurate simulation on the grid orthogonal in energy and pitch angle. We propose a new invariant V ≡ μ ⋅ (K + 0.5)2 with the new grid orthogonal in (V, K, L*), that is more convenient for 2‐D and 3‐D simulations. The 2‐D and 3‐D simulations on the (V, K, L*) grid produce the same result as the two‐grid method with a grid resolution of only 101 × 101 points in V and K.

Evolution of the plasma sheet electron pitch angle distribution by whistler-mode chorus waves in non-dipole magnetic fields

Abstract. We present a detailed numerical study on the effects of a non-dipole magnetic field on the Earth's plasma sheet electron distribution and its implication for diffuse auroral precipitation. Use of the modified bounce-averaged Fokker-Planck equation developed in the companion paper by Ni et al. (2012) for 2-D non-dipole magnetic fields suggests that we can adopt a numerical scheme similar to that used for a dipole field, but should evaluate bounce-averaged diffusion coefficients and bounce period related terms in non-dipole magnetic fields. Focusing on nightside whistler-mode chorus waves at L = 6, and using various Dungey magnetic models, we calculate and compare of the bounce-averaged diffusion coefficients in each case. Using the Alternative Direction Implicit (ADI) scheme to numerically solve the 2-D Fokker-Planck diffusion equation, we demonstrate that chorus driven resonant scattering causes plasma sheet electrons to be scattered much faster into loss cone in a non-dipole field than a dipole. The electrons subject to such scattering extends to lower energies and higher equatorial pitch angles when the southward interplanetary magnetic field (IMF) increases in the Dungey magnetic model. Furthermore, we find that changes in the diffusion coefficients are the dominant factor responsible for variations in the modeled temporal evolution of plasma sheet electron distribution. Our study demonstrates that the effects of realistic ambient magnetic fields need to be incorporated into both the evaluation of resonant diffusion coefficients and the calculation of Fokker-Planck diffusion equation to understand quantitatively the evolution of plasma sheet electron distribution and the occurrence of diffuse aurora, in particular at L > 5 during geomagnetically disturbed periods when the ambient magnetic field considerably deviates from a magnetic dipole.

Bounce-averaged Fokker-Planck diffusion equation in non-dipolar magnetic fields with applications to the Dungey magnetosphere

Abstract. We perform a detailed derivation of the bounce-averaged relativistic Fokker-Planck diffusion equation applicable to arbitrary magnetic field at a constant Roederer L. The form of the bounce-averaged diffusion equation is found regardless of details of the mirror geometry, suggesting that the numerical schemes developed for solving the modified two-dimensional (2-D) Fokker-Planck equation in a magnetic dipole should be feasible for similar computation efforts on modeling wave-induced particle diffusion processes in any non-dipolar magnetic field. However, bounce period related terms and bounce-averaged diffusion coefficients are required to be computed in realistic magnetic fields. With the application to the Dungey magnetosphere that is controlled by the intensity of southward interplanetary magnetic field (IMF), we show that with enhanced southward IMF the normalized bounce period related term decreases accordingly, and bounce-averaged diffusion coefficients cover a broader range of electron energy and equatorial pitch angle with a tendency of increased magnitude and peaking at lower energies. The compression of the Dungey magnetosphere can generally produce scattering loss of plasma sheet electrons ~100 keV on a timescale shorter than that in a dipolar field, and induce momentum diffusion at high pitch angles closer to 90°. Correspondingly, the strong diffusion rate drops considerably as a product of changes in both the equatorial loss cone and the bounce period. The extent of differences in all the parameters introduced by the southward IMF intensification also becomes larger for a field line with higher equatorial crossing. With the derived general formulism of bounce-averaged diffusion equation for arbitrary 2-D magnetic field, our results confirm the need for the adoption of realistic magnetic fields to perform accurate determination of electron resonant scattering rates and precise multi-dimensional diffusion simulations of magnetospheric electron dynamics.

Simulation of the acceleration of relativistic electrons in the inner magnetosphere using RCM-VERB coupled codes

[1] Radiation belt dynamics have been modeled by the modified Fokker-Planck diffusion equation with sources from the low-energy plasma sheet population and losses to the atmosphere and magnetopause. We perform a coupled simulation of the Rice Convection Model (RCM) and Versatile Electron Radiation Belt (VERB) code. The RCM models magnetospheric convection and provides a low-energy electron seed population for the VERB diffusion code simulations of the Earth's radiation belts. VERB simulations are driven by the realistic time-dependent electron seed population and by the Kp index, which is used to specify rates of diffusion by ultralow frequency (ULF) and very low frequency wave activity and, therefore, diffusion processes. Radial diffusion is produced by ULF waves, while pitch angle and energy diffusion are produced by chorus waves outside the plasmasphere and by hiss waves inside the plasmasphere. The results of the simulation indicate that storm time enhanced magnetospheric convection combined with radial diffusion can bring electrons with tens of keV energy close to the Earth and can affect electron fluxes at 3–4 RE. These electrons can be further accelerated locally by chorus waves to MeV energies. Furthermore, outward radial diffusion smooths out the peak of the high-energy fluxes and produces MeV electron enhancement around geosynchronous orbit (6–7 RE) despite the absence of local electron acceleration in that region. Our coupled simulations indicate that local acceleration in the inner magnetosphere may be a dominant source of relativistic electrons that reach geosynchronous orbit.

Modeling for precipitation loss of ring current protons by electromagnetic ion cyclotron waves

We examine the precipitation loss of ring current protons produced by the electromagnetic ion cyclotron (EMIC) waves at the location L=3.5 where the symmetrical ring current primarily occurs. Using a recently introduced hybrid finite difference (HFD) method, we solve the standard Fokker–Planck diffusion equation with incorporation of the cross diffusion terms that were often ignored in the previous work. We show that EMIC waves can produce efficient pitch angle scattering of energetic (tens of keV) protons, yielding a rapid decrement in the flux with a time scale of a few hours, consistent with observational data. We also demonstrate that omission of the cross diffusion terms generally overestimates the decrement in the flux, typically by a factor of ∼ 2 after a few hours at lower pitch angles, suggesting that the cross terms also play an important role in precipitation loss of ring current protons.

Evolution of Ring Current Protons Induced by Electromagnetic Ion Cyclotron Waves

We investigate the evolution of the phase space density (PSD) of ring current protons induced by electromagnetic ion cyclotron (EMIC) waves at the location L = 3.5, calculate the diffusion coefficients in pitch angle and momentum, and solve the standard two-dimensional Fokker–Planck diffusion equation. The pitch angle diffusion coefficient is found to be larger than the momentum diffusion coefficient by a factor of about 103 or above at lower pitch angles. We show that EMIC waves can produce efficient pitch angle scattering of energetic (≈ 100 keV) protons, yielding a rapid decrement in PSD, typically by a factor of ≈ 10 within a few hours, consistent with observational data. This result further supports previous findings that wave-particle interaction is responsible for the rapid ring current decay.

Exact solutions for a generalized nonlinear fractional Fokker–Planck equation

This paper is devoted to investigating a generalized nonlinear Fokker–Planck diffusion equation with external force and absorption. We first investigate the integer nonlinear anomalous diffusion, and we obtain the corresponding exact solution expressed by q -exponential function. The solutions of nonlinear diffusion equation with one-fractional derivative and multi-fractional derivative are also studied in detail, and the solutions can have a compact behavior or a long tailed behavior.

Discussing the processes constraining the Jovian synchrotron radio emission's features

Our recent analysis and understanding of the Jovian synchrotron radio emission with a radiation-belt model is presented. In this work, the electron population is determined by solving the Fokker–Planck diffusion equation and considering different physical processes. The results of the modeling are first compared to in situ particle data, brightness distributions, radio spectrum, and beaming curves to verify the simulated particle distributions. The dynamics of high-energy electrons in Jupiter's inner magnetosphere and their related radio emission are then examined. The results demonstrate that the Jovian moons set the extension and intensity of the synchrotron emission's brightness distribution along the magnetic equator. Simulations show that moons and dust both control the transport toward the planet by significantly reducing the abundance of particles constrained to populate, near the equator and inside 1.8 Jovian radii, the innermost region of the magnetosphere. Due to interactions with dust and synchrotron mechanism, radiation-belt electrons are moved along field lines, between Metis (1.79 Jovian radii) and Amalthea (2.54 Jovian radii), toward high latitudes. The quantity of particles transported away from the equator is sufficient to produce measurable secondary radio emissions. Among all the phenomena acting in the inner magnetosphere, the moons (Amalthea and Thebe) are the primary moderator for the radiation's intensity at high latitudes. Moon losses also affect the characteristics of the total radio flux with longitude. The sweeping effect amplifies the 10-h modulation of the beaming curve's amplitude while energy resonances occurring near Amalthea and Thebe belong to phenomena adjusting it to the right level. Interactions with dust do not significantly constrain radio spectrum features. Resonances near Amalthea and Thebe are responsible for the Jovian radio spectrum's particular slope.

Sea-urchin feeding fronts

Sea-urchin feeding fronts are a striking example of spatial pattern formation in an ecological system. If it is assumed that urchins are asocial, and that they move randomly, then the formation of these dense fronts is an apparent paradox. The key lies in observations that urchins move further in areas where their algal food is less plentiful. This naturally leads to the accumulation of urchins in areas with abundant algae. If urchin movement is represented as a random walk, with a step size that depends on algal concentration, then their movement may be described by a Fokker–Planck diffusion equation. For certain combinations of algal growth and urchin grazing, traveling wave solutions are obtained. Two-dimensional simulations of urchin algal dynamics show that an initially uniformly distributed urchin population, grazing on an alga with a smoothly varying density, may form a propagating front separating two sharply delineated regions. On one side of the front algal density is uniformly low, and on the other side of the front algal density is uniformly high. Bounds on when stable fronts will form are obtained in terms of urchin density and grazing, and algal growth.

MOLECULAR TRANSPORT AND FLUIDICS IN CARBON NANOTUBE

In this article we review recent developments in molecular transport and fluidics in carbon nanotube (CNT)-based nanochannels. Atomic molecular dynamics simulations and theoretical studies based on Fokker–Planck diffusion equation on the transport of large and long polymer molecules in CNTs are the focus of the article. Fast translocation and diffusion processes of large molecules in CNTs are reviewed and discussed, considering the effects of interfacial interactions and molecular conformations and structures at interface. The transport features for multiple molecules diffusing through CNTs are also discussed.

Scale-Dependent Subsurface Dispersion: A Fractal-Based Stochastic Model

A stochastic model that can simulate non-Fickian (scale-dependent) subsurface dispersion by imposing long-range correlation structure within a particle-based transport scheme is detailed here. The model uses increments from self-similar random fractal processes. These processes [fractional Brownian motions (FBMs)] are characterized by an index H, which is related to the correlation properties of the function. Temporally correlated FBMs that mimic the faster-than-Fickian growth observed in contaminant plumes in heterogeneous subsurface porous media are chosen. It is shown how FBMs can be incorporated within traditional Langevin-based formulations of the random-walk particle tracking models to describe mass displacements. A Fokker-Planck diffusion equation with a time-dependent tensor and its solution are presented to describe FBM anomalous dispersion. A relationship of the model's parameters to measurable porous media parameters is given.

Stochastic diffusivity and heat transport in the presence of a radial dependence of the perturbed magnetic field in the reversed field pinch

Stochastic diffusivity in the presence of island overlapping can be numerically estimated following the radial evolution of an ensemble of magnetic field lines, starting from a given initial position, when they have travelled for a length of the order of the perturbed magnetic field correlation length along the main magnetic field. In this paper we show that, caused by the radial dependence of the magnetic perturbation and due to the short correlation length in reversed field pinches (RFPs), different `local' diffusion rates at different radii originate.The numerical results describing the stochastic diffusion process, obtained for typical RFP configurations and magnetic field perturbations, are presented and compared with an analytical WKB approximated solution of the relevant one-dimensional Fokker-Planck diffusion equation. The magnetic diffusivity profile is then inserted into a heat diffusion equation for the electrons in which the heat flux is balanced by the ohmic input power. In this way the electron temperature profile is numerically obtained in collisionless and collisional regimes.

Fractal Cracking of Concrete: Parameterization of Spatial Diffusion

Concrete cracks on the tension face of beams in uniaxial bending are interpreted as a non-Fickian diffusive phenomenon resulting from a self-affine random fractal process. It is shown how a complete spatial description of the cracking geometry can be found from experimental data using both a (Hurst) scaling exponent and a diffusion-type coefficient. Once determined experimentally, these parameters are used to synthesize cracking patterns using fractional Brownian motion functions. In addition, it is shown how an effective Fokker-Planck diffusion equation can be used to describe the spatial geometry of the cracking.