Fokker-Planck Model Research Articles

Nonlinear inhomogeneous Fokker–Planck models: Energetic-variational structures and long-time behavior

Inspired by the modeling of grain growth in polycrystalline materials, we consider a nonlinear Fokker–Plank model, with inhomogeneous diffusion and with variable mobility parameters. We develop large time asymptotic analysis of such nonstandard models by reformulating and extending the classical entropy method, under the assumption of periodic boundary condition. In addition, illustrative numerical tests are presented to highlight the essential points of the current analytical results and to motivate future analysis.

Monte Carlo stochastic Galerkin methods for non-Maxwellian kinetic models of multiagent systems with uncertainties

In this paper, we focus on the construction of a hybrid scheme for the approximation of non-Maxwellian kinetic models with uncertainties. In the context of multiagent systems, the introduction of a kernel at the kinetic level is useful to avoid unphysical interactions. The methods here proposed, combine a direct simulation Monte Carlo (DSMC) in the phase space together with stochastic Galerkin (sG) methods in the random space. The developed schemes preserve the main physical properties of the solution together with accuracy in the random space. The consistency of the methods is tested with respect to surrogate Fokker–Planck models that can be obtained in the quasi-invariant regime of parameters. Several applications of the schemes to non-Maxwellian models of multiagent systems are reported.

Active Precipitation of Radiation Belt Electrons Using Rocket Exhaust Driven Amplification (REDA) of Man-Made Whistlers.

Ground‐based very low frequency (VLF) transmitters located around the world generate signals that leak through the bottom side of the ionosphere in the form of whistler mode waves. Wave and particle measurements on satellites have observed that these man‐made VLF waves can be strong enough to scatter trapped energetic electrons into low pitch angle orbits, causing loss by absorption in the lower atmosphere. This precipitation loss process is greatly enhanced by intentional amplification of the whistler waves using a newly discovered process called rocket exhaust driven amplification (REDA). Satellite measurements of REDA have shown between 30 and 50 dB intensification of VLF waves in space using a 60 s burn of the 150 g/s thruster on the Cygnus satellite that services the International Space Station. This controlled amplification process is adequate to deplete the energetic particle population on the affected field lines in a few minutes rather than the multi‐day period it would take naturally. Numerical simulations of the pitch angle diffusion for radiation belt particles use the UCLA quasi‐linear Fokker Planck model to assess the impact of REDA on radiation belt remediation of newly injected energetic electrons. The simulated precipitation fluxes of energetic electrons are applied to models of D‐region electron density and bremsstrahlung X‐rays for predictions of the modified environment that can be observed with satellite and ground‐based sensors.

Second-order analysis of Fokker–Planck ensemble optimal control problems

Ensemble optimal control problems governed by a Fokker-Planck equation with space-time dependent controls are investigated. These problems require the minimisation of objective functionals of probability type and aim at determining robust control mechanisms for the ensemble of trajectories of the stochastic system defining the Fokker-Planck model. In this work, existence of optimal controls is proved and a detailed analysis of their characterization by first- and second-order optimality conditions is presented. For this purpose, the well-posedness of the Fokker-Planck equation, and new estimates concerning an inhomogeneous Fokker-Planck model are discussed, which are essential to prove the necessary regularity and compactness of the control-to-state ma p appearing in the first-and second-order analysis.

Mesoscopic Boltzmann model equations for thermally perfect gases

We propose a way to derive Bhatnagar–Gross–Krook and Fokker–Planck models of the Boltzmann equation for rarefied flows of thermally perfect gases. These models can allow for various internal energies (rotation, vibration, electronic), which is required for high temperature flows, like in atmospheric reentry problems. However, our models do not contain any internal degrees of freedom; instead, they are accounted for by a mesoscopic approach. The molecular velocity is the only kinetic variable in the models that makes their computational complexity similar to that of simple monatomic gases. Moreover, we prove that these models satisfy conservation and entropy properties (H-theorem), and we derive their corresponding compressible Euler and Navier–Stokes asymptotics gases.

Physics-informed deep learning characterizes morphodynamics of Asian soybean rust disease

Medicines and agricultural biocides are often discovered using large phenotypic screens across hundreds of compounds, where visible effects of whole organisms are compared to gauge efficacy and possible modes of action. However, such analysis is often limited to human-defined and static features. Here, we introduce a novel framework that can characterize shape changes (morphodynamics) for cell-drug interactions directly from images, and use it to interpret perturbed development of Phakopsora pachyrhizi, the Asian soybean rust crop pathogen. We describe population development over a 2D space of shapes (morphospace) using two models with condition-dependent parameters: a top-down Fokker-Planck model of diffusive development over Waddington-type landscapes, and a bottom-up model of tip growth. We discover a variety of landscapes, describing phenotype transitions during growth, and identify possible perturbations in the tip growth machinery that cause this variation. This demonstrates a widely-applicable integration of unsupervised learning and biophysical modeling.

Estimate of pre-thermal quench non-thermal electron density profile during Ar pellet shutdowns of low-density target plasmas in DIII-D

The radial density profile of pre-thermal quench (pre-TQ) early-time non-thermal (hot) electrons is estimated by combining electron cyclotron emission and soft x-ray data during the rapid shutdown of low-density (ne≲1019 m−3) DIII-D target plasmas with cryogenic argon pellet injection. This technique is mostly limited in these experiments to the pre-TQ phase and quickly loses validity during the TQ. Two different cases are studied: a high (10 keV) temperature target and a low (4 keV) temperature target. The results indicate that early-time, low-energy (∼10 keV) hot electrons form ahead of the argon pellet as it enters the plasma, affecting the pellet ablation rate; it is hypothesized that this may be caused by rapid cross field transport of argon ions ahead of the pellet or by rapid cross field transport of hot electrons. Fokker–Planck modeling of the two shots suggests that the hot electron current is quite significant during the pre-TQ phase (up to 50% of the total current). Comparison between modeled pre-TQ hot electron current and post-TQ hot electron current inferred from avalanche theory suggests that hot electron current increases during the high-temperature target TQ but decreases during the low-temperature target TQ. The uncertainties in this estimate are large; however, if true, this suggests that TQ radial loss of hot electron current could be larger than previously estimated in DIII-D.

Gaussian Quantum Markov Semigroups on a One-Mode Fock Space: Irreducibility and Normal Invariant States

We consider the most general Gaussian quantum Markov semigroup on a one-mode Fock space, discuss its construction from the generalized GKSL representation of the generator. We prove the known explicit formula on Weyl operators, characterize irreducibility and its equivalence to a Hörmander type condition on commutators and establish necessary and sufficient conditions for existence and uniqueness of normal invariant states. We illustrate these results by applications to the open quantum oscillator and the quantum Fokker-Planck model.

An Eulerian Vlasov-Fokker–Planck algorithm for spherical implosion simulations of inertial confinement fusion capsules

We present a numerical algorithm that enables a phase-space adaptive Eulerian Vlasov–Fokker–Planck (VFP) simulation of inertial confinement fusion (ICF) capsule implosions. The approach relies on extending a recent mass, momentum, and energy conserving phase-space moving-mesh adaptivity strategy to spherical geometry. In configuration space, we employ a mesh motion partial differential equation (MMPDE) strategy while, in velocity space, the mesh is expanded/contracted and shifted with the plasma’s evolving temperature and drift velocity. The mesh motion is dealt with by transforming the underlying VFP equations into a computational (logical) coordinate, with the resulting inertial terms carefully discretized to ensure conservation. To deal with the spatial and temporally varying dynamics in a spherically imploding system, we have developed a novel nonlinear stabilization strategy for MMPDE in the configuration space. The strategy relies on a nonlinear optimization procedure that optimizes between mesh quality and the volumetric rate change of the mesh to ensure both accuracy and stability of the solution. Implosions of ICF capsules are driven by several boundary conditions: (1) an elastic moving wall boundary; (2) a time-dependent Maxwellian Dirichlet boundary; and (3) a pressure-driven Lagrangian boundary. Of these, the pressure-driven Lagrangian boundary driver is new to our knowledge. The implementation of our strategy is verified through a set of test problems, including the Guderley and Van-Dyke implosion problems — the first-ever reported using a Vlasov–Fokker–Planck model.

Analytic insights into nonlocal energy transport. III. steady state Fokker Planck theory in spherical and planar geometry

This paper develops an approximate steady state Fokker Planck theory for energetic electron transport in a spherical laser fusion target. First, we apply the theory to only a small population of electrons, which is specified at the outset. Then, one determines the nonlocal electron energy flux of these electrons at a particular position. These energetic electrons may either come from the tail of the thermal distribution function or be generated by an instability. This paper develops two approximate methods of solution, which we call the “characteristic method” and “sparse eigenfunction.” The former works only in planar geometry and the latter in both planar and spherical geometry. Comparison of the two methods in planar geometry shows that even though the approximations are very different, they give about the same result, increasing their credibility. For the example we have chosen, spherical effects are not important for electrons from the tail of the distribution function but may well be for instability generated electrons, which have much higher energy. Comparing planar to spherical, one finds an additional spherical barrier protecting the fuel. It turns out that the associated fuel preheat a Fokker Planck model predicts is considerably less than that predicted by the Krook models as developed at both NRL and other places.

Scattering by whistler-mode waves during a quiet period perturbed by substorm activity

We study the dynamics of radiation belt electrons during a 10-day quiet period perturbed by substorm activity and preceding a high-speed stream (HSS), aiming at a global description of the radiation belts in L-shell, L in [2, 6], and energy [0.1, 10] MeV. We combine Van Allen Probes observations and Fokker-Planck numerical simulations of pitch-angle diffusion. The Fokker-Planck model uses event-driven pitch angle diffusion coefficients from whistler-mode waves, built from the wave properties and the ambient plasma density measurements from the Van Allen Probes. We first find this event has some similar characteristics to regular quiet times previously studied; a widely extended plasmasphere within which we observe strong and varying whistler-mode waves. These ambient conditions lead to strong pitch-angle scattering, which contributes to the creation of a wide slot region as well as a significant decay of the outer radiation belts, which are observed and qualitatively well simulated. In addition, we find the substorm activity causes short duration (within ± 4h) decay of the plasma density and a lowering amplitude of the whistler-mode waves within the plasmasphere, both causing opposite effects in terms of pitch angle diffusion. This leads to a diminution of pitch-angle diffusion at the time of the main substorm activity. Conversely, whistler-mode waves become enhanced in the time periods between the substorm injections. All effects cumulated, we find an enhancement of pitch angle diffusion by whistler-mode waves above L~4.7 during the 10-day period. This directly relates to the combination of quietness and substorm activity which allows pitch angle diffusing of up to 1 MeV electrons in the outer belt. Relativistic electrons of 1–2 MeV remain trapped in the outer belt, from L~4.7 to L~5.2, forming, in both the observations and the simulations, a distinct pocket of remnant electrons.

Study of suprathermal electron dynamics during electron cyclotron current drive using hard x-ray measurements in the TCV tokamak

Suprathermal electron dynamics in real and momentum spaces plays an important role in the physics of electron cyclotron current drive (ECCD), in particular for predicting the wave driven current. In the TCV tokamak, a hard x-ray spectroscopy system equipped with digital pulse detection enables detailed studies of the creation and relaxation of suprathermal electrons. The time evolution of hard x-ray spatial profiles and energy spectra is observed utilizing a minimally perturbative ECCD modulation technique. Time-dependent Fokker-Planck modeling combined with a hard x-ray synthetic diagnostic is used to interpret experimental results and test various suprathermal electron transport models. The comparison suggests a dependency of the radial transport on the EC wave power.

Bifurcation and stability of downflowing gyrotactic micro-organism suspensions in a vertical pipe

Abstract

A novel hybrid DSMC-Fokker Planck algorithm implemented to rarefied gas flows

In this article, a new hybrid approach combining Fokker Planck (FP) and Direct Simulation Monte Carlo (DSMC) is introduced. The FP approach, which is an approximation of the Boltzmann equation, has been recently introduced for modeling rarefied gas flows. Compared to standard molecular methods like DSMC, it has a lower computational cost. However, the FP model may provide erroneous results in modeling some flow features, including vortex region and shock waves. The DSMC approach is accurate enough; yet, it has a high computational cost, especially at low Knudsen/low Mach number conditions. The purpose of this article is to suggest an optimized hybrid algorithm to benefit from high-speed modeling and sufficient accuracy simultaneously. We show a modified form of the gradient length Knudsen number, i.e., KnGL/Kn, must be considered to define the range of accuracy of the FP method. We used KnGL/Kn as a parameter for switching between DSMC and FP. The cavity flow and the nozzle flow are considered for the investigation of the proposed hybrid algorithm. The dependence of the accuracy and computational cost on KnGL/Kn is reported. The results demonstrate that the optimized KnGL/Kn is between 3 and 4, which gives suitable computational cost and acceptable accuracy.

How whistler mode hiss waves and the plasmasphere drive the quiet decay of radiation belts electrons following a geomagnetic storm

We show how an extended period of quiet solar wind conditions contributes to a quiet state of the plasmasphere that expands up to L ∼ 5.5, which creates the perfect conditions for wave-particle interactions between the radiation belt electrons and whistler-mode hiss waves. The correlation between the hiss waves and the plasma density is direct with hiss wave power increasing with plasma density, while it was generally assumed that these quantities can be specified independently. Whistler-mode hiss waves pitch angle diffuse and ultimately scatter freshly injected electrons into the atmosphere until the slot region is formed between the inner and outer belt and the outer belt is drastically reduced. In this study, we use and combine Van Allen Probes observations and Fokker-Planck numerical simulations. The Fokker-Planck model uses consistent event-driven pitch angle diffusion coefficients from whistler-mode hiss waves. Observations and simulations allow us to reach a global understanding of the variations in the trapped electron population with time, space, energy, and pitch angle that is based on the existing theory of quasi-linear wave-particle interactions. We show, for instance, the outer belt is pitch-angle homogeneous, which is explained by the event-driven diffusion coefficients that are roughly constant for equatorial pitch angle α 0∼<60°, E>100 keV, 3.5<L<Lpp∼6. The impact of this work is to bring an improved understanding of the belt evolution based on the integration of high quality and highly temporally and spatially resolved measurements that are integrated in modern computations. We also propose the event-driven method as an accurate method (within ×2) to predict the electron flux decay after storms.

A kinetic Fokker–Planck approach for modeling variable hard-sphere gas mixtures

Kinetic Fokker–Planck (FP) methods for modeling rarefied gas flows have received increasing attention over the last few years. However, formulating such models for realistic multi-species gases is still an open subject of research. Therefore, in this letter, we develop a kinetic FP model for describing gas mixtures with particles interacting according to the variable hard-sphere interaction potential. In accordance with the kinetic FP framework, a stochastic solution algorithm is employed in order to solve the model on a particle level. Different test cases are carried out, and the performance of the proposed method is compared with the direct simulation Monte Carlo algorithm.

Fokker–Planck model for binary mixtures

Abstract

Applying a Statistical Model to the Observed Texture Evolution of Fatigued Metal Films

Reliably modeling fatigue induced degradation of metal films requires a consistent mathematical description of the physically relevant damage driving forces. The processes, which trigger deformation, damage and eventually failure, have to be included and combined in a physically meaningful way to obtain a valid model. Therefore, it is essential to consider the material response to applied stress, as the microstructural changes are responsible for the initialization of the degradation process. In this work, the material response of a polycrystalline copper film under repetitively applied thermo-mechanical stress is investigated by means of a test structure. The changes in the microstructure are studied by means of electron backscatter diffraction data, where the focus is laid on the grain boundary character distribution - a distribution of relative lengths of grain boundaries and corresponding misorientations. It is investigated whether the data can be described by a Fokker-Planck model that accounts for the system’s energy. Parameter estimation is performed on basis of a Maximum Likelihood approach, which is supplemented by an adjoint method to efficiently compute the gradient of the loss function with respect to the parameters. Additionally, uncertainty quantification of the estimates is required, which is done by computing the credible sets of the parameters. To this end, the results of local approximations, Markov Chain Monte Carlo simulations and profile likelihoods are compared.

Master equation approach for modeling diatomic gas flows with a kinetic Fokker-Planck algorithm

In recent years the kinetic Fokker-Planck approach for modeling gas flows has become increasingly popular. In the Fokker-Planck ansatz the collision integral of the Boltzmann equation is approximated by a Fokker-Planck operator in velocity space. Instead of solving the resulting Fokker-Planck equation directly, the underlying random process is modeled, which leads to an efficient stochastic solution algorithm. Despite the attention to the Fokker-Planck ansatz, the modeling of polyatomic gases has been addressed only in a few works. In this paper a scheme is presented to extend arbitrary monatomic Fokker-Planck models to model polyatomic species. A master equation approach is used to model internal energy relaxation, but instead of solving the master equation directly, the underlying random process is simulated. Three different models are suggested to describe internal particle energies as continuous scalars or as a set of discrete energy levels. The proposed models are applied on different test cases to demonstrate their accuracy. Within the bounds of expectations, a very good agreement with reference DSMC simulations is achieved.

The relativistic Vlasov–Maxwell–Fokker–Planck system in the whole space

We are concerned with the Cauchy problem for the relativistic Vlasov–Maxwell–Fokker–Planck system in the whole space. For perturbative initial data with suitable regularity and integrability, we obtain the global classical solutions and the large time decay rates. For the proof, a new structure of the linearized relativistic Fokker–Planck operator is observed so that the coercivity estimates are established, crucial for the optimal large time decay rates is the pivotal Sobolev inequalities and interpolation inequalities as well as the structure of the coupled system. Moreover, it is shown that the energy without any weights decays in time with the polynomial rates varying directly with the derivatives of the solution, and the lower order derivatives of the solution except the magnetic field are further proved to decay faster than the optimal ones which coincide with the linearized equations without electro and magnetic fields, such a new phenomenon occurs only in the case of the relativistic Fokker–Planck model.