Vlasov Fokker Planck Equation Research Articles

Incompressible Navier–Stokes limit from nonlinear Vlasov–Fokker–Planck equation

The aim of this paper is to justify the rigorous derivation of the incompressible Navier–Stokes equations from the nonlinear Vlasov–Fokker–Planck (VFP) equation with a constant temperature. Under the incompressible Navier–Stokes scaling, we first establish the global existence of regular solutions to the rescaled nonlinear VFP equation near the Maxwellian, obtaining some uniform bound estimates. We then show the strong convergence of solution to the nonlinear VFP equation towards the incompressible Navier–Stokes system.

A physics-informed deep learning description of Knudsen layer reactivity reduction

A physics-informed neural network (PINN) is used to evaluate the fast ion distribution in the hot spot of an inertial confinement fusion target. The use of tailored input and output layers to the neural network is shown to enable a PINN to learn the parametric solution to the Vlasov–Fokker–Planck equation in the absence of any synthetic or experimental data. As an explicit demonstration of the approach, the specific problem of Knudsen layer fusion yield reduction is treated. Here, the predictions from the Vlasov–Fokker–Planck PINN are used to provide a non-perturbative solution of the fast ion tail in the vicinity of the hot spot, thus allowing the spatial profile of the fusion reactivity to be evaluated for a range of collisionalities and hot spot conditions. Excellent agreement is found between the predictions of the Vlasov–Fokker–Planck PINN and the results from traditional numerical solvers with respect to both the energy and spatial distribution of fast ions and the fusion reactivity profile, demonstrating that the Vlasov–Fokker–Planck PINN provides an accurate and efficient means of determining the impact of Knudsen layer yield reduction across a broad range of plasma conditions.

Theories derived from Haissinski equation and their applications to electron storage rings

As a stationary solution of the Vlasov–Fokker–Planck equation, the Haissinski equation predicts the equilibrium line density of a bunch that circulates in a storage ring for a given wake function. This paper shows that some equations regarding the centroid shift of the bunch, the peak position of the bunch profile, bunch length, and the impedance from the bunch profile can be derived from the Haissinski equation in a self-consistent manner. Specifically, a generalized quadratic equation for potential-well bunch lengthening is derived to accommodate any arbitrary impedance model (that is, the impedance spectrum under consideration can take any form). This expands upon Zotter’s cubic equation, which is primarily applicable to inductive impedance. The equations derived in this paper are tested using computed impedance models for some electron storage rings, showing machine-dependent properties of impedance effects. We conclude that these equations can be employed in electron storage rings to effectively bridge the gap between impedance computations and beam-based measurements.

Exponential Entropy Dissipation for Weakly Self-Consistent Vlasov–Fokker–Planck Equations

Exponential Entropy Dissipation for Weakly Self-Consistent Vlasov–Fokker–Planck Equations

An extended Vlasov–Fokker–Planck approach for kinetic simulations of laser plasmas

Vlasov–Fokker–Planck simulation codes occupy an important niche in modeling laser-produced plasmas, since they are well suited to studying the effect of collisions on electron kinetic phenomena, especially energy transport. One of the most important elements of energy transport is the absorption of laser light by the plasma; however, simulating this in detail requires resolving oscillations of the laser light, whose characteristic timescale is orders of magnitude shorter than the simulation time needed to study transport physics. For this reason, most Vlasov–Fokker–Planck codes used to study electron transport in laser plasmas rely on simplified models of the laser–plasma coupling. Their underlying assumptions nominally preclude their use for modeling laser light having short-scale structure in space or time, such as broadband lasers. In this work, we derive a more general computational framework suitable for arbitrarily structured laser fields. Our approach is based on an extended set of Vlasov–Fokker–Planck equations that separately solve for the low- and high-frequency plasma response. We implement these extended Vlasov–Fokker–Planck equations in the spherical harmonic code K2 and demonstrate the performance of the method on several laser absorption test problems, with particular attention to the judicious selection of time steps, time integrators, and spherical harmonic truncation, according to the intensity and spectrum of the laser light under consideration. Comparison with the widely used Langdon absorption operator shows the Langdon operator performs remarkably well for predicting laser heating in the simple cases considered here, even in situations that would seem to violate its underlying assumptions.

Global weak solutions to a quantum kinetic-fluid model with large initial data

In this paper, we consider a quantum kinetic-fluid model in one-dimensional torus. This model consists of the Vlasov–Fokker–Planck equation coupled with the compressible quantum Navier–Stokes equations via a nonlinear friction force depends on the density of the fluid. The quantum fluid with degenerate viscosity is assumed to be isentropic (adiabatic coefficient γ>1). We establish the global weak solutions to this model with arbitrary large initial data. This is the first result on existence of solutions to kinetic-fluid models with density-dependent friction force and degenerate viscosity.

Steady-state solutions of split beams in electron storage rings

Recently, a novel operation method for synchrotron light sources with transversely split beams has been explored to fulfill the rising demand for flexible and high-throughput X-ray sources required in such diverse fields as time-resolved X-ray spectroscopy, molecular chemistry in organic cells, high-resolution medical imaging, quantum materials science or sustainable energy research. Within that novel operation mode, additional stable regions are produced in the horizontal phase space by operating an electron storage ring on a resonance that is driven by the nonlinear sextupole or octupole magnets. In the longitudinal phase space, a similar split can be produced by introducing an oscillation of the synchrotron phase via a modulation of the phase of the radiofrequency resonator. Strong radiation damping in electron storage rings, however, has to be overcome before additional regions in phase space can become populated by particles and form stable islands. This damping mechanism changes the dynamics of the system and causes diffusion between the different islands in phase space, raising the question what kind of equilibrium state exists in the asymptotic temporal limit. In this paper, a finite-differences approximation in rotating action-angle coordinates is used to solve the Vlasov–Fokker–Planck equation and to study the obtained equilibrium states for the longitudinal as well as the transverse case. The number of solution vectors and the magnitude of the corresponding singular values of the matrix of the underlying finite-differences equation are used as abstract indicators to define the required parameter set that provides stable additional beamlets. As a consequence, the beamlets have a stability that is close to that of the main beam in terms of diffusion caused by the radiation damping and quantum excitation.

Quantified overdamped limit for kinetic Vlasov–Fokker–Planck equations with singular interaction forces

We establish a quantified overdamped limit for kinetic Vlasov–Fokker–Planck equations with nonlocal interaction forces. We provide explicit bounds on the error between solutions of that kinetic equation and the limiting equation, which is known under the names of aggregation-diffusion equation or McKean–Vlasov equation. Introducing an intermediate system via a coarse-graining map, we quantitatively estimate the error between the spatial densities of the Vlasov–Fokker–Planck equation and the intermediate system in the Wasserstein distance of order 2. We then derive an evolution-variational-like inequality for Wasserstein gradient flows which allows us to quantify the error between the intermediate system and the corresponding limiting equation. Our strategy only requires weak integrability of the interaction potentials, thus in particular it includes the quantified overdamped limit of the kinetic Vlasov–Poisson–Fokker–Planck system to the aggregation-diffusion equation with either repulsive electrostatic or attractive gravitational interactions.

The Vlasov–Fokker–Planck equation with high dimensional parametric forcing term

We consider the Vlasov–Fokker–Planck equation with random electric field where the random field is parametrized by countably many infinite random variables due to uncertainty. At the theoretical level, with suitable assumption on the anisotropy of the randomness, adopting the technique employed in elliptic PDEs (Cohen and DeVore in Acta Numerica 24:1–159, 2015) , we prove the best N approximation in the random space enjoys a convergence rate, which depends on the summability of the coefficients of the random variable, higher than the Monte-Carlo method. For the numerical method, based on the adaptive sparse polynomial interpolation (ASPI) method introduced in Chkifa et al. (Found Comput Math 14:601–603, 2014), we develop a residual based adaptive sparse polynomial interpolation (RASPI) method which is more efficient for multi-scale linear kinetic equation, when using numerical schemes that are time dependent and implicit. Numerical experiments show that the numerical error of the RASPI decays faster than the Monte-Carlo method and is also dimension independent.

Quantitative estimate of the overdamped limit for the Vlasov–Fokker–Planck systems

This note adapts a probabilistic approach to establish a quantified estimate of the overdamped limit for the Vlasov–Fokker–Planck equation towards the aggregation–diffusion equation, which in particular includes cases of the Newtonian type singular forces. The proofs are based on the investigation of the weak convergence of the corresponding stochastic differential equations (SDEs) of Mckean type in the continuous path space. We show that one can recover the same (actually stronger) overdamped limit result as in Choi and Tse (2020) under the same assumptions.

Asymptotic analysis for a Vlasov–Fokker–Planck/Navier–Stokes system in a bounded domain

We study an asymptotic analysis of a coupled system of kinetic and fluid equations. More precisely, we deal with the nonlinear Vlasov–Fokker–Planck equation coupled with the compressible isentropic Navier–Stokes system through a drag force in a bounded domain with the specular reflection boundary condition for the kinetic equation and homogeneous Dirichlet boundary condition for the fluid system. We establish a rigorous hydrodynamic limit corresponding to strong noise and local alignment force. The limiting system is a type of two-phase fluid model consisting of the isothermal Euler system and the compressible Navier–Stokes system. Our main strategy relies on the relative entropy argument based on the weak–strong uniqueness principle. For this, we provide a global-in-time existence of weak solutions for the coupled kinetic-fluid system. We also show the existence and uniqueness of strong solutions to the limiting system in a bounded domain with the kinematic boundary condition for the Euler system and Dirichlet boundary condition for the Navier–Stokes system.

On the α-Navier–Stokes–Vlasov and the α-Navier–Stokes–Vlasov–Fokker–Planck equations

We consider the α-Navier–Stokes equations coupled with a Vlasov type equation to model the flow of an incompressible fluid containing small particles. We prove the existence of global weak solutions to the coupled system subject to periodic boundary conditions. Moreover, we investigate the regularity of weak solutions and the uniqueness of regular solutions. The convergence of its solutions to that of the Navier–Stokes–Vlasov equations when α tends to zero is also established. The results are extended to the model with the diffusion of spray, i.e., to the α-Navier–Stokes–Vlasov–Fokker–Planck equations.

Global weak solutions for a Vlasov–Fokker–Planck/Navier–Stokes system with nonhomogeneous boundary data

In this paper, we consider a kinetic-fluid model with nonhomogeneous Dirichlet boundary data in a 3D bounded domain. This model consists of a Vlasov–Fokker–Planck equation coupled with the compressible Navier–Stokes equations via a friction force. We establish the global existence of weak solutions to it for the isentropic fluid (adiabatic coefficient $$\gamma > \frac{3}{2}$$ ) with large initial data, and large velocity and density at the inflow boundary.

Variational integrators for stochastic dissipative Hamiltonian systems

Abstract Variational integrators are derived for structure-preserving simulation of stochastic forced Hamiltonian systems. The derivation is based on a stochastic discrete Hamiltonian, which approximates a type-II stochastic generating function for the stochastic flow of the Hamiltonian system. The generating function is obtained by introducing an appropriate stochastic action functional and considering a stochastic generalization of the deterministic Lagrange–d’Alembert principle. Our approach presents a general methodology to derive new structure-preserving numerical schemes. The resulting integrators satisfy a discrete version of the stochastic Lagrange–d’Alembert principle, and in the presence of symmetries, they also satisfy a discrete counterpart of Noether’s theorem. Furthermore, mean-square and weak Lagrange–d’Alembert Runge–Kutta methods are proposed and tested numerically to demonstrate their superior long-time numerical stability and energy behaviour compared to nongeometric methods. The Vlasov–Fokker–Planck equation is considered as one of the numerical test cases, and a new geometric approach to collisional kinetic plasmas is presented.

Asymptotic analysis for a homogeneous bubbling regime Vlasov–Fokker–Planck/Navier–Stokes system

The evolution of a cloud of particles in a compressible fluid can be modeled with a Vlasov–Fokker–Planck equation for the distribution function of the particles coupled with Navier–Stokes or Euler equations for the density and velocity of the fluid. Formal calculations have established the convergence of solution to the mesoscopic model to solutions to the macroscopic Navier–Stokes or Euler model coupled with a Smoluchowski equation as the ratio of the settling time for the microscopic velocity fluctuation of the particles to the characteristic macroscopic time scale goes to zero. This paper provides a rigorous asymptotic analysis for a homogeneous mesoscopic fluid–particle interaction model for particles dispersed in a compressible fluid is provided for the bubbling regime. A relative entropy inequality for a mixed hyperbolic/parabolic system of equations is employed.

Network dynamics on graphops

In this article we report on a novel way to incorporate complex network structure into the analysis of interacting particle systems. More precisely, it is well-known that in well-mixed/homogeneous/all-to-all-coupled systems, one may derive mean-field limit equations such as Vlasov–Fokker–Planck equations (VFPEs). A mesoscopic VFPE describes the probability of finding a single vertex/particle in a certain state, forming a bridge between microscopic statistical physics and macroscopic fluid-type approximations. One major obstacle in this framework is to incorporate complex network structures into limiting equations. In many cases, only heuristic approximations exist, or the limits rely on particular classes of integral operators. In this paper, we notice that there is a much more elegant, and profoundly more general, way available due to recent progress in the theory of graph limits. In particular, we show how one may easily enter complex network dynamics via graphops (graph operators) into VFPEs.

Modulation of Galactic Cosmic Rays by Plasma Disturbances Propagating Through the Local Interstellar Medium in the Outer Heliosheath

Abstract The modulation of cosmic rays by a propagating plasma disturbance, a global merged interaction region (GMIR), in the heliosheath is simulated using a Vlasov–Fokker–Planck equation for the transport of energetic particles with significant anisotropy. The prescribed plasma structure of the GMIR contains a shock front and plasma rarefaction region behind the shock, which propagate through a simplified paramagnetic shielding model of the heliosheath. When a GMIR goes through the heliospheric magnetic field in the inner heliosheath, its modulation effects on cosmic rays are consistent with typical Forbush decreases. When a GMIR goes through the interstellar magnetic field in the outer heliosheath, only cosmic rays with large pitch angles with respect to the magnetic field vector (cosine values close to zero) are modulated by it. The difference is due to the very weak scattering of particles by the interstellar turbulence. Particles trapped in the rarefied magnetic field inside a GMIR suffer a significant amount of adiabatic cooling, which results in a considerable intensity decrease and a bidirectional anisotropy. The simulation result can be used to explain what Voyager 1 observed in the very local interstellar medium. Depending on the strength of plasma compression inside a GMIR, some cosmic rays may be accelerated, but the GMIR effect on the cosmic-ray intensity is much weaker than that due to adiabatic cooling because particles have only a brief interaction with a GMIR without trapping.

The Linearized Vlasov and Vlasov–Fokker–Planck Equations in a Uniform Magnetic Field

We study the linearized Vlasov equations and the linearized Vlasov-Fokker-Planck equations in the weakly collisional limit in a uniform magnetic field. In both cases, we consider periodic confinement and Maxwellian (or close to Maxwellian) backgrounds. In the collisionless case, for modes transverse to the magnetic field, we provide a precise decomposition into a countably infinite family of standing waves for each spatial mode. These are known as Bernstein modes in the physics literature, though the decomposition is not an obvious consequence of any existing arguments that we are aware of. We show that other modes undergo Landau damping. In the presence of collisions with collision frequency $\nu \ll 1$, we show that these modes undergo uniform-in-$\nu$ Landau damping and enhanced collisional relaxation at the time-scale $O(\nu^{-1/3})$. The modes transverse to the field are uniformly stable and exponentially thermalize on the time-scale $O(\nu^{-1})$. Most of the results are proved using Laplace transform analysis of the associated Volterra equations, whereas a simple case of Yan Guo's energy method for hypocoercivity of collision operators is applied for stability in the collisional case.

Propagation of chaos for the Vlasov–Poisson–Fokker–Planck equation with a polynomial cut-off

We consider a [Formula: see text]-particle system interacting through the Newtonian potential with a polynomial cut-off in the presence of noise in velocity. We rigorously prove the propagation of chaos for this interacting stochastic particle system. Taking the cut-off like [Formula: see text] with [Formula: see text] in the force, we provide a quantitative error estimate between the empirical measure associated to that [Formula: see text]-particle system and the solutions of the [Formula: see text]-dimensional Vlasov–Poisson–Fokker–Planck (VPFP) system. We also study the propagation of chaos for the Vlasov–Fokker–Planck equation with less singular interaction forces than the Newtonian one.

Vlasov–Fokker–Planck equation: stochastic stability of resonances and unstable manifold expansion

We investigate the dynamics close to a homogeneous stationary state of the Vlasov equation in one dimension, in presence of a small dissipation modeled by a Fokker–Planck operator. When the stationary state is stable, we show the stochastic stability of Landau poles. When the stationary state is unstable, depending on the relative size of the dissipation and the unstable eigenvalue, we find three distinct nonlinear regimes: for a very small dissipation, the system behaves as the pure Vlasov equation; for a strong enough dissipation, the dynamics presents similarities with a standard dissipative bifurcation; in addition, we identify an intermediate regime interpolating between the two previous ones. The nonlinear analysis relies on an unstable manifold expansion, performed using Bargmann representation for the functions and operators analyzed. The resulting series are estimated with Mellin transform techniques.