Fokker-Planck Kinetic Equation Research Articles

Exponential Stability and Hypoelliptic Regularization for the Kinetic Fokker–Planck Equation with Confining Potential

This paper is concerned with a modified entropy method to establish the large-time convergence towards the (unique) steady state, for kinetic Fokker–Planck equations with non-quadratic confinement potentials in whole space. We extend previous approaches by analyzing Lyapunov functionals with non-constant weight matrices in the dissipation functional (a generalized Fisher information). We establish exponential convergence in a weighted H1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^1$$\\end{document}-norm with rates that become sharp in the case of quadratic potentials. In the defective case for quadratic potentials, i.e. when the drift matrix has non-trivial Jordan blocks, the weighted L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-distance between a Fokker–Planck-solution and the steady state has always a sharp decay estimate of the order O((1+t)e-tν/2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal O\\big ( (1+t)e^{-t\ u /2}\\big )$$\\end{document}, with ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document} the friction parameter. The presented method also gives new hypoelliptic regularization results for kinetic Fokker–Planck equations (from a weighted L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-space to a weighted H1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^1$$\\end{document}-space).

Numerical approximation of kinetic Fokker–Planck equations with specular reflection boundary conditions

This work is devoted to the analysis of a numerical approximation to a general multi-dimensional kinetic Fokker–Planck (FP) equation with reaction and source terms and subject to specular reflection boundary conditions. This numerical approximation is based on splitting the kinetic FP model into a transport equation in space and a FP diffusive model in the velocity coordinates. The former is discretized by a Kurganov-Tadmor finite-volume scheme, while the latter is approximated by a generalized Chang & Cooper finite-volume method. Time integration is performed by a strong stability-preserving Runge-Kutta method where the reaction and source terms are accommodated with a Strang splitting technique and the use of a Magnus integrator. It is proved that the resulting numerical solution method is conservative and positive preserving, in the case where the continuous model has these properties, and it is second-order accurate in time and in phase space in the L1-norm, subject to a CFL condition. Results of numerical experiments are reported that validate these theoretical results.

Fractional diffusion for Fokker–Planck equation with heavy tail equilibrium: An à la Koch spectral method in any dimension

In this paper, we extend the spectral method developed (Dechicha and Puel (2023)) to any dimension d ⩾ 1, in order to construct an eigen-solution for the Fokker–Planck operator with heavy tail equilibria, of the form ( 1 + | v | 2 ) − β 2 , in the range β ∈ ] d , d + 4 [. The method developed in dimension 1 was inspired by the work of H. Koch on nonlinear KdV equation (Nonlinearity 28 (2015) 545). The strategy in this paper is the same as in dimension 1 but the tools are different, since dimension 1 was based on ODE methods. As a direct consequence of our construction, we obtain the fractional diffusion limit for the kinetic Fokker–Planck equation, for the correct density ρ : = ∫ R d f d v, with a fractional Laplacian κ ( − Δ ) β − d + 2 6 and a positive diffusion coefficient κ.

Improving the Convergence Rates for the Kinetic Fokker–Planck Equation by Optimal Control

Improving the Convergence Rates for the Kinetic Fokker–Planck Equation by Optimal Control

Biogenic Nanomagnetic Carriers Derived from Magnetotactic Bacteria: Magnetic Parameters of Magnetosomes Inside Magnetospirillum spp.

Magnetic parameters of magnetosomes inside the bacteria of MSR-1, LBB-42, AMB-1, SP-1, BB-1, and SO-1 strains of the genus Magnetospirillum fixed by 5% formalin in the nutrient medium were estimated by measurements of the nonlinear longitudinal response to a weak ac magnetic field (NLR-M2) at room temperature. For the BB-1, MSR-1, and AMB-1 strains, the measurements of the electron magnetic resonance (EMR) spectra with the special X-band spectrometer for wide-line registration were also carried out. To trace the evolution of the magnetic state of the magnetosomes during the long-term storage, freshly prepared samples (“new”) and samples after a year of storage at 4 °C (“old”) were studied. The assessment of the state of the bacteria ensemble in the medium after the long-term storage was carried out for one typical strain (BB-1) using atomic force microscopy. The stable single-domain state of magnetic centers in the magnetosomes indicating their proximity to a superparamagnetic (SPM) regime was found at the scan frequency 0.02 Hz of the steady magnetic field. This allowed a semi-quantitative analysis of M2 data to be carried out with the formalism based on the numerical solution of the kinetic Fokker–Planck equation for SPM particles. Processing the NLR-M2 data demonstrated the presence of two kinds of magnetosomes in both the “new” and “old” samples: (i) those with the large magnetic moment (the “heavy”, monodisperse mode) and (ii) those with the comparatively small magnetic moment (the “light”, highly dispersed mode). The EMR spectra were formed mostly by the “heavy” fraction for both samples. The presence of two peaks in the spectra evidenced the presence of conventional uniaxial magnetic anisotropy in the magnetosomes. The appearance of one or two additional peaks in the spectra in the “old” fraction of some strains implied their instability at the long-term storage, even when fixed by formalin and sealed in the nitrogen atmosphere.

Bounds and long term convergence for the voltage-conductance kinetic system arising in neuroscience

<p style='text-indent:20px;'>The voltage-conductance equation determines the probability distribution of a stochastic process describing a fluctuation-driven neuronal network arising in the visual cortex. Its structure and degeneracy share many common features with the kinetic Fokker-Planck equation, which has attracted much attention recently.</p><p style='text-indent:20px;'>We prove an <inline-formula><tex-math id="M5">\begin{document}$ L^\infty $\end{document}</tex-math></inline-formula> bound on the steady state solution and the long term convergence of the evolution problem towards this stationary state. Despite the hypoellipticity property, the difficulty is to treat the boundary conditions that change type along the boundary. This leads us to use specific weights in the Alikakos iterations and adapt the relative entropy method.</p>

Time averages for kinetic Fokker-Planck equations

We consider kinetic Fokker-Planck (or Vlasov-Fokker-Planck) equations on the torus with Maxwellian or fat tail local equilibria. Results based on weak norms have recently been achieved by S. Armstrong and J.-C. Mourrat in case of Maxwellian local equilibria. Using adapted Poincaré and Lions-type inequalities, we develop an explicit and constructive method for estimating the decay rate of time averages of norms of the solutions, which covers various regimes corresponding to subexponential, exponential and superexponential (including Maxwellian) local equilibria. As a consequence, we also derive hypocoercivity estimates, which are compared to similar results obtained by other techniques.

A Galerkin type method for kinetic Fokker-Planck equations based on Hermite expansions

A Galerkin type method for kinetic Fokker-Planck equations based on Hermite expansions

Semigroup decay for the linearized kinetic ellipsoidal Fokker-Planck equation

In this paper, we deal with the semigroup decay for the linearized kinetic ellipsoidal Fokker-Planck equation in the torus. This model is an extension of the nonlinear kinetic Fokker-Planck equation in order to give a correct Prandtl number in the Navier-Stokes limit. Due to the diffusion coefficient is replaced by a non diagonal temperature tensor, this makes the linearized operator for the nonlinear kinetic ellipsoidal Fokker-Planck equation with more complicated form. By taking advantage of the H1 type hypocoercivity techniques, we prove that the solutions converge exponential to the equilibrium with explicit rates.

Quasilinear theory of general electromagnetic fluctuations including discrete particle effects for magnetized plasmas: General analysis

The general quasilinear Fokker–Planck kinetic equation for the gyrophase-averaged plasma particle distribution functions in magnetized plasmas is derived, making no restrictions on the energy of the particles and on the frequency of the electromagnetic fluctuations and avoiding the often made Coulomb approximation of the electromagnetic interactions. The inclusion of discrete particle effects breaks the dichotomy of nonlinear kinetic plasma theory divided into the test particle and the test fluctuation approximation because it provides expression of both the non-collective and collective electromagnetic fluctuation spectra in terms of the plasma particle distribution functions. Within the validity of the quasilinear approach, the resulting full quasilinear transport equation can be regarded as a determining nonlinear equation for the time evolution of the plasma particle distribution functions.

Theoretical Studies of Nonlinear Relaxation Electrophysical Phenomena in Dielectrics with Ionic–Molecular Chemical Bonds in a Wide Range of Fields and Temperatures

This paper is devoted to the development of generalized (for a wide range of fields (100 kV/m–1000 MV/m) and temperatures (0–1500 K) in the radio frequency range (1 kHz–500 MHz)) methods for the theoretical investigation of the physical mechanism of nonlinear kinetic phenomena during the establishment of the relaxation polarization, due to the diffusion motion of the main charge carriers in dielectrics with ionic–molecular chemical bonds (hydrogen-bonded crystals (HBC), including layered silicates, crystalline hydrates and corundum–zirconium ceramics (CZC), etc.) in an electric field. The influence of the nonlinearities equations of the initial phenomenological model of dielectric relaxation (in HBC-proton relaxation) on the mechanism for the formation of volume–charge polarization in solid dielectrics is analyzed. The solutions for the nonlinear kinetic Fokker–Planck equation, together with the Poisson equation, for the model of blocked electrodes are built in an infinite approximation (including all orders k of smallness without dimensional parameters) of perturbation theory for an arbitrary order r of the frequency harmonic of an alternating external polarizing field. It has been established that the polarization nonlinearities in ion-molecular dielectrics, already detected at the fundamental frequency, are interpreted in the mathematical model (for the first time in this work) as interactions of the relaxation modes of the volume charge density calculated on different orders of spatial Fourier harmonics. At the fundamental frequency of the field, an analytical generalized expression is written for complex dielectric permittivity (CDP), which is expressed analytically in terms of special relaxation parameters, which are quite complex real functions in the fields of frequency and temperature. The theoretical CDP and the dielectric loss tangent spectra studied depend on the nature of the relaxation processes in the selected temperature range (Maxwell and diffusion relaxation; thermally activated and tunneling relaxation), which is relevant from the point of view of choosing exact calculation formulas when analyzing the optimal operating modes of functional elements (based on dielectrics and their composites) for circuits of instrumentation, radio engineering and power equipment in real industrial production.

Global $${L}_{p}$$ Estimates for Kinetic Kolmogorov–Fokker–Planck Equations in Nondivergence Form

We study the degenerate Kolmogorov equations (also known as kinetic Fokker-Planck equations) in nondivergence form. The leading coefficients $a^{ij}$ are merely measurable in $t$ and satisfy the vanishing mean oscillation (VMO) condition in $x, v$ with respect to some quasi-metric. We also assume boundedness and uniform nondegeneracy of $a^{ij}$ with respect to $v$. We prove global a priori estimates in weighted mixed-norm Lebesgue spaces and solvability results. We also show an application of the main result to the Landau equation. Our proof does not rely on any kernel estimates.

LOG-TRANSFORM AND THE WEAK HARNACK INEQUALITY FOR KINETIC FOKKER-PLANCK EQUATIONS

AbstractThis article deals with kinetic Fokker–Planck equations with essentially bounded coefficients. A weak Harnack inequality for nonnegative super-solutions is derived by considering their log-transform and adapting an argument due to S. N. Kružkov (1963). Such a result rests on a new weak Poincaré inequality sharing similarities with the one introduced by W. Wang and L. Zhang in a series of works about ultraparabolic equations (2009, 2011, 2017). This functional inequality is combined with a classical covering argument recently adapted by L. Silvestre and the second author (2020) to kinetic equations.

Early-stage Coronal Hard X-Ray Source in Solar Flares in the Collapsing Trap Model

A bright hard X-ray coronal source observed at the early stage of solar flares is considered. The plasma density in a quiet corona is not enough to explain the hard X-ray bremsstrahlung radiation. The generally accepted concept of increasing plasma density in the looptop is associated with the effect of evaporation of hot chromosphere plasma. We discuss the increase in plasma density at the looptop at the early stage of a flare, due to magnetic loop contraction during the relaxation of the magnetic field (the so-called collapsing trap model). In this case, the increase in the plasma density at the looptop occurs on a timescale of seconds–tens of seconds, while the process of plasma evaporation increases the plasma density for much longer. The Fokker–Planck kinetic equation for accelerated electrons with a betatron and Fermi terms is solved numerically. We calculate increases in the energy of the accelerated electrons, the energy spectrum, and the pitch-angle anisotropy due to betatron and Fermi first-order acceleration. For a collapse time of 8 s, the total energy of the accelerated electrons increases by ∼20%–200%, depending on the model parameters. The ratio of the looptop/total hard X-ray flux at 29–58 keV increases by 15%–30% in the collapsing trap model. It is shown that this model can explain the appearance of bright coronal hard X-ray sources in the first seconds–tens of seconds after the hard X-ray flux starts growing.

A probabilistic study of the kinetic Fokker–Planck equation in cylindrical domains

We consider classical solutions to the kinetic Fokker–Planck equation on a bounded domain \({\mathcal {O}} \subset ~{\mathbb {R}}^d\) in position, and we obtain a probabilistic representation of the solutions using the Langevin diffusion process with absorbing boundary conditions on the boundary of the phase-space cylindrical domain \(D = {\mathcal {O}} \times {\mathbb {R}}^d\). Furthermore, a Harnack inequality, as well as a maximum principle, are provided on D for solutions to this kinetic Fokker–Planck equation, together with the existence of a smooth transition density for the associated absorbed Langevin process. This transition density is shown to satisfy an explicit Gaussian upper-bound. Finally, the continuity and positivity of this transition density at the boundary of D are also studied. All these results are in particular crucial to study the behavior of the Langevin diffusion process when it is trapped in a metastable state defined in terms of positions.

A self-consistent hybrid model of kinetic striations in low-current argon discharges

A self-consistent hybrid model of standing and moving striations was developed for low-current DC discharges in noble gases. We introduced the concept of surface diffusion in phase space (r, u) (where u denotes the electron kinetic energy) described by a tensor diffusion in the nonlocal Fokker–Planck kinetic equation for electrons in the collisional plasma. Electrons diffuse along surfaces of constant total energy ɛ = u − eφ(r) between energy jumps in inelastic collisions with atoms. Numerical solutions of the 1d1u kinetic equation for electrons were obtained by two methods and coupled to ion transport and Poisson solver. We studied the dynamics of striation formation in Townsend and glow discharges in argon gas at low discharge currents using a two-level excitation-ionization model and a ‘full-chemistry’ model, which includes stepwise and Penning ionization. Standing striations appeared in Townsend and glow discharges at low currents, and moving striations were obtained for the discharge currents exceeding a critical value. These waves originate at the anode and propagate towards the cathode. We have seen two types of moving striations with the two-level and full-chemistry models, which resemble the s and p striations previously observed in the experiments. Simulations indicate that processes in the anode region could control moving striations in the positive column plasma. The developed model helps clarify the nature of standing and moving striations in DC discharges of noble gases at low discharge currents and low gas pressures.

Stable and Efficient Petrov--Galerkin Methods for a Kinetic Fokker--Planck Equation

We propose a stable Petrov-Galerkin discretization of a kinetic Fokker-Planck equation constructed in such a way that uniform inf-sup stability can be inferred directly from the variational formulation. Inspired by well-posedness results for parabolic equations, we derive a lower bound for the dual inf-sup constant of the Fokker-Planck bilinear form by means of stable pairs of trial and test functions. The trial function of such a pair is constructed by applying the kinetic transport operator and the inverse velocity Laplace-Beltrami operator to a given test function. For the Petrov-Galerkin projection we choose an arbitrary discrete test space and then define the discrete trial space using the same application of transport and inverse Laplace-Beltrami operator. As a result, the spaces replicate the stable pairs of the continuous level and we obtain a well-posed numerical method with a discrete inf-sup constant identical to the inf-sup constant of the continuous problem independently of the mesh size. We show how the specific basis functions can be efficiently computed by low-dimensional elliptic problems, and confirm the practicability and performance of the method for a numerical example.

Lagrangian dual framework for conservative neural network solutions of kinetic equations

<p style='text-indent:20px;'>In this paper, we propose a novel conservative formulation for solving kinetic equations via neural networks. More precisely, we formulate the learning problem as a constrained optimization problem with constraints that represent the physical conservation laws. The constraints are relaxed toward the residual loss function by the Lagrangian duality. By imposing physical conservation properties of the solution as constraints of the learning problem, we demonstrate far more accurate approximations of the solutions in terms of errors and the conservation laws, for the kinetic Fokker-Planck equation and the homogeneous Boltzmann equation.</p>

Operator-splitting schemes for degenerate, non-local, conservative-dissipative systems

<p style='text-indent:20px;'>In this paper, we develop a natural operator-splitting variational scheme for a general class of non-local, degenerate conservative-dissipative evolutionary equations. The splitting-scheme consists of two phases: a conservative (transport) phase and a dissipative (diffusion) phase. The first phase is solved exactly using the method of characteristic and DiPerna-Lions theory while the second phase is solved approximately using a JKO-type variational scheme that minimizes an energy functional with respect to a certain Kantorovich optimal transport cost functional. In addition, we also introduce an entropic-regularisation of the scheme. We prove the convergence of both schemes to a weak solution of the evolutionary equation. We illustrate the generality of our work by providing a number of examples, including the kinetic Fokker-Planck equation and the (regularized) Vlasov-Poisson-Fokker-Planck equation.</p>

The Dirichlet problem for Kolmogorov-Fokker-Planck type equations with rough coefficients

We establish the existence and uniqueness, in bounded as well as unbounded Lipschitz type cylinders of the forms UX×VY,t and Ω×Rm×R, of weak solutions to Cauchy-Dirichlet problems for the strongly degenerate parabolic operatorL:=∇X⋅(A(X,Y,t)∇X)+X⋅∇Y−∂t, assuming that A=A(X,Y,t)={ai,j(X,Y,t)} is a real m×m-matrix valued, measurable function such that A(X,Y,t) is symmetric, bounded and uniformly elliptic. Subsequently we solve the continuous Dirichlet problem and establish the representation of the solution using associated parabolic measures. The paper is motivated, through our recent studies [11–13], by a growing need and interest to gain a deeper understanding of the Dirichlet problem for the operator L in Lipschitz type domains. The key idea underlying our results is to prove, along the lines of Brezis and Ekeland [3,4], and inspired by the recent work of S. Armstrong and J-C. Mourrat [2] concerning variational methods for the kinetic Fokker-Planck equation, that the solution can be obtained as the minimizer of a uniformly convex functional.