Fokker-Planck Operator Research Articles

The Fokker–Planck–Boltzmann equation in low regularity space

We prove the global existence and the exponential time decay rates of the mild solutions to the Fokker–Planck–Boltzmann equation near Maxwellians if initial data satisfy some smallness in the function space Lk1LT∞Lv2. Compared with previous results, there are two features in this manuscript. The first is that we don’t need the velocity weight estimates in the work (Duan et al., 2021) with the help of the Fokker–Planck operator. The second is that we only need the low regularity space Lk1LT∞Lv2 rather than Hx,v4 in the work (Xiong et al., 2014). Our analysis is based on Lk1LT∞Lv2 function space introduced by the work of Duan et al. (2021) and the coercivity property of Fokker–Planck operator.

Analytic Solution of an Active Brownian Particle in a Harmonic Well.

We provide an analytical solution for the time-dependent Fokker-Planck equation for a two-dimensional active Brownian particle trapped in an isotropic harmonic potential. Using the passive Brownian particle as basis states we show that the Fokker-Planck operator becomes lower diagonal, implying that the eigenvalues are unaffected by the activity. The propagator is then expressed as a combination of the equilibrium eigenstates with weights obeying exact iterative relations. We show that for the low-order correlation functions, such as the positional autocorrelation function, the recursion terminates at finite order in the Péclet number, allowing us to generate exact compact expressions and derive the velocity autocorrelation function and the time-dependent diffusion coefficient. The nonmonotonic behavior of latter quantities serves as a fingerprint of the nonequilibrium dynamics.

An alternative expression for ad hoc field collision model

An alternative form of ad hoc collision model to the Fokker–Planck field collision operator is proposed. The model is so constructed that the results of exact Fokker–Planck field collision, applied to the low-order Legendre and associated Laguerre polynomials, are used to impose self-adjointness first and then collisional conservation laws. The model is shown to be actually identical to the conventional ad hoc model in the l = 0 and 1 parts in Legendre polynomial expansion in an isothermal case, while it is not limited to extend to the higher order, in contrast to the conventional ad hoc model, in which the collisional conservation laws are imposed primarily, using the exact test collision form of the Fokker–Planck operator. Furthermore, it is discussed that the proposed model is identified to the lowest order in an expansion of formal Hirshman–Sigmar operator, while there is a difference between the formal and conventional Hirshman–Sigmar operators. Numerical examples demonstrate that the proposed model can yield comparable results to those by the exact Fokker–Planck and conventional Hirshman–Sigmar operators.

Neural-network based collision operators for the Boltzmann equation

Kinetic gas dynamics in rarefied and moderate-density regimes have complex behavior associated with collisional processes. These processes are generally defined by convolution integrals over a high-dimensional space (as in the Boltzmann operator), or require evaluating complex auxiliary variables (as in Rosenbluth potentials in Fokker-Planck operators) that are challenging to implement and computationally expensive to evaluate. In this work, we develop a data-driven neural network model that augments a simple and inexpensive BGK collision operator with a machine-learned correction term, which improves the fidelity of the simple operator with a small overhead to overall runtime. The composite collision operator has a tunable fidelity and, in this work, is trained using and tested against a direct-simulation Monte-Carlo (DSMC) collision operator.

Study of the Kramers–Fokker–Planck quadratic operator with a constant magnetic field

We study the quadratic Kramers–Fokker–Planck operator with a constant magnetic field and a quadratic potential. We describe the exact expression of the norm of the semi-group associated with the operator near the equilibrium. At this level, explicit and accurate estimates of this norm are shown in small and long times and uniform-in-time estimates are shown when the magnetic parameter b tends to infinity.

Perturbative expansion of irreversible work in symmetric and asymmetric processes.

The systematic expansion method of the solution of the Fokker-Planck equationis developed by generalizing the formulation proposed in [J. Phys. A: Math. Theor. 50, 325001 (2017)10.1088/1751-8121/aa7af4]. Using this method, we obtain an alternative formula to calculate the mean work perturbatively which is applicable to systems with degeneracy in the eigenvalues of the Fokker-Planck operator. This method enables us to study how the geometrical symmetry affects thermodynamic description of a Brownian particle. To illustrate the application of the derived theory, we consider the Fokker-Planck equationwith a two-dimensional harmonic potential. To investigate the effect of symmetry of the potential, we study thermodynamic properties in symmetric and asymmetric deformation processes of the potential: the rotational symmetry of the harmonic potential is held in the former, but it is broken in the latter. Optimized deformations in these processes are defined by minimizing mean work. Comparing these optimized processes, we find that the difference between the symmetric and asymmetric processes is maximized when the deformation time of the potential is given by a critical time which is characterized by the relaxation time of the Fokker-Planck equation. This critical time in the mean work is smaller than that of the change of the mean energy because of the hysteresis effect in the irreversible processes.

Nonlinear Fokker-Planck collision operator in Rosenbluth form for gyrokinetic simulations using discontinuous Galerkin method

A gyroaveraged nonlinear collision operator is formulated based on the Fokker-Planck operator in the Rosenbluth-MacDonald-Judd (RMJ) potential form and implemented for the gyrokinetic simulations with the discontinuous Galerkin scheme. The divergence structure of the original RMJ form is carefully preserved throughout the formulation to guarantee the density conservation while neglecting the finite Larmor radius effect. The B-spline finite element method is used to calculate the Rosenbluth potentials for the nonlinear collision operator. In addition to the nonlinear collision operator, linear and Dougherty collision models are also implemented to assess the benefits and drawbacks of each model. For the conservation of the parallel momentum and energy, we adopt a simple advection-diffusion model which numerically enforces the conservation of physical quantities. From bump-on-tail relaxation tests, the monotonically increasing entropy in time and conservation properties are demonstrated for the developed collision operator. Also, a few theoretical predictions for the neoclassical physics such as the neoclassical heat flux, poloidal flow and collisional damping of zonal flow are successfully reproduced by numerical simulations.

Non-Hermitian adiabatic perturbation theory of topological quantization of the average velocity of a magnetic skyrmion under thermal fluctuations

We study the two-dimensional motion of a magnetic skyrmion driven by a ratchetlike polarized electric current that is periodic in both space and time. Some general cases are considered in each of which in the low-temperature and adiabatic limit, regardless of the details of the driving current, the time and statistical average velocity along any direction is topologically quantized as a Chern number multiplied by a basic unit. We make two approaches, one based on identifying the drift direction and the other based on the non-Hermitian adiabatic perturbation theory developed for the Fokker-Planck operator. Both approach apply in the case of periodicity along the direction of the driving current and homogeneity in the transverse direction for which the analytical result is confirmed by our numerical simulation on the constituent spins, and a convenient experiment is proposed.

Improved multispecies Dougherty collisions

The Dougherty model Fokker–Planck operator is extended to describe nonlinear full- $f$ ( f is the distribution function) collisions between multiple species in plasmas. Simple relations for cross-species primitive moments are developed which obey conservation laws, and reproduce familiar velocity and temperature relaxation rates. This treatment of multispecies Dougherty collisions, valid for arbitrary mass ratios, avoids unphysical temperatures and satisfies the $H$ -theorem (H is related to the entropy) unlike an analogous Bhatnagar–Gross–Krook operator. Formulas for both a Cartesian velocity space and a gyroaveraged operator are provided for use in Vlasov as well as long-wavelength gyrokinetic models. We present an algorithm for the discontinuous Galerkin discretization of this operator, and provide results from relaxation and Landau damping benchmarks.

Tensor-Based Parameter Reduction of Dynamic Load Models With Variable Frequency Drive

Accurate load modeling is critical for credible power system stability analysis. Because of the increasing complexity in modern system load, establishing a more comprehensive load model and performing accurate parameter estimation are the two biggest challenges in composite load modeling. In this paper, an induction motor with variable frequency drive (IM-VFD) together with the traditional ZIP+IM model is utilized to describe the dynamic characteristics of the system load. We estimate the time-varying probabilistic distributions of all states and parameters to describe their stochastic characteristics, thereby representing the real-time varying features of an actual load. The Fokker-Planck operator is applied to linearize the updated load model in tensor format. We then use tensor decomposition to reduce the computational and storage costs for calculating load model parameters. A sensitivity indicator is used to analyze the sensitivities of the estimated parameter distributions for subsequent model reduction. Simulations demonstrate that the updated load model has better performance than the traditional ZIP+IM model.

Effective Hamiltonians and Lagrangians for Conditioned Markov Processes at Large Volume

When analysing statistical systems or stochastic processes, it is often interesting to ask how they behave given that some observable takes some prescribed value. This conditioning problem is well understood within the linear operator formalism based on rate matrices or Fokker-Planck operators, which describes the dynamics of many independent random walkers. Relying on certain spectral properties of the biased linear operators, guaranteed by the Perron-Frobenius theorem, an effective process can be found such that its path probability is equivalent to the conditional path probability. In this paper, we extend those results for nonlinear Markov processes that appear when the many random walkers are no longer independent, and which can be described naturally through a Lagrangian-Hamiltonian formalism within the theory of large deviations at large volume. We identify the appropriate spectral problem as being a Hamilton-Jacobi equation for a biased Hamiltonian, for which we conjecture that two special global solutions exist, replacing the Perron-Frobenius theorem concerning the positivity of the dominant eigenvector. We then devise a rectification procedure based on a canonical gauge transformation of the biased Hamiltonian, yielding an effective dynamics in agreement with the original conditioning. Along the way, we present simple examples in support of our conjecture, we examine its consequences on important physical objects such as the fluctuation symmetries of the biased and rectified processes as well as the dual dynamics obtained through time-reversal. We apply all those results to simple independent and interacting models, including a stochastic chemical reaction network and a population process called the Brownian Donkey.

Hybrid stochastic differential systems in pharmacokinetics

AbstractModelling patient medication poor compliance combined with the inherent uncertainty in a bioavailability trial leads recognizably to the fact that plasma concentration-time curve must be treated as a random process. We suggest for modelling a general class of hybrid stochastic differential systems (HSDS). Roughly speaking, a HSDS is a piecewise diffusion process with jumps of two types: spontaneous and predictable. The essential tasks for us afterwards are to identify the model ingredients and to derive its bearings; of interest are a couple of operators associated to every Markov process: the generating operator and the Fokker–Planck operator. The model induces a Fokker–Planck–Kolmogorov equation along with moment equations, and computations based on direct solutions of the latter make it possible to study the variability of the concentration around its mean as compared to the full compliance case and to assess the effect of some parameters such as the intake and elimination rates.

Consistency and convergence for a family of finite volume discretizations of the Fokker–Planck operator

We introduce a family of various finite volume discretization schemes for the Fokker–Planck operator, which are characterized by different Stolarsky weight functions on the edges. This family particularly includes the well-established Scharfetter–Gummel discretization as well as the recently developed square-root approximation (SQRA) scheme. We motivate this family of discretizations both from the numerical and the modeling point of view and provide a uniform consistency and error analysis. Our main results state that the convergence order primarily depends on the quality of the mesh and in second place on the choice of the Stolarsky weights. We show that the Scharfetter–Gummel scheme has the analytically best convergence properties but also that there exists a whole branch of Stolarsky means with the same convergence quality. We show by numerical experiments that for small convection the choice of the optimal representative of the discretization family is highly non-trivial, while for large gradients the Scharfetter–Gummel scheme stands out compared to the others.

Sensitivity analysis of the non-linear Fokker–Planck equations with uncertainty

In this paper, we study sensitivity analysis of the non-linear Fokker–Planck equations with uncertainty. Modeled by random variable, the uncertainty enters the solution through initial data. By deriving Lyapunov-type inequalities, under some smallness assumptions on the initial data, we have the regularity of the solution in the random space. The nonlinearity in the model brings new difficulties to the energy estimates, which is resolved by using careful weighted-in-v energy estimates suitable for Fokker–Planck operators. The analysis is important for understanding of the sensitivity of the system under random perturbations, and for the study of convergence rates of popular numerical methods for uncertainty quantification based on polynomial chaos expansions.

Kramers–Fokker–Planck operators with homogeneous potentials

In this article we establish a global subelliptic estimate for Kramers–Fokker–Planck operators with homogeneous potentials V(q) under some conditions, involving in particular the control of the eigenvalues of the Hessian matrix of the potential. Our subelliptic lower bounds are the optimal ones up to some logarithmic correction.

The Approximate and Analytic Solutions of the Time-Fractional Intermediate Diffusion Wave Equation Associated with the Fokker–Planck Operator and Applications

In this paper, the time-fractional wave equation associated with the space-fractional Fokker–Planck operator and with the time-fractional-damped term is studied. The concept of the Green function is implemented to drive the analytic solution of the three-term time-fractional equation. The explicit expressions for the Green function G3(t) of the three-term time-fractional wave equation with constant coefficients is also studied for two physical and biological models. The explicit analytic solutions, for the two studied models, are expressed in terms of the Weber, hypergeometric, exponential, and Mittag–Leffler functions. The relation to the diffusion equation is given. The asymptotic behaviors of the Mittag–Leffler function, the hypergeometric function 1F1, and the exponential functions are compared numerically. The Grünwald–Letnikov scheme is used to derive the approximate difference schemes of the Caputo time-fractional operator and the Feller–Riesz space-fractional operator. The explicit difference scheme is numerically studied, and the simulations of the approximate solutions are plotted for different values of the fractional orders.

Maximal estimates for the Fokker–Planck operator with magnetic field

We consider the Fokker–Planck operator with a strong external magnetic field. We show a maximal type estimate on this operator using a nilpotent approach on vector field polynomial operators and including the notion of representation of a Lie algebra. This estimate makes it possible to give an optimal characterization of the domain of the closure of the considered operator.

Eigenfunctions of the Perron–Frobenius operator and the finite-time Lyapunov exponents in uniformly hyperbolic area-preserving maps

The subleading eigenvalues and associated eigenfunctions of the Perron–Frobenius operator for two-dimensional area-preserving maps are numerically investigated. We closely examine the validity of the so-called Ulam method, a numerical scheme believed to provide eigenvalues and eigenfunctions of the Perron–Frobenius operator, both for linear and nonlinear maps on the torus. For the nonlinear case, the second-largest eigenvalues and the associated eigenfunctions of the Perron–Frobenius operator are investigated by calculating the Fokker–Planck operator with sufficiently small diffusivity. On the basis of numerical schemes thus established, we find that eigenfunctions for the subleading eigenvalues exhibit spatially inhomogeneous patterns, especially showing localization around the region where unstable manifolds are sparsely running. Finally, such spatial patterns of the eigenfunction are shown to be very close to the distribution of the maximal finite-time Lyapunov exponents.

Resetting with stochastic return through linear confining potential

We consider motion of an overdamped Brownian particle subject to stochastic resetting in one dimension. In contrast to the usual setting where the particle is instantaneously reset to a preferred location (say, the origin), here we consider a finite time resetting process facilitated by an external linear potential V(x) = λ|x|(λ > 0). When resetting occurs, the trap is switched on and the particle experiences a force −∂ x V(x) which helps the particle to return to the resetting location. The trap is switched off as soon as the particle makes a first passage to the origin. Subsequently, the particle resumes its free diffusion motion and the process keeps repeating. In this set-up, the system attains a non-equilibrium steady state. We study the relaxation to this steady state by analytically computing the position distribution of the particle at all time and then analyzing this distribution using the spectral properties of the corresponding Fokker–Planck operator. As seen for the instantaneous resetting problem, we observe a ‘cone spreading’ relaxation with travelling fronts such that there is an inner core region around the resetting point that reaches the steady state, while the region outside the core still grows ballistically with time. In addition to the unusual relaxation phenomena, we compute the large deviation functions (LDFs) associated to the corresponding probability density and find that the LDFs describe a dynamical transition similar to what is seen previously in case of instantaneous resetting. Notably, our method, based on spectral properties, complements the existing renewal formalism and reveals the intricate mathematical structure responsible for such relaxation phenomena. We verify our analytical results against extensive numerical simulations.

Large deviations of the Lyapunov exponent in 2D matrix Langevin dynamics with applications to one-dimensional Anderson localization models

For the 2D matrix Langevin dynamics that correspond to the continuous-time limit of the products of some 2 × 2 random matrices, the finite-time Lyapunov exponent can be written as an additive functional of the associated Riccati process submitted to some Langevin dynamics on the infinite periodic ring. Its large deviations properties can thus be analyzed from two points of view that are equivalent in the end by consistency but give different perspectives. In the first approach, one starts from the large deviations at level 2.5 for the joint probability of the empirical density and empirical current of the Riccati process, and one then performs the appropriate Euler–Lagrange optimization in order to compute the cumulant generating function of the Lyapunov exponent. In the second approach, this cumulant generating function is obtained from the spectral analysis of the appropriate tilted Fokker–Planck operator. The associated conditioned process obtained via the generalization of Doob’s h-transform allows one to clarify the equivalence with the first approach. Finally, applications to one-dimensional Anderson localization models are described in order to explicitly obtain the first cumulants of the finite-size Lyapunov exponent.