Fokker-Planck Research Articles

Structure‐preserving numerical methods for Fokker–Planck equations

AbstractA common way to numerically solve Fokker–Planck equations is the Chang–Cooper method in space combined with one of the Euler methods in time. However, the explicit Euler method is only conditionally positive, leading to severe restrictions on the time step to ensure positivity. On the other hand, the implicit Euler method is robust but nonlinearly implicit. Instead, we propose to combine the Chang–Cooper method with unconditionally positive Patankar‐type time integration methods, since they are unconditionally positive, robust for stiff problems, only linearly implicit, and also higher‐order accurate. We describe the combined approach, analyze it, and present a relevant numerical example demonstrating advantages compared to schemes proposed in the literature.

Nonnegative tensor train for the multicomponent Smoluchowski equation

Nonnegative tensor train for the multicomponent Smoluchowski equation

Exactly solvable inhomogeneous fermion systems

Abstract 15 exactly solvable inhomogeneous (spinless) fermion systems on one-dimensional lattices are constructed explicitly based on the discrete orthogonal polynomials of Askey scheme, e.g. the Krawtchouk, Hahn, Racah, Meixner, q-Racah polynomials. The Schrödinger and Heisenberg equations are solved explicitly, as the entire set of the eigenvalues and eigenstates are known explicitly. The ground state two point correlation functions are derived explicitly. The multi point correlation functions are obtained by Wick’s Theorem. Corresponding 15 exactly solvable XX spin systems are also displayed. They all have nearest neighbour interactions. The exact solvability of Schrödinger equation means that of the corresponding Fokker-Planck equation. This leads to 15 exactly solvable Birth and Death fermions and 15 Birth and Death spin models. These provide plenty of materials for calculating interesting quantities, e.g. entanglement entropy, etc.

Analysis of a stochastic SEIS epidemic model motivated by Black–Karasinski process: Probability density function

This paper examines a stochastic SEIS epidemic model motivated by Black–Karasinski process. First, it is shown that Black–Karasinski process is a both biologically and mathematically reasonable assumption compared with existing stochastic modeling methods. By analyzing the diffusion structure of the model and solving the relevant Kolmogorov–Fokker–Planck equation, a complete characterization for explicitly approximating the stationary density function near some quasi-positive equilibria is provided. Then for the deterministic model, the basic reproduction number and related asymptotic stability are studied. Finally, several numerical examples are given to substantiate our theoretical findings.

QSLE-v1.0: A New Software Package for the Calculation of Coupled Quantum-Classical Dynamics in Condensed Phases Based on a Stochastic Approach.

Recently, a stochastic version of the quantum-classical Liouville equation has been proposed [Campeggio, J.; Cortivo, R.; Zerbetto, M. J. Chem. Phys. 2023, 158, 244104], to compute the coupled quantum-classical dynamics of molecules in condensed phases. The approach, called quantum-stochastic Liouville equation (QSLE), is based on coupling the time evolution of electronic states to a stochastic description of the relevant (classical) nuclear coordinates. Natural internal coordinates are used, i.e., bond lengths, bond angles, and dihedral angles. The approach is tailored to directly propagate the populations of the electronic states over time, based on a classical Fokker-Planck equation for the nuclear degrees of freedom coupled to a master equation for the jumps among the electronic states. The QSLE is a multiscale approach requiring many ingredients to be assembled in order to carry out the numerical solution. To make the approach accessible, a software package that handles the main (and most cumbersome) details of the numerical workflow has been implemented into the software package QSLE-v1.0, which is introduced in the present paper. Here, it is considered the case of one torsional nuclear coordinate and two nonadiabatic electronic potential energy surfaces. This is a very basic model for interpreting photoisomerization or charge transfer phenomena, but despite its simplicity, it can be applied even in complex systems where the relevant quantum/classical parts affecting the phenomena under study are highly localized. A sand-box model system for describing photoisomerization processes is reported to demonstrate the usage of the software package. QSLE-v1.0 is open source and distributed under the GPLv2.0 license.

Information geometry of the Otto metric

AbstractWe introduce the dual of the mixture connection with respect to the Otto metric which represents a new kind of exponential connection. This provides a dual structure consisting of the mixture connection, the Otto metric as a Riemannian metric, and the new exponential connection. We derive the geodesic equation of this exponential connection, which coincides with the Kolmogorov forward equation of a gradient flow. We then derive the canonical contrast function of the introduced dual structure.

Stationary Distribution and Density Function for a High-Dimensional Stochastic SIS Epidemic Model with Mean-Reverting Stochastic Process

This paper explores a high-dimensional stochastic SIS epidemic model characterized by a mean-reverting, stochastic process. Firstly, we establish the existence and uniqueness of a global solution to the stochastic system. Additionally, by constructing a series of appropriate Lyapunov functions, we confirm the presence of a stationary distribution of the solution under R0s>1. Taking 3D as an example, we analyze the local stability of the endemic equilibrium in the stochastic SIS epidemic model. We introduce a quasi-endemic equilibrium associated with the endemic equilibrium of the deterministic system. The exact probability density function around the quasi-stable equilibrium is determined by solving the corresponding Fokker–Planck equation. Finally, we conduct several numerical simulations and parameter analyses to demonstrate the theoretical findings and elucidate the impact of stochastic perturbations on disease transmission.

Run-and-tumble particles in slit geometry as a splitting probability problem

Run-and-tumble particles confined between two walls seem like a simple enough problem to possess analytical tractability. Yet, to date no satisfactory analysis is available for dimensions higher than one. This work contributes to the theoretical understanding of this system by reinterpreting it as a splitting probability problem. Such reinterpretation permits us to formulate the problem as the integral equation, rather than a more standard formulation based on the Fokker–Planck equation. In addition to providing an analogy with another phenomenon, the reinterpretation permits a new type of analysis, yields useful results, and offers some analytical tractability.

An investigation of firm size distributions involving the growth functions

Following the ideas of prospect theory, a class of growth functions is used to characterize deterministic variations of firm size, which portrays the asymmetric efforts of the firm to achieve the desired size. Considering the differences in the ability of different firm size to cope with uncertainties, the Boltzmann equation for the evolution of firm size is constructed. Utilizing a suitable scaling limit, the Fokker–Planck equation is acquired and its explicit steady-state solution is derived. Our results illustrate that different choices of parameters in the growth function lead to various statistical laws for firm size, such as the Amoroso distribution, the lognormal distribution and Zipf’s law. Under certain conditions, inequality for the distribution of firm size decreases as firm size increases. The numerical analyses are presented to illustrate our results.

Relativistic diffusion model for hadron production in p -Pb collisions at energies available at the CERN Large Hadron Collider

We investigate charged-hadron production in relativistic heavy-ion collisions of asymmetric systems within a nonequilibrium-statistical framework. Calculated centrality-dependent pseudorapidity distributions for p-Pb collisions at sNN=5.02 and 8.16 TeV are compared with data from the Large Hadron Collider. Our approach combines a relativistic diffusion model with formulations based on quantum chromodynamics while utilizing numerical solutions of a Fokker-Planck equation to account for the shift and broadening of the fragmentation sources for particle-production with respect to the stopping (net-baryon) rapidity distributions. To represent the centrality dependence of charged-hadron production in asymmetric systems over a broad region of pseudorapidities, the consideration and precise modeling of the fragmentation sources, along with the central gluon-gluon source, is found to be essential. Specifically, this results in an inversion of the particle-production amplitude from backward to forward dominance when transitioning from central to peripheral collisions, in agreement with recent ATLAS and ALICE p-Pb data at sNN=5.02TeV. Published by the American Physical Society 2024

Reconstruction of intra- and extra-neurite conductivity tensors via conductivity at Larmor frequency and DWI data patterns

The developed magnetic resonance electrical properties tomography (MREPT) techniques visualize the internal conductivity distribution at Larmor frequency by measuring the B1 transceive phase data. In biological tissues, electrical conductivity is influenced by ion concentrations and mobility. To visualize the anisotropic conductivity tensor of biological tissues, we use the Einstein–Smoluchowski equation, which links the diffusion coefficient to particle mobility. By assuming a correlation between ion mobility and water diffusivity, we aim to decompose the internal isotropic conductivity at Larmor frequency into anisotropic conductivity tensors within the intra- and extra-neurite compartments. The multi-compartment spherical mean technique (MC-SMT), utilizing both high and low b-value diffusion-weighted imaging (DWI) data, characterizes the diffusion of water molecules within and across the intra- and extra-neurite compartments by analyzing the microstructural intricacies and the foundational architectural arrangement of the brain’s tissues. By analyzing the relationships between the measured DWI data, the microscopic diffusion signal, and the fiber orientation distribution function, we predict the DWI data for the intra- and extra-neurite compartments using spherical harmonic decomposition. Using the predicted DWI data for the intra- and extra-neurite compartments, we develop a conductivity tensor imaging method that operates specifically within the separated compartments. Human brain experiments, involving four healthy volunteers and a brain tumor patient, were performed to assess and confirm the reliability of the proposed method.

Fokker-Planck Central Moment Lattice Boltzmann Method for Effective Simulations of Fluid Dynamics

We present a new formulation of the central moment lattice Boltzmann (LB) method based on a minimal continuous Fokker-Planck (FP) kinetic model, originally proposed for stochastic diffusive-drift processes (e.g., Brownian dynamics), by adapting it as a collision model for the continuous Boltzmann equation (CBE) for fluid dynamics. The FP collision model has several desirable properties, including its ability to preserve the quadratic nonlinearity of the CBE, unlike that based on the common Bhatnagar-Gross-Krook model. Rather than using an equivalent Langevin equation as a proxy, we construct our approach by directly matching the changes in different discrete central moments independently supported by the lattice under collision to those given by the CBE under the FP-guided collision model. This can be interpreted as a new path for the collision process in terms of the relaxation of the various central moments to “equilibria”, which we term as the Markovian central moment attractors that depend on the products of the adjacent lower order moments and a diffusion coefficient tensor, thereby involving of a chain of attractors; effectively, the latter are nonlinear functions of not only the hydrodynamic variables, but also the non-conserved moments; the relaxation rates are based on scaling the drift coefficient by the order of the moment involved. The construction of the method in terms of the relevant central moments rather than via the drift and diffusion of the distribution functions directly in the velocity space facilitates its numerical implementation and analysis. We show its consistency to the Navier-Stokes equations via a Chapman-Enskog analysis and elucidate the choice of the diffusion coefficient based on the second order moments in accurately representing flows at relatively low viscosities or high Reynolds numbers. We will demonstrate the accuracy and robustness of our new central moment FP-LB formulation, termed as the FPC-LBM, using the D3Q27 lattice for simulations of a variety of flows, including wall-bounded turbulent flows. We show that the FPC-LBM is more stable than other existing LB schemes based on central moments, while avoiding numerical hyperviscosity effects in flow simulations at relatively very low physical fluid viscosities through a refinement to a model founded on kinetic theory.

Elucidating the Mechanism of Helium Evaporation from Liquid Water.

We investigate the evaporation of trace amounts of helium solvated in liquid water using molecular dynamics simulations and theory. Consistent with experimental observations, we find a super-Maxwellian distribution of kinetic energies of evaporated helium. This excess of kinetic energy over typical thermal expectations is explained by an effective continuum theory of evaporation based on a Fokker-Planck equation, parametrized molecularly by a potential of mean force and position-dependent friction. Using this description, we find that helium evaporation is strongly influenced by the friction near the interface, which is anomalously small near the Gibbs dividing surface due to the ability of the liquid-vapor interface to deform around the gas particle. Our reduced description provides a mechanistic interpretation of trace gas evaporation as the motion of an underdamped particle in a potential subject to a viscous environment that varies rapidly across the air-water interface. From it we predict the temperature dependence of the excess kinetic energy of evaporation, which is yet to be measured.

Dynamical behaviors of a stochastic SIVS epidemic model with the Ornstein-Uhlenbeck process and vaccination of newborns.

In this paper, we study a stochastic SIVS infectious disease model with the Ornstein-Uhlenbeck process and newborns with vaccination. First, we demonstrate the theoretical existence of a unique global positive solution in accordance with this model. Second, adequate conditions are inferred for the infectious disease to die out and persist. Then, by classic Lynapunov function method, the stochastic model is allowed to obtain the sufficient condition so that the stochastic model has a stationary distribution represents illness persistence in the absence of endemic equilibrium. Calculating the associated Fokker-Planck equations yields the precise expression of the probability density function for the linearized system surrounding the quasi-endemic equilibrium. In the end, the theoretical findings are shown by numerical simulations.

An evaluation of the hybrid Fokker–Planck-DSMC approach for high-speed rarefied gas flows

The Direct Simulation Monte Carlo (DSMC) method has been widely used for simulations of rarefied gas flows. It has been proven that the numerical solution of the DSMC method converges to the Boltzmann equation, providing a solid physical foundation for high Knudsen numbers. However, many aerospace engineering problems include both low and high-density regions in a single domain, making the DSMC method computationally expensive. Over the past decade, the particle Fokker–Planck (FP) method has been studied as a means to reduce computational costs near the continuum regime. While enhancing efficiency, the FP method loses physical accuracy at high Knudsen numbers. Aiming for universal accuracy and efficiency across the whole range of Knudsen numbers, the hybrid FP-DSMC method has been studied. Nevertheless, consistent comparisons among the DSMC, FP, and hybrid FP-DSMC methods have received limited attention so far. This paper presents a consistent comparative study of the DSMC, FP, and hybrid FP-DSMC methods to assess the accuracy and efficiency of the hybrid FP-DSMC method. The hybrid FP-DSMC solver is developed based on the open-source SPARTA framework. The benchmark problems include supersonic flow in a planar nozzle, hypersonic flow around a cylinder, and hypersonic flow around a THAAD-like missile. The results demonstrate that the hybrid FP-DSMC method can be more efficient than the DSMC method while being more accurate than the FP method. The speed-up achieved by the FP-DSMC method ranged from 1.5 to 14 times compared to the converged DSMC simulation.

Stochastic Memristor Modeling Framework Based on Physics-Informed Neural Networks

In this paper, we present a framework of modeling memristor noise for circuit simulators using physics-informed neural networks (PINNs). The variability of the memristor that is directly related to the neuromorphic system can be handled with this approach. The memristor noise model is transformed into a Fokker–Planck equation (FPE) from a probabilistic perspective. The translated equations are physically interpreted through the PINN. The weights and biases extracted from the PINN are implemented in Verilog-A through simple operations. The characteristics of the stochastic system under the noise are obtained by integrating the probability density function. This approach allows for the unification of different memristor models and the analysis of the effects of noise.

Collocation Finite Element Method for the Fractional Fokker–Planck Equation

ABSTRACTIn this study, the approximate results of the fractional Fokker–Planck equations have been investigated. First, finite element schemes have been obtained using collocation finite element method based on the trigonometric quintic B‐spline basis functions. Then, the present method is tested on two fundamental problems having appropriate initial conditions. The newly obtained numerical results contained the error norms and for various temporal and spatial steps are compared with the exact ones and other solutions. More accurate results have been obtained for large numbers of spatial and temporal elements.

Applying the Adomian Method to Solve the Fokker–Planck Equation: A Case Study in Astrophysics

The objective of this study is to model astrophysical systems using the nonlinear Fokker–Planck equation, with the Adomian method chosen for its iterative and precise solutions in this context, applying boundary conditions relevant to data from the Rossi X-ray Timing Explorer (RXTE). The results include analysis of 156 X-ray intensity distributions from X-ray binaries (XRBs), exhibiting long-tail profiles consistent with Tsallis q-Gaussian distributions. The corresponding q-values align with the principles of Tsallis thermostatistics. Various diffusion hypotheses—classical, linear, nonlinear, and anomalous—are examined, with q-values further supporting Tsallis thermostatistics. Adjustments in the parameter α (related to the order of fractional temporal derivation) reveal the extent of the memory effect, strongly correlating with fractal properties in the diffusive process. Extending this research to other XRBs is both possible and recommended to generalize the characteristics of X-ray scattering and electromagnetic waves at different frequencies originating from similar astronomical objects.

High-order schemes of exponential time differencing for stiff systems with nondiagonal linear part

Exponential time differencing methods are a power tool for high-performance numerical simulation of computationally challenging problems in condensed matter physics, fluid dynamics, chemical and biological physics, where mathematical models often possess fast oscillating or decaying modes—in other words, are stiff systems. Practical implementation of these methods for the systems with nondiagonal linear part of equations is exacerbated by infeasibility of an analytical calculation of the exponential of a nondiagonal linear operator; in this case, the coefficients of the exponential time differencing scheme cannot be calculated analytically. We suggest an approach, where these coefficients are numerically calculated with auxiliary problems. We rewrite the high-order Runge–Kutta type schemes in terms of the solutions to these auxiliary problems and practically examine the accuracy and computational performance of these methods for a heterogeneous Cahn–Hilliard equation, a sixth-order spatial derivative equation governing pattern formation in the presence of an additional conservation law, and a Fokker–Planck equation governing macroscopic dynamics of a network of neurons.

Inertial amplification as a performance enhancement method for snap-through vibration energy harvester

In recent years, there has been a lot of interest in exploiting the bistable behavior of snap-through systems to harvest energy from vibration sources. The efficient operation of any bistable VEH depends on its ability to exhibit large-amplitude interwell motion. Under weak ambient excitation, bistable VEH performs marginally because of the confinement of motion to a single well. Frequency up-conversion, multi-stability, and adaptive techniques are some of the performance enhancement strategies suggested for bistable-VEH. Considering the VEH's space constraints, the above designs are hard to implement in practical cases. This study introduces an inertial amplification mechanism (IAM) as a simple passive strategy to enhance the performance of a snap-through VEH, a concept not explored in previous studies. The addition of IAM increases the effective mass without increasing the physical mass and thereby enhances energy harvesting, especially from weak ambient excitation sources. The dynamics and performance of the enhanced snap-through VEH are investigated analytically and numerically under harmonic and random excitations. The harmonic balance method (HBM) derives the frequency-amplitude relationship, which shows a hardening behavior and an increase in bandwidth. The effective potential method provides a closed-form expression for the joint probability density function (Joint PDF), which is governed by the Fokker-Planck equation. The joint PDF shows a transition from bimodal to unimodal with an increase in the value of the geometrical parameter. The stochastic averaging method is employed to obtain the stationary probability density function, which defines the long-term dynamics of the VEH. The effects of noise intensity, mass ratio, and inertial amplifier angle on the dynamics are investigated. Finally, the performance of the proposed VEH is compared with a conventional snap-through VEH, an equivalent linear VEH, and a multistable VEH under harmonic and random excitation conditions. The findings suggest that the snap-through VEH with the IAM has advantages over the linear and multistable nonlinear VEH in terms of extracting energy from low-intensity harmonic and random excitation sources. This simple augmentation strategy preserves the original system's bistability, eliminating the need for the complex design of a multistable VEH.