Fokker-Planck Model Research Articles

A graded mesh technique for numerical approximation of a multi-term Caputo time-fractional Fokker-Planck equation in 2D space

This paper focuses on the development of an efficient numerical approach for solving a two-dimensional multi-term Caputo time fractional Fokker-Planck (TFFP) model. The solution of such problem, in general, shows a weak singularity at the time origin. A numerical technique based on a graded time mesh is proposed to handle the singular behaviour of the solution. The multi-term Caputo time fractional derivatives in the TFFP model are discretized by means of the L1 scheme on the nonuniform mesh, while a high-order compact alternating direction implicit finite difference scheme is designed to approximate the spatial derivatives. Convergence and stability analysis of the suggested method is analyzed. Two numerical examples subjected to smooth and nonsmooth solutions are presented to corroborate the theoretical results and illustrate the applicability of the method. The results obtained by the proposed graded mesh technique are compared with the results obtained by the uniform mesh technique.

A spatial interpolation for a stochastic particle Fokker–Planck model using a polynomial reconstruction

The stochastic particle Fokker–Planck (FP) model describes the behavior of rarefied gases while reducing the computational cost compared to the direct simulation Monte Carlo (DSMC) method, particularly for gas flows in the continuum regime. Many studies using FP models rely on cell-averaged macroscopic properties to update particle velocities, limiting spatial resolution in regions with large macroscopic gradients. To overcome this limitation, this paper introduces a spatial interpolation method based on the polynomial reconstruction. This method provides more accurate estimations of macroscopic properties using cell-averaged values and allows for extension to higher-order spatial accuracy. The spatial interpolation method is evaluated through three numerical simulations: Couette flow, lid-driven cavity flow, and hypersonic flow over a flat plate. The results demonstrate that the polynomial reconstruction method significantly improves accuracy. The second-order polynomial reconstruction method consistently outperforms the first-order polynomial reconstruction method, while the fourth-order polynomial reconstruction method does not consistently surpass the second-order polynomial reconstruction method due to challenges in boundary treatment. The study also examines accuracy improvements by interpolating a combined property of the viscous stress and density in the hypersonic flow over a flat plate, where large viscous stress gradients are present. The result demonstrates that interpolating the combined property enhances the overall accuracy of flow predictions by capturing large gradients.

Exact time-dependent thermodynamic relations for a Brownian particle moving in a ratchet potential coupled with quadratically decreasing temperature.

The thermodynamic relations for a Brownian particle moving in a discrete ratchet potential coupled with quadratically decreasing temperature are explored as a function of time. We show that this thermal arrangement leads to a higher velocity (lower efficiency) compared to a Brownian particle operating between hot and cold baths, and a heat bath where the temperature linearly decreases along with the reaction coordinate. The results obtained in this study indicate that if the goal is to design a fast-moving motor, the quadratic thermal arrangement is more advantageous than the other two thermal arrangements. In contrast, the entropy, entropy production rate, and entropy extraction rate are significantly larger in the case of a quadratically decreasing temperature compared to the linearly decreasing temperature case and piecewise constant temperature case. Furthermore, the thermodynamic features of a system consisting of several Brownian ratchets arranged in a complex network are explored. The theoretical findings exhibit that as the network size increases, the entropy, entropy production, and entropy extraction of the system also increase, showing that these thermodynamic quantities exhibit extensive property. As a result, as the number of lattice sizes increases, thermodynamic relations such as entropy, entropy production, and entropy extraction also step up, confirming that these complex networks cannot be reduced to a corresponding one-dimensional lattice. However, in the long time limit, thermodynamic relations such as velocity, entropy production rate, and entropy extraction rate become independent of the network size. These results are also confirmed via a continuum Fokker-Planck model for the overdamped case.

A Consistent Kinetic Fokker–Planck Model for Gas Mixtures

We propose a general multi-species Fokker–Planck model. We prove consistency of our model: conservation properties, positivity of all temperatures, H-Theorem and the shape of equilibrium as Maxwell distributions with the same mean velocity and temperature. Moreover, we derive the usual macroscopic equations from the kinetic two-species BGK model and compute explicitly the exchange terms of momentum and energy.

A multiscale stochastic particle method based on the Fokker-Planck model for nonequilibrium gas flows

The Fokker-Planck (FP) model characterizes the intermolecular collisions as continuous stochastic processes in velocity space. Compared to the Direct Simulation Monte Carlo (DSMC) method, which is grounded in the Boltzmann equation, the stochastic particle method based on the FP model, referred to as the SP-FP method, demonstrates a relative improvement in computational efficiency. However, the traditional SP-FP method, utilizing operator splitting scheme, encounters analogous limitations to DSMC, thereby limiting the application of large temporal-spatial discretization for simulating near-continuum and continuum flows. In this study, two critical advancements are introduced to improve the traditional SP-FP method. First, the exact integration scheme of the ellipsoidal-statistical FP (ESFP) model is derived, enabling the replication of the theoretical solutions for homogeneous relaxation with any time step. Furthermore, leveraging the integration scheme of the ESFP model, we propose a Multiscale Stochastic Particle (MSP) method. The MSP method quantifies the numerical errors stemming from the operator splitting scheme and corrects them in the relaxation step. This capability allows the MSP method to be employed with larger temporal-spatial discretization for inhomogeneous problems, achieving second-order accuracy while retaining implementation simplicity. Application of the MSP method to various benchmark problems, including Couette flow, Fourier flow, Poiseuille flow, Sod tube flow, Lid-driven cavity flow, and flow through a slit, demonstrates its promise for simulating multiscale nonequilibrium gas flows with high computational efficiency and accuracy.

On stability and regularity for subdiffusion equations involving delays

We study a class of nonlocal evolution equations involving time-varying delays which is employed to depict subdiffusion processes. The global solvability, stability and regularity are shown by using the resolvent theory, nonlocal Halanay inequality, fixed point argument and embeddings of fractional Sobolev spaces. Our result is applied to the nonlocal Fokker–Planck model with nonlinear force fields.

Fokker-Planck modeling of the stochastic dynamics of a Rijke tube.

We derive and numerically validate a low-order oscillator model to capture the stochastic dynamics of a prototypical thermoacoustic system (a Rijke tube) undergoing a subcritical Hopf bifurcation in the presence of additive noise. We find that on the fixed-point branch before the bifurcation, the system is dominated by the first duct mode, and the Fokker-Planck solution for the first Galerkin mode can adequately predict the stochastic dynamics of the overall system. We also find that this analytical framework predicts well the dominant mode on the limit-cycle branch, but underperforms in the hysteretic bistable zone where the role of nonlinearities is more pronounced. Besides offering new insights into stochastic thermoacoustic behavior, this study shows that an analytical framework based on the Fokker-Planck equation can facilitate the early detection of thermoacoustic instabilities in a Rijke-tube model subjected to noise.

A second-order particle Fokker-Planck model for rarefied gas flows

The direct simulation Monte Carlo (DSMC) method has become a powerful tool for studying rarefied gas flows. However, for the DSMC method to be effective, the cell size must be smaller than the mean free path, and the time step smaller than the mean collision time. These constraints make it difficult to use the DSMC method in multiscale rarefied gas flows. Over the past decade, the particle Fokker-Planck (FP) method has been studied to address computational cost issues in the near-continuum regime. To capture the main features of the Boltzmann equation, various FP models have been proposed, such as the quadratic entropic FP (Quad-EFP) and the ellipsoidal statistical FP (ESFP). Nevertheless, few studies have clearly demonstrated that the FP method offers a computational advantage over the DSMC method without sacrificing accuracy. This is because conventional particle FP methods have employed first-order accuracy schemes. The present study proposes a unified stochastic particle ESFP (USP-ESFP) model. This model improves the accuracy of shear stress and heat flux predictions. Additionally, a spatial interpolation scheme is introduced to the particle FP method. The numerical test cases include relaxation problem, Couette flows, Poiseuille flows, velocity perturbation, and hypersonic flows around a cylinder. The results show that the USP-ESFP model agrees well with both analytical and DSMC results. Furthermore, the USP-ESFP model is found to be less sensitive to cell size and time step than the DSMC method, resulting in a factor of four speed-up for the considered hypersonic flow around a cylinder.

On the increased interfacial resistance of hydrogen in carbon nanotube arrays and its effect on gas mixture separation

We outline a surface scattering kernel for rarefied gas flows through ideally ordered nanomaterials, such as high aspect ratio carbon nanotubes. The derived model allows for a comparison of the tangential momentum accommodation coefficient, and, hence, the total effective friction, for different species of gases as a function of the particle diameter. This surface kernel is incorporated with a Fokker–Planck model as an approximation to transport of a rarefied gas through ideally ordered carbon nanotubes. The results of this analysis predict that H2 experiences higher friction in such systems in comparison with larger molecules such as CH4. The results are proposed as a potential explanation of the reduced gas transport of hydrogen gas in nanoporous systems.

Suspension Bridge Estimation Method using the Fokker-Planck Model

The failure of the suspension bridge has been known since the beginning of the bridge collapse. Most of these failures form the basis of current engineering knowledge. One of the factors of failure is human-made factors related to the calculation of the bridge estimate. This paper presents an indirect estimation method using numerical simulation using finite elements by analyzing the Fokker-Planck model when dynamic excitation is associated with bridge loads. The results show that the Fokker-Planck model's homogeneous form can take into account the solution for the bridge analysis approach. It leads to a stable state when giving mass variations to the model. The indirect estimation method using finite elements can estimate the cable tension with controllable weak damping. It can be concluded that the method in this study is more accurate and convenient for the application technique.

A Mixed FEM for a Time-Fractional Fokker–Planck Model

A Mixed FEM for a Time-Fractional Fokker–Planck Model

A stochastic Fokker–Planck–Master model for diatomic rarefied gas flows

The direct simulation Monte Carlo (DSMC) method is widely used for numerical solutions of the Boltzmann equation. However, the associated computational cost becomes prohibitive in the near-continuum regime. To address this limitation, the particle-based Fokker–Planck (FP) method has been extensively studied in the past decade. The FP equation, which describes Brownian motion, does not require resolution of the collisional time and length scales. While several monatomic FP models have been proposed, the modeling of diatomic gases within the FP framework has received limited attention. In this paper, we propose a new diatomic kinetic model, named the Fokker–Planck–Master (FPM) model, which can accurately describe energy exchanges between translational-rotational and translational-vibrational modes. The FPM model combines the FP equation to describe the evolution of translational and rotational modes, and the master equation to describe the evolution of the vibrational modes. The numerical test cases include relaxation problems, Couette flows, and hypersonic flows past a vertical flat plate. The results demonstrate that the FPM model shows good agreement with both analytical and DSMC solutions.

Exploring the Origin of Stars on Bound and Unbound Orbits Causing Tidal Disruption Events

Tidal disruption events (TDEs) provide a clue to the properties of a central supermassive black hole (SMBH) and an accretion disk around it, and to the stellar density and velocity distributions in the nuclear star cluster surrounding the SMBH. Deviations of TDE light curves from the standard occurring at a parabolic encounter with the SMBH depend on whether the stellar orbit is hyperbolic or eccentric and the penetration factor (β, the tidal disruption radius to the orbital pericenter ratio). We study the orbital parameters of bound and unbound stars being tidally disrupted by comparison of direct N-body simulation data with an analytical model. Starting from the classical steady-state Fokker–Planck model of Cohn & Kulsrud, we develop an analytical model of the number density distribution of those stars as a function of orbital eccentricity (e) and β. To do so, fittings of the density and velocity distribution of the nuclear star cluster and of the energy distribution of tidally disrupted stars are required and obtained from N-body data. We confirm that most of the stars causing TDEs in a spherical nuclear star cluster originate from the full loss-cone region of phase space, derive analytical boundaries in eccentricity-β space, and find them confirmed by N-body data. Since our limiting eccentricities are much smaller than critical eccentricities for full accretion or the full escape of stellar debris, we conclude that those stars are only very marginally eccentric or hyperbolic, close to parabolic.

Confined run-and-tumble model with boundary aggregation: Long-time behavior and convergence to the confined Fokker–Planck model

The motile micro-organisms such as Escherichia coli, sperm, or some seaweed are usually modeled by self-propelled particles that move with the run-and-tumble process. Individual-based stochastic models are usually employed to model the aggregation phenomenon at the boundary, which is an active research field that has attracted a lot of biologists and biophysicists. Self-propelled particles at the microscale have complex behaviors, while characteristics at the population level are more important for practical applications but rely on individual behaviors. Kinetic PDE models that describe the time evolution of the probability density distribution of the motile micro-organisms are widely used. However, how to impose the appropriate boundary conditions that take into account the boundary aggregation phenomena is rarely studied. In this paper, we propose the boundary conditions for a 2D confined run-and-tumble model (CRTM) for self-propelled particle populations moving between two parallel plates with a run-and-tumble process. The proposed model satisfies the relative entropy inequality and thus long-time convergence. We establish the relation between CRTM and the confined Fokker–Planck model (CFPM) studied in [J. Fu, B. Perthame and M. Tang, Fokker–Plank system for movement of micro-organism population in confined environment, J. Statist. Phys. 184 (2021) 1–25]. We prove theoretically that when the tumble is highly forward peaked and frequent enough, CRTM converges asymptotically to the CFPM. A numerical comparison of the CRTM with aggregation and CFPM is given. The time evolution of both the deterministic PDE model and individual-based stochastic simulations are displayed, which match each other well.

Diffusive Limit of the Vlasov–Poisson–Fokker–Planck Model: Quantitative and Strong Convergence Results

Diffusive Limit of the Vlasov–Poisson–Fokker–Planck Model: Quantitative and Strong Convergence Results

An efficient jump-diffusion approximation of the Boltzmann equation

A jump-diffusion process along with a particle scheme is devised as an accurate and efficient particle solution to the Boltzmann equation. The proposed process (hereafter Gamma-Boltzmann model) is devised to match the evolution of all moments up to the heat fluxes while attaining the correct Prandtl number of 2/3 for monatomic gas with Maxwellian molecular potential. This approximation model is not subject to issues associated with the previously developed Fokker-Planck (FP) based models; such as having wrong Prandtl number, limited applicability, or requiring estimation of higher-order moments. An efficient particle solution to the proposed Gamma-Boltzmann model is devised and compared computationally to the direct simulation Monte Carlo and the cubic FP model (Gorji et al. (2011) [10]) in several test cases including Couette flow and lid-driven cavity. The simulation results indicate that the Gamma-Boltzmann model yields a good approximation of the Boltzmann equation, provides a more accurate solution compared to the cubic FP in the limit of a low number of particles, and remains computationally feasible even in dense regimes.

Critical assessment of various particle Fokker–Planck models for monatomic rarefied gas flows

In the past decade, the particle-based Fokker–Planck (FP) method has been extensively studied to reduce the computational costs of the direct simulation Monte Carlo method for near-continuum flows. The FP equation describes a continuous stochastic process through the combined effects of systematic forces and random fluctuations. A few different FP models have been proposed to fulfill consistency with the Boltzmann equation, but a comprehensive comparative study is needed to assess their performance. The present paper investigates the accuracy and efficiency of four different FP models—Cubic-FP, ellipsoidal-statistical FP (ES-FP), and quadratic entropic FP (Quad-EFP)—under rarefied conditions. The numerical test cases include one-dimensional Couette and Fourier flows and an argon flow past a cylinder at supersonic and hypersonic velocities. It is found that the Quad-EFP model gives the best accuracy in low-Mach internal flows, whereas the ES-FP model performs best at predicting shock waves. In terms of numerical efficiency, the Linear-FP and ES-FP models run faster than the Cubic-FP and Quad-EFP models due to their simple algebraic nature. However, it is observed that the computational advantages of the FP models diminish as the spatiotemporal resolution becomes smaller than the collisional scales. In order to take advantage of their numerical efficiency, high-order joint velocity-position integration schemes need to be devised to ensure the accuracy of FP models with very coarse resolution.

Derivation of the Fractional Fokker–Planck Equation for Stable Lévy with Financial Applications

This paper aims to propose a generalized fractional Fokker–Planck equation based on a stable Lévy stochastic process. To develop the general fractional equation, we will use the Lévy process rather than the Brownian motion. Due to the Lévy process, this fractional equation can provide a better description of heavy tails and skewness. The analytical solution is chosen to solve the fractional equation and is expressed using the H-function to demonstrate the indicator entropy production rate. We model market data using a stable distribution to demonstrate the relationships between the tails and the new fractional Fokker–Planck model, as well as develop an R code that can be used to draw figures from real data.

Local well-posedness of a nonlinear Fokker–Planck model

Noise or fluctuations play an important role in the modeling and understanding of the behavior of various complex systems in nature. Fokker–Planck equations are powerful mathematical tools to study behavior of such systems subjected to fluctuations. In this paper we establish local well-posedness result of a new nonlinear Fokker–Planck equation. Such equations appear in the modeling of the grain boundary dynamics during microstructure evolution in the polycrystalline materials and obey special energy laws.

Fokker–Planck modeling of many-agent systems in swarm manufacturing: asymptotic analysis and numerical results

Fokker–Planck modeling of many-agent systems in swarm manufacturing: asymptotic analysis and numerical results