Time-dependent Fokker-Planck Equation Research Articles

Protocols for trainable and differentiable quantum generative modeling

We propose an approach for learning probability distributions as differentiable quantum circuits (DQC) that enable efficient quantum generative modeling (QGM) and synthetic data generation. Contrary to existing QGM approaches, we perform training of a DQC-based model, where data is encoded in a latent space with the proposed phase feature map of exponential capacity. This is followed by a trainable quantum circuit, forming the model. We then map the trained model to the bit basis using a fixed unitary transformation, in this case corresponding to a quantum Fourier transform circuit. It allows fast sampling from parametrized distributions using a single-shot readout. Importantly, latent space training provides models that are automatically differentiable, and we show how samples from solutions of stochastic differential equations (SDEs) can be accessed by solving stationary and time-dependent Fokker-Planck equations with a quantum protocol. Our approach opens a route to multidimensional generative modeling with qubit registers explicitly correlated via a (fixed) entangling layer. In this case quantum computers can offer advantage as efficient samplers, which perform complex inverse transform sampling enabled by the fundamental laws of quantum mechanics. On a technical side the advances are multiple, as we introduce the phase feature map, analyze its properties, and develop frequency-taming techniques that include qubitwise training and feature map sparsification. Published by the American Physical Society 2024

Probability flow solution of the Fokker–Planck equation

The method of choice for integrating the time-dependent Fokker–Planck equation (FPE) in high-dimension is to generate samples from the solution via integration of the associated stochastic differential equation (SDE). Here, we study an alternative scheme based on integrating an ordinary differential equation that describes the flow of probability. Acting as a transport map, this equation deterministically pushes samples from the initial density onto samples from the solution at any later time. Unlike integration of the stochastic dynamics, the method has the advantage of giving direct access to quantities that are challenging to estimate from trajectories alone, such as the probability current, the density itself, and its entropy. The probability flow equation depends on the gradient of the logarithm of the solution (its ‘score’), and so is a-priori unknown. To resolve this dependence, we model the score with a deep neural network that is learned on-the-fly by propagating a set of samples according to the instantaneous probability current. We show theoretically that the proposed approach controls the Kullback–Leibler (KL) divergence from the learned solution to the target, while learning on external samples from the SDE does not control either direction of the KL divergence. Empirically, we consider several high-dimensional FPEs from the physics of interacting particle systems. We find that the method accurately matches analytical solutions when they are available as well as moments computed via Monte-Carlo when they are not. Moreover, the method offers compelling predictions for the global entropy production rate that out-perform those obtained from learning on stochastic trajectories, and can effectively capture non-equilibrium steady-state probability currents over long time intervals.

Artificial neural network solver for time-dependent Fokker–Planck equations

Stochastic differential equations (SDEs) play a crucial role in various applications for modeling systems that have either random perturbations or chaotic dynamics at faster time scales. The time evolution of the probability distribution of a stochastic differential equation is described by the Fokker–Planck equation, which is a second order parabolic partial differential equation (PDE). Previous work combined artificial neural networks and Monte Carlo data to solve stationary Fokker–Planck equations. This paper extends this approach to time dependent Fokker–Planck equations. The main focus is on the investigation of algorithms for training a neural network that has multi-scale loss functions. Additionally, a new approach for collocation point sampling is proposed. A few 1D and 2D numerical examples are demonstrated.

Time evolution of electron distributions to bimodal steady states for electrons dilutely dispersed in theinert gases Ar, Kr, and Xe with deep Ramsauer Townsend minima in themomentum transfer cross section

The current paper considers the thermalization of an ensemble of electrons under the influence of an external electric field and dilutely dispersed in one of the inert gas moderators, Argon, Krypton or Xenon for which the electron momentum transfer cross sections have deep Ramsauer-Townsend minima. As a consequence, the steady state electron distribution functions are bimodal over a small range of external electric field strengths. The current work is directed towards the time evolution of the electron distribution function determined from the numerical solution of the Fokker-Planck equation. The kinetic theory of electrons dilutely dispersed in a heat bath of atoms at temperature T b has a very long history. The solution of the Fokker-Planck equation can be expressed as a sum of exponentials of the form where λ n are the eigenvalues of the Fokker-Planck operator. Alternatively, a finite difference algorithm is used to solve the time dependent Fokker-Planck equation to give the time dependent electron energy distribution function. We demonstrate the evolution of the initial Maxwellian into a nonequilibrium bimodal distribution which cannot be rationalized with either the Gibbs-Boltzmann entropy or the Tsallis nonextensive entropy. Instead, the time dependent approach of an initial Maxwellian to the bimodal distribution is described in terms of the Kullback-Leibler entropy. We also demonstrate the inapplicability of the Boltzmann entropy nor the Tsallis entropy for a model system with a power law momentum transfer cross section of the form, σ(x) = σ 0/x p , where is the reduced speed. This model with p = 2 is also employed to demonstrate a steady-state Kappa distribution which features prominently in space physics and other fields. For p > 2, we show distribution functions that increase without bound analogous to runaway electrons. The steady nonequilibrium distributions are interpreted as solutions of a Pearson ordinary differential equation.

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.

Physically guided deep learning solver for time-dependent Fokker–Planck equation

The Fokker–Planck (FP) equation is of significant relevance in stochastic dynamics. To solve the FP equation, this paper proposes a physically guided deep learning-based method. The proposed method uses physical modeling as the constraint for guiding the deep neural networks to learn the solutions of the FP equation from unlabeled data. Four numerical examples were used to verify the accuracy and effectiveness of the proposed method for solving linear FP equations and nonlinear FP equations. Moreover, we examined the robustness of the trained model. Finally, extra observation data was used to replace the boundary conditions and initial conditions in complicated forms. Meanwhile, the effect of missing boundary conditions or initial conditions to the proposed method was further analyzed. The results demonstrated that observation data can be used as promising substitution for the initial and boundary conditions while the proposed method can still obtain accurate numerical solutions in the absence of some conditions.

Numerical solution to 3D bilinear Fokker-Planck control problem

A characterization and numerical scheme to control problem governed by three-dimensional (3D) time-dependent Fokker-Planck (FP) equation is presented. We formulate a control formulation that controls the drift of the stochastic (FP) process. In this way, the probability density function (PDF) attains a specific configuration. Moreover, a FP control strategy for collective motion is investigated and first-order optimality conditions are presented. On staggered grids, the Chang-Cooper (CC) discretization scheme that ensures the positivity, second-order accuracy, and conservativeness to the FP equation is employed, to the discretized state (resp. adjoint) system. Furthermore, a line search strategy is applied to update the control variable. Results of numerical experiments show the efficiency of the proposed numerical scheme to stochastic (FP) control problems.

Swimmer dynamics in externally driven fluid flows: The role of noise

We theoretically investigate the effect of random fluctuations on the motion of elongated microswimmers near hydrodynamic transport barriers in externally-driven fluid flows. Focusing on the two-dimensional hyperbolic flow, we consider the effects of translational and rotational diffusion as well as tumbling, i.e. sudden jumps in the swimmer orientation. Regardless of whether diffusion or tumbling are the primary source of fluctuations, we find that noise significantly increases the probability that a swimmer crosses one-way barriers in the flow, which block the swimmer from returning to its initial position. We employ an asymptotic method for calculating the probability density of noisy swimmer trajectories in a given fluid flow, which produces solutions to the time-dependent Fokker-Planck equation in the weak-noise limit. This procedure mirrors the semiclassical approximation in quantum mechanics and similarly involves calculating the least-action paths of a Hamiltonian system derived from the swimmer's Fokker-Planck equation. Using the semiclassical technique, we compute (i) the steady-state orientation distribution of swimmers with rotational diffusion and tumbling and (ii) the probability that a diffusive swimmer crosses a one-way barrier. The semiclassical results compare favorably with Monte Carlo calculations.

Solving Time Dependent Fokker-Planck Equations Via Temporal Normalizing Flow

In this work, we propose an adaptive learning approach based on temporal normalizing flows for solving time-dependent Fokker-Planck (TFP) equations. It is well known that solutions of such equations are probability density functions, and thus our approach relies on modelling the target solutions with the temporal normalizing flows. The temporal normalizing flow is then trained based on the TFP loss function, without requiring any labeled data. Being a machine learning scheme, the proposed approach is mesh-free and can be easily applied to high dimensional problems. We present a variety of test problems to show the effectiveness of the learning approach.

Solving Time Dependent Fokker-Planck Equations via Temporal Normalizing Flow

In this work, we propose an adaptive learning approach based on temporal normalizing flows for solving time-dependent Fokker-Planck (TFP) equations. It is well known that solutions of such equations are probability density functions, and thus our approach relies on modelling the target solutions with the temporal normalizing flows. The temporal normalizing flow is then trained based on the TFP loss function, without requiring any labeled data. Being a machine learning scheme, the proposed approach is mesh-free and can be easily applied to high dimensional problems. We present a variety of test problems to show the effectiveness of the learning approach.

The Influence of the Hydrodynamic Response of the Flare Plasma on the Kinetics of an Accelerated Electron Beam and Hard X-rays

The transfer of accelerated electrons in the plasma of flaring loops is accompanied by a hydrodynamic response of the heated plasma. The dynamics of the accelerated electron beam was determined from the solution of the one-dimensional, time-dependent relativistic Fokker-Planck equation, and the hydrodynamic response of the thermal plasma was studied by integrating a system of one-dimensional equations using the numerical RADYN code. Thus, the electron distribution function and thermal plasma parameters are changing self-consisting. In this problem the most important values are the energy flux, the duration of continuous injection, and the accelerated electrons spectrum. The most significant for the beam—thermal plasma system is the process of plasma evaporation. The upflows of the plasma (evaporation) increases the looptop plasma density comparing with steady-state value and thus reduces the Coulomb mean free path of fast electrons over time. For impulsive events, we consider heating plasma by electron beams with the energy flux of 3 × 1010, 1011 erg cm–2 s–1 in the energy band from 25 keV up to 10 MeV, and duration of 20 s, which is quite consistent with M-X GOES class flares. It is shown that for impulsive events, the effect of plasma “evaporation” for the specified values of the energy flux, isotropic pitch-angle distributions of the accelerated electrons, and power–low energy spectra leads to an increase of the plasma density in the magnetic loop on one-two orders near the transition region only. The effect of plasma evaporation is insignificant and does not affect the hard X-rays from the top of the loop for the considered parameters.

Exact solutions of Fokker–Planck equation via the Nikiforov–Uvarov method

We propose a powerful approach to provide the exact solutions of the time-dependent Fokker–Planck equation (FPE) for a given pair of drift and diffusion functions in stochastic phenomena. First, we briefly review Nikiforov–Uvarov mathematical method and then apply it to consider three important examples. Subsequently, the probability distribution functions of FPE are obtained in terms of special orthogonal functions for three cases, as well as the corresponding eigenvalues are derived. Several applications are proposed and it is shown that the results of our approach are in good agreement with those obtained by other methods.

Two-level difference scheme for the two-dimensional Fokker–Planck equation

In this paper, we propose a two-level difference scheme for solving the two-dimensional Fokker–Planck equation. This equation is a parabolic type equation which governs the time evolution of probability density function of the stochastic processes. In addition, these equations preserve positivity and conservation. The Chang–Cooper discretization scheme is used, which ensures second-order accuracy, positiveness, and satisfies the conservation of the total probability. In particular, we investigate a two-level scheme with factor-three coarsening strategy. With coarsening by a factor-of-three we obtained simplified inter-grid transfer operators and thus have a significant reduction in CPU time. Numerical experiments are performed to validate efficiency of the proposed Chang–Cooper two-level algorithms to stationary and time-dependent Fokker–Planck equations, respectively.

Approximation to a Fokker-Planck equation for the Brownian motor.

An approximate method is proposed and numerically confirmed for a time-dependent Fokker-Planck equation (FPE) modeling the Brownian motor. The situation addressed in this paper is that the potential is spatially asymmetric and multiplicatively modulated. The approximate solution is composed of Sturm-Liouville eigenfunctions. While some time-independent partial differential equations and those perturbed around them have been known to be reduced to ordinary differential equations with a few degrees of freedom, the time-dependent FPE dealt with in this paper is also found to be reduced to an ordinary differential equation.

Kappa and other nonequilibrium distributions from the Fokker-Planck equation and the relationship to Tsallis entropy.

This paper considers two nonequilibrium model systems described by linear Fokker-Planck equations for the time-dependent velocity distribution functions that yield steady state Kappa distributions for specific system parameters. The first system describes the time evolution of a charged test particle in a constant temperature heat bath of a second charged particle. The time dependence of the distribution function of the test particle is given by a Fokker-Planck equation with drift and diffusion coefficients for Coulomb collisions as well as a diffusion coefficient for wave-particle interactions. A second system involves the Fokker-Planck equation for electrons dilutely dispersed in a constant temperature heat bath of atoms or ions and subject to an external time-independent uniform electric field. The momentum transfer cross section for collisions between the two components is assumed to be a power law in reduced speed. The time-dependent Fokker-Planck equations for both model systems are solved with a numerical finite difference method and the approach to equilibrium is rationalized with the Kullback-Leibler relative entropy. For particular choices of the system parameters for both models, the steady distribution is found to be a Kappa distribution. Kappa distributions were introduced as an empirical fitting function that well describe the nonequilibrium features of the distribution functions of electrons and ions in space science as measured by satellite instruments. The calculation of the Kappa distribution from the Fokker-Planck equations provides a direct physically based dynamical approach in contrast to the nonextensive entropy formalism by Tsallis [J. Stat. Phys. 53, 479 (1988)JSTPBS0022-471510.1007/BF01016429].

Evolution of moments and correlations in nonrenewal escape-time processes.

The theoretical description of nonrenewal stochastic systems is a challenge. Analytical results are often not available or can be obtained only under strong conditions, limiting their applicability. Also, numerical results have mostly been obtained by ad hoc Monte Carlo simulations, which are usually computationally expensive when a high degree of accuracy is needed. To gain quantitative insight into these systems under general conditions, we here introduce a numerical iterated first-passage time approach based on solving the time-dependent Fokker-Planck equation (FPE) to describe the statistics of nonrenewal stochastic systems. We illustrate the approach using spike-triggered neuronal adaptation in the leaky and perfect integrate-and-fire model, respectively. The transition to stationarity of first-passage time moments and their sequential correlations occur on a nontrivial time scale that depends on all system parameters. Surprisingly this is so for both single exponential and scale-free power-law adaptation. The method works beyond the small noise and time-scale separation approximations. It shows excellent agreement with direct Monte Carlo simulations, which allow for the computation of transient and stationary distributions. We compare different methods to compute the evolution of the moments and serial correlation coefficients (SCCs) and discuss the challenge of reliably computing the SCCs, which we find to be very sensitive to numerical inaccuracies for both the leaky and perfect integrate-and-fire models. In conclusion, our methods provide a general picture of nonrenewal dynamics in a wide range of stochastic systems exhibiting short- and long-range correlations.

Modeling the capture rate by a radially oscillating spherical bubble. A bio-mimetic model for studying the mechanically-mediated uptake by cells

Abstract The surface of living cells constitutes a dynamic environment submitted to complex oscillatory motions. Oscillations may modify the uptake of incoming molecules. In this study we explored a bio-mimetic system formed by oscillating bubbles suspended in a sea of randomly distributed diffusants. We investigated by a time-dependent Fokker–Planck equation the effect of the periodic motions on the adsorption of a diffusant onto the moving bubble surface. We introduced both direct interactions between the diffusant and the fluctuating surface and indirect interactions due to the hydrodynamic motions around a vibrating surface. Results are expressed in terms of oscillation frequencies and amplitudes. An overall reduction of the bound diffusant at the bubble surface was observed.

THE FORMATION OF KAPPA-DISTRIBUTION ACCELERATED ELECTRON POPULATIONS IN SOLAR FLARES

Driven by recent RHESSI observations of confined loop-top hard X-ray sources in solar flares, we consider stochastic acceleration of electrons in the presence of Coulomb collisions. If electron escape from the acceleration region can be neglected, the electron distribution function is determined by a balance between diffusive acceleration and collisions. Such a scenario admits a stationary solution for the electron distribution function that takes the form of a kappa distribution. We show that the evolution toward this kappa distribution involves a "wave front" propagating forwards in velocity space, so that electrons of higher energy are accelerated later; the acceleration time scales with energy according to $\tau_{\rm acc} \sim E^{3/2}$. At sufficiently high energies escape from the finite-length acceleration region will eventually dominate. For such energies, the electron velocity distribution function is obtained by solving a time-dependent Fokker-Planck equation in the "leaky-box" approximation. Solutions are obtained in the limit of a small escape rate from an acceleration region that can effectively be considered a thick target.

Weiss mean-field approximation for multicomponent stochastic spatially extended systems.

We develop a mean-field approach for multicomponent stochastic spatially extended systems and use it to obtain a multivariate nonlinear self-consistent Fokker-Planck equation defining the probability density of the state of the system, which describes a well-known model of autocatalytic chemical reaction (brusselator) with spatially correlated multiplicative noise, and to study the evolution of probability density and statistical characteristics of the system in the process of spatial pattern formation. We propose the finite-difference method for the numerical solving of a general class of multivariate nonlinear self-consistent time-dependent Fokker-Planck equations. We illustrate the accuracy and reliability of the method by applying it to an exactly solvable nonlinear Fokker-Planck equation (NFPE) for the Shimizu-Yamada model [Prog. Theor. Phys. 47, 350 (1972)] and nonlinear Fokker-Planck equation [Desai and Zwanzig, J. Stat. Phys. 19, 1 (1978)] obtained for a nonlinear stochastic mean-field model introduced by Kometani and Shimizu [J. Stat. Phys. 13, 473 (1975)]. Taking the problems indicated above as an example, the accuracy of the method is compared with the accuracy of Hermite distributed approximating functional method [Zhang et al., Phys. Rev. E 56, 1197 (1997)]. Numerical study of the NFPE solutions for a stochastic brusselator shows that in the region of Turing bifurcation several types of solutions exist if noise intensity increases: unimodal solution, transient bimodality, and an interesting solution which involves multiple "repumping" of probability density through bimodality. Additionally, we study the behavior of the order parameter of the system under consideration and show that the second type of solution arises in the supercritical region if noise intensity values are close to the values appropriate for the transition from bimodal stationary probability density for the order parameter to the unimodal one.

Time dependent leptonic modeling of Fermi II processes in the jets of flat spectrum radio quasars

In this paper, we discuss the light-curve features of various flaring scenarios in a time-dependent leptonic model for low-frequency-peaked blazars. The quasar 3C273 is used as an illustrative example. Our code takes into account Fermi-II acceleration and all relevant electron cooling terms, including the external radiation fields generally found to be important in the modeling of the SEDs of FSRQs, as well as synchrotron self absorption and γγ pair-production. General parameters are constrained through a fit to the average spectral energy distribution (SED) of the blazar by numerically solving the time-dependent Fokker–Planck equation for the electron evolution in a steady-state situation. We then apply perturbations to several input parameters (magnetic field, particle injection luminosity, acceleration time scale) to simulate flaring events and compute time-dependent SEDs and light curves in representative energy bands (radio, optical, X-rays, γ-rays). Time lags between different bands are evaluated using a discrete cross correlation analysis. We find that Fermi-II acceleration has a significant effect on the distributions and that flaring events caused by increased acceleration efficiency of the Fermi II process will produce a correlation between the radio, optical and γ-ray bandpasses, but an anti-correlation between these three bandpasses and the X-ray band, with the X-rays lagging behind the variations in other bands by up to several hours.