Fokker-Planck Research Articles

X-ray dark-field via spectral propagation-based imaging

Dark-field X-ray imaging visualizes scattering from unresolved microstructure. Most dark-field imaging techniques rely on crystals or structured illumination, but recent work has shown that dark-field effects are observable in straightforward propagation-based imaging (PBI). Based on the single-material X-ray Fokker–Planck equation with an a priori dark-field energy dependence, we propose an algorithm to extract phase and dark-field effects from dual-energy PBI images. We successfully apply the dark-field retrieval algorithm to simulated and experimental dual-energy data, and show that by accounting for dark-field effects, projected thickness reconstruction is improved compared to the classic Paganin algorithm. With the emergence of spectral detectors, the method could enable single-exposure dark-field imaging of dynamic and living samples.

Moment neural network and an efficient numerical method for modeling irregular spiking activity.

Continuous rate-based neural networks have been widely applied to modeling the dynamics of cortical circuits. However, cortical neurons in the brain exhibit irregular spiking activity with complex correlation structures that cannot be captured by mean firing rate alone. To close this gap, we consider a framework for modeling irregular spiking activity, called the moment neural network, which naturally generalizes rate models to second-order moments and can accurately capture the firing statistics of spiking neural networks. We propose an efficient numerical method that allows for rapid evaluation of moment mappings for neuronal activations without solving the underlying Fokker-Planck equation. This allows simulation of coupled interactions of mean firing rate and firing variability of large-scale neural circuits while retaining the advantage of analytical tractability of continuous rate models. We demonstrate how the moment neural network can explain a range of phenomena including diverse Fano factor in networks with quenched disorder and the emergence of irregular oscillatory dynamics in excitation-inhibition networks with delay.

Insights of infected Schwann cells extinction and inherited randomness in a stochastic model of leprosy

Investigating disease progression, transmission of infection and impacts of Multidrug Therapy (MDT) to inhibit demyelination in leprosy involves a certain amount of difficulty in terms of the in-built uncertain complicated and complex intracellular cell dynamical interactions. To tackle this scenario and to elucidate a more realistic, rationalistic approach of examining the infection mechanism and associated drug therapeutic interventions, we propose a four-dimensional ordinary differential equation-based model. Stochastic processes has been employed on this deterministic system by formulating the Kolmogorov forward equation introducing a transition state and the quasi-stationary distribution, exact distribution analysis have been investigated which allow us to estimate an expected time to extinction of the infected Schwann cells into the human body more prominently. Additionally, to explore the impact of uncertainty in the key intracellular factors, the stochastic system is investigated incorporating random perturbations and environmental noises in the disease dissemination, proliferation and reinfection rates. Rigorous numerical simulations validating the analytical outcomes provide us significant novel insights on the progression of leprosy and unravelling the existing major treatment complexities. Analytical experiments along with the simulations utilizing Monte-Carlo method and Euler–Maruyama scheme involving stochasticity predicts that the bacterial density is underestimated due to the recurrence of infection and suggests that maintaining a drug-efficacy rate in the range 0.6−0.8 would be substantially efficacious in eradicating leprosy.

Markov generators as non-Hermitian supersymmetric quantum Hamiltonians: spectral properties via bi-orthogonal basis and singular value decompositions

Continuity equations associated with continuous-time Markov processes can be considered as Euclidean Schrödinger equations, where the non-Hermitian quantum Hamiltonian H=divJ is naturally factorized into the product of the divergence operator div and the current operator J . For non-equilibrium Markov jump processes in a space of N configurations with M links and C=M−(N−1)⩾1 independent cycles, this factorization of the N × N Hamiltonian H=I†J involves the incidence matrix I and the current matrix J of size M × N, so that the supersymmetric partner H^=JI† governing the dynamics of the currents existing on the M links is of size M × M. To better understand the relations between the spectral decompositions of these two Hamiltonians H=I†J and H^=JI† with respect to their bi-orthogonal basis of right and left eigenvectors that characterize the relaxation dynamics towards the steady state and the steady currents, it is useful to analyze the properties of the singular value decomposition of the two rectangular matrices I and J of size M × N and the interpretations in terms of discrete Helmholtz decompositions. This general framework concerning Markov jump processes can be adapted to non-equilibrium diffusion processes governed by Fokker–Planck equations in dimension d, where the number N of configurations, the number M of links and the number C=M−(N−1) of independent cycles become infinite, while the two matrices I and J become first-order differential operators acting on scalar functions to produce vector fields.

Boltzmann's H-function for molecules with orientational degrees of freedom: Emergence of unique features.

Boltzmann's H-function H(t), often regarded as an analog of time-dependent entropy, holds a venerable place in the history of science. However, accurate numerical evaluation of H(t) for particles other than atoms is rare. To remove this lacuna, we generalize Boltzmann's H-function to a gas of molecules with orientational degrees of freedom and evaluate H(t) from the time-dependent joint probability distribution function f(p, L, t) for linear (p) and angular (L) momenta, evolving from an initial nonequilibrium state, by molecular dynamics simulations. We consider both prolate- and oblate-shaped particles, interacting via the well-known Gay-Berne potential and obtain the relaxation of the generalized molecular H(t) from initial (t = 0) nonequilibrium states. In the long-time limit, the H function saturates to its exact equilibrium value, which is the sum of translational and rotational contributions to the respective entropies. Both the translational and rotational components of H(t) decay nearly exponentially with time; the rotational component is more sensitive to the molecular shape that enters through the aspect ratio. A remarkable rapid decrease in the rotational relaxation time is observed as the spherical limit is approached, in a way tantalizingly reminiscent of Hu-Zwanzig hydrodynamic prediction with the slip boundary condition. In addition, we obtain H(t) analytically by solving the appropriate translational and rotational Fokker-Planck equation and obtain a modest agreement with simulations. We observe a remarkable signature of translation-rotation coupling as a function of molecular shape, captured through a physically meaningful differential term that quantifies the magnitude of translation-rotation coupling.

Testing for Markovian character of transfer of fluctuations in solar wind turbulence on kinetic scales.

We apply statistical analysis to search for processes responsible for turbulence in physical systems. In our previous studies, we have shown that solar wind turbulence in the inertial range of large magnetohydrodynamic scales exhibits Markov properties. We have recently extended this approach on much smaller kinetic scales. Here we are testing for the Markovian character of stochastic processes in a kinetic regime based on magnetic field and velocity fluctuations in the solar wind, measured onboard the Magnetospheric Multiscale (MMS) mission: behind the bow shock, inside the magnetosheath, and near the magnetopause. We have verified that the Chapman-Kolmogorov necessary conditions for Markov processes is satisfied for local transfer of energy between the magnetic and velocity fields also on kinetic scales. We have confirmed that for magnetic fluctuations, the first Kramers-Moyal coefficient is linear, while the second term is quadratic, corresponding to drift and diffusion processes in the resulting Fokker-Planck equation. It means that magnetic self-similar turbulence is described by generalized Ornstein-Uhlenbeck processes. We show that for the magnetic case, the Fokker-Planck equationleads to the probability density functions of the kappa distributions, which exhibit global universal scale invariance with a linear scaling and lack of intermittency. On the contrary, for velocity fluctuations, higher order Kramers-Moyal coefficients should be taken into account and hence scale invariance is not observed. However, the nonextensity parameter in Tsallis entropy provides a robust measure of the departure of the system from equilibrium. The obtained results are important for a better understanding of the physical mechanism governing turbulent systems in space and laboratory.

From the Fokker–Planck equation to a contact Hamiltonian system

Abstract The Fokker–Planck equation is one of the fundamental equations in nonequilibrium statistical mechanics, and this equation is known to be derived from the Wasserstein gradient flow equation with a free energy. This gradient flow equation describes relaxation processes and is formulated on a Riemannian manifold. Meanwhile contact Hamiltonian systems are also known to describe relaxation processes. Hence a relation between these two equations is expected to be clarified, which gives a solid foundation in geometric statistical mechanics. In this paper a class of contact Hamiltonian systems is derived from a class of the Fokker–Planck equations on Riemannian manifolds. In the course of the derivation, the Fokker–Planck equation is shown to be written as a diffusion equation with a weighted Laplacian without any approximation, which enables to employ a theory of eigenvalue problems.

Relaxation and Noise-Driven Oscillations in a Model of Mitotic Spindle Dynamics

During cell division, the mitotic spindle moves dynamically through the cell to position the chromosomes and determine the ultimate spatial position of the two daughter cells. These movements have been attributed to the action of cortical force generators which pull on the astral microtubules to position the spindle, as well as pushing events by these same microtubules against the cell cortex and plasma membrane. Attachment and detachment of cortical force generators working antagonistically against centring forces of microtubules have been modelled previously (Grill et al. in Phys Rev Lett 94:108104, 2005) via stochastic simulations and mean-field Fokker–Planck equations (describing random motion of force generators) to predict oscillations of a spindle pole in one spatial dimension. Using systematic asymptotic methods, we reduce the Fokker–Planck system to a set of ordinary differential equations (ODEs), consistent with a set proposed by Grill et al., which can provide accurate predictions of the conditions for the Fokker–Planck system to exhibit oscillations. In the limit of small restoring forces, we derive an algebraic prediction of the amplitude of spindle-pole oscillations and demonstrate the relaxation structure of nonlinear oscillations. We also show how noise-induced oscillations can arise in stochastic simulations for conditions in which the mean-field Fokker–Planck system predicts stability, but for which the period can be estimated directly by the ODE model and the amplitude by a related stochastic differential equation that incorporates random binding kinetics.

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.

Impact of torque on active Brownian particle: exact moments in two and three dimensions

Abstract In this work, we investigate the effects of torque, accounting for translational diffusion, on active Brownian particles (ABPs) in two (2d) and three (3d) dimensions. The torque is equivalent to chirality in 2d, well known as chiral active Brownian particles (cABPs), whereas in 3d, this is introduced as an external torque on the ABPs. Despite the inherent complexity in solving the Fokker–Planck equation, we demonstrate a Laplace transform method to precisely calculate the temporal evolution of various dynamic moments. Our analysis yields explicit expressions for multiple moments, such as the second and fourth moments of displacement, revealing the impact of persistence and chirality/torque. These moments exhibit oscillatory behavior, and excess kurtosis indicates deviations from the Gaussian distribution during intermediate time intervals.

Efficient generation of barrier crossing trajectories using approximate Brownian bridges.

We examine continuous random walks that are conditioned to reach one region before another. These conditioned processes are used to generate stochastic trajectories for barrier crossing events, which are generally rare and difficult to sample. The processes are generated using a Brownian bridge technique, resulting in near perfect sampling efficiency without accruing error in the conditional statistics of the process. The construction requires the hitting probability or committer function, which is a solution to the backward Fokker-Planck equation, a partial-differential equationthat can be difficult to solve through general means. Therefore, we derive a one-dimensional approximation through asymptotic methods for barrier crossing trajectories. We show that this approximation has a simple analytical representation and approaches the true solution as the barrier height increases. Brownian bridge trajectories generated with this approximate solution are then shown to result in accurate conditional statistics when used in conjunction with importance sampling, even in the case when potential energy barriers are not large. We show this idea's effectiveness by simulating rare events in a stochastic chemical reaction network (Schögl reaction) with multiple steady states. This methodology shows great promise for future implementation to simulate rare barrier crossing events for a wide variety of physical processes.

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.

Extinction and stationary distribution of a novel SIRS epidemic model with general incidence rate and Ornstein–Uhlenbeck process

We propose, in this paper, a novel stochastic SIRS epidemic model to characterize the effect of uncertainty on the distribution of infectious disease, where the general incidence rate and Ornstein–Uhlenbeck process are also introduced to describe the complexity of disease transmission. First, the existence and uniqueness of the global nonnegative solution of our model is obtained, which is the basis for the discussion of the dynamical behavior of the model. And then, we derive a sufficient condition for exponential extinction of infectious diseases. Furthermore, through constructing a Lyapunov function and using Fatou’s lemma, we obtain a sufficient criterion for the existence and ergodicity of a stationary distribution, which implies the persistence of the disease. In addition, the specific form of the density function of the model near the quasiendemic equilibrium is proposed by solving the corresponding Fokker–Planck equation and using some relevant algebraic equation theory. Finally, we explain the above theoretical results through some numerical simulations.

The First Boundary-Value Problem for the Fokker–Planck Equation with One Spatial Variable

The First Boundary-Value Problem for the Fokker–Planck Equation with One Spatial Variable

Weak noise approximation for the Kolmogorov forward equation for a leaky integrate-and-fire neuron subject to stochastic stimulation

We develop a weak noise approximation for the Kolmogorov forward equation governing the dynamics of a leaky integrate-and-fire neuron subject to white noise. Although being very simple, our approximation provides accurate results as far the magnitude of noise-induced fluctuations Δ remains much smaller than the distance A between the mean potential (center of mass) and the excitation threshold. The error for the firing rate is <3% if A/Δ>3 for the stationary stimuli and if A/Δ>5 for time-varying stimuli.

Gelation and Localization in Multicomponent Coagulation with Multiplicative Kernel Through Branching Processes

The multicomponent coagulation equation is a generalization of the Smoluchowski coagulation equation, where the size of a particle is described by a vector. Similar to the original Smoluchowski equation, the multicomponent coagulation equation exhibits gelation behavior when supplied with a multiplicative kernel. Additionally, a new type of behaviour called localization is observed due to the multivariate nature of the particle size distribution. Here we extend the branching process representation technique, which we introduced to study differential equations in our previous work, and apply it to find a concise probabilistic solution of the multicomponent coagulation equation supplied with monodisperse initial conditions. We also provide short proofs for the gelation time and characterisation the localization phenomenon.

Entropy Estimate for Degenerate SDEs with Applications to Nonlinear Kinetic Fokker–Planck Equations

Entropy Estimate for Degenerate SDEs with Applications to Nonlinear Kinetic Fokker–Planck Equations

Variational methods for the kinetic Fokker–Planck equation

Variational methods for the kinetic Fokker–Planck equation

Markovian dynamics for a quantum/classical system and quantum trajectories

Quantum trajectory techniques have been used in the theory of open systems as a starting point for numerical computations and to describe the monitoring of a quantum system in continuous time. We extend this technique to develop a general approach to the dynamics of quantum/classical hybrid systems. By using two coupled stochastic differential equations, we can describe a classical component and a quantum one which have their own intrinsic dynamics and which interact with each other. A mathematically rigorous construction is given, under the restriction of having a Markovian joint dynamics and of involving only bounded operators on the Hilbert space of the quantum component. An important feature is that, if the interaction allows for a flow of information from the quantum component to the classical one, necessarily the dynamics is dissipative. We show also how this theory is connected to a suitable hybrid dynamical semigroup, which reduces to a quantum dynamical semigroup in the purely quantum case and includes Liouville and Kolmogorov–Fokker–Planck equations in the purely classical case. Moreover, this semigroup allows to compare the proposed stochastic dynamics with various other proposals based on hybrid master equations. Some simple examples are constructed in order to show the variety of physical behaviors which can be described; in particular, a model presenting hidden entanglement is introduced.

Dynamics of nucleation in binary alloys

ABSTRACT Nucleation in binary alloys is studied in the capillary approximation of classical theory. By allowing both the size of the cluster and its composition to vary, the phase transition is investigated in a two-dimensional space. This parametrisation allows to derive diffusivities of the Fokker-Planck equation for the distribution function of clusters from the Cahn–Hilliard equation. It is shown how kinetics, together with thermodynamics, determine the direction of the nucleation current as well as the magnitude of the nucleation rate. The properties of the critical clusters and the direction of the nucleation current at the saddle point are derived for a generic model, from the binodal line to the spinodal limit. The critical clusters are found to exhibit properties that are very similar to that of non-classical theories. Once moderate supersaturation is reached, the interplay between kinetics and thermodynamics is found to invalidate the classical picture by modifying the direction of the nucleation current. The consequences on the magnitude of the rate of nucleation are discussed. The model is then applied to decomposition of FeCr solid solutions and is shown to constitute a reasonable sharp-interface approximation of the diffuse-interface theory of nucleation for the determination of both the cluster properties and the nucleation rate.