One-dimensional Fokker-Planck Equation Research Articles

A Deterministic Adjoint-Based Semi-Analytical Algorithm for Fast Response Change Computations in Proton Therapy

In this paper we propose a solution to the need for a fast particle transport algorithm in Online Adaptive Proton Therapy capable of cheaply, but accurately computing the changes in patient dose metrics as a result of changes in the system parameters. We obtain the proton phase-space density through the product of the numerical solution to the one-dimensional Fokker-Planck equation and the analytical solution to the Fermi-Eyges equation. Moreover, a corresponding adjoint system was derived and solved for the adjoint flux. The proton phase-space density together with the adjoint flux and the metric (chosen as the energy deposited by the beam in a variable region of interest) allowed assessing the accuracy of our algorithm to different perturbation ranges in the system parameters and regions of interest. The algorithm achieved negligible errors () for small Hounsfield unit (HU) perturbation ranges (–40 HU to 40 HU) and small to moderate errors (3% to 17%) – in line with the well-known limitation of adjoint approaches – for large perturbation ranges (–400 HU to 400 HU) in the case of most clinical interest where the region of interest surrounds the Bragg peak. Given these results coupled with the capability of further improving the timing performance it can be concluded that our algorithm presents a viable solution for the specific purpose of Online Adaptive Proton Therapy.

Flux Reconstruction Schemes for Fokker-Planck Equations with Drift-Admitting Jumps

We develop in this work flux reconstruction (FR) schemes for one-dimensional Fokker-Planck equations with drift-admitting jumps, which have applications in describing the propagators of piecewise-smooth stochastic differential equations. Since the propagators are nonsmooth at the jumps of the drift, difficulties arise in finding the corresponding solutions not only theoretically but also numerically. To be more specific, the main difficulties lie in the fact that two matching conditions have to be imposed simultaneously at each jump, i.e., where the the propagator and the probability current are continuous. In this work, we show that the FR method is an ideal choice for designing numerical schemes for solving this problem. The corresponding FR schemes are given in details. Some benchmark examples are also employed to validate the proposed schemes numerically.

Efficient numerical approximation of a non-regular Fokker–Planck equation associated with first-passage time distributions

In neuroscience, the distribution of a decision time is modelled by means of a one-dimensional Fokker–Planck equation with time-dependent boundaries and space-time-dependent drift. Efficient approximation of the solution to this equation is required, e.g., for model evaluation and parameter fitting. However, the prescribed boundary conditions lead to a strong singularity and thus to slow convergence of numerical approximations. In this article we demonstrate that the solution can be related to the solution of a parabolic PDE on a rectangular space-time domain with homogeneous initial and boundary conditions by transformation and subtraction of a known function. We verify that the solution of the new PDE is indeed more regular than the solution of the original PDE and proceed to discretize the new PDE using a space-time minimal residual method. We also demonstrate that the solution depends analytically on the parameters determining the boundaries as well as the drift. This justifies the use of a sparse tensor product interpolation method to approximate the PDE solution for various parameter ranges. The predicted convergence rates of the minimal residual method and that of the interpolation method are supported by numerical simulations.

One-Dimensional Fokker–Planck Equations and Functional Inequalities for Heavy Tailed Densities

We present and discuss connections between the problem of trend to equilibrium for one-dimensional Fokker–Planck equations modeling socio-economic problems, and one-dimensional functional inequalities of the type of Poincaré, Wirtinger and logarithmic Sobolev, with weight, for probability densities with polynomial tails. As main examples, we consider inequalities satisfied by inverse Gamma densities, taking values on mathbb R_+, and Cauchy-type densities, taking values on mathbb R.

On a class of Fokker–Planck equations with subcritical confinement

We study the relaxation to equilibrium for a class of linear one-dimensional Fokker–Planck equations characterized by a particular subcritical confinement potential. An interesting feature of this class of Fokker–Planck equations is that, for any given probability density $e(x)$, the diffusion coefficient can be built to have $e(x)$ as steady state. This representation of the equilibrium density can be fruitfully used to obtain one-dimensional Wirtinger-type inequalities and to recover, for a sufficiently regular density $e(x)$, a polynomial rate of convergence to equilibrium. Numerical results then confirm the theoretical analysis, and allow to conjecture that convergence to equilibrium with positive rate still holds for steady states characterized by a very slow polynomial decay at infinity.

Evolutionary dynamics and eigenspectrum of confluent Heun equation

We consider a biological population evolving under the joint action of selection, mutation and random genetic drift. The evolutionary dynamics are described by a one-dimensional Fokker–Planck equation whose eigenfunctions obey a confluent Heun equation. These eigenfunctions are expanded in an infinite series of orthogonal Jacobi polynomials and the expansion coefficients are found to obey a three-term recursion equation. Using scaling ideas, we obtain an expression for the expansion coefficients and an analytical estimate of the number of terms required in the series for an accurate determination of the eigenfunction. The eigenvalue spectrum is studied using a perturbation theory for weak selection and numerically for strong selection. In the latter case, we find that the eigenvalue for the first excited state exhibits a sharp transition: for mutation rate below one, the eigenvalue increases linearly with increasing mutation rate and then remains a constant; higher eigenvalues are found to display a more complex behavior.

Coercivity, hypocoercivity, exponential time decay and simulations for discrete Fokker–Planck equations

In this article, we propose and study several discrete versions of homogeneous and inhomogeneous one-dimensional Fokker-Planck equations. In particular, for these dis-cretizations of velocity and space, we prove the exponential convergence to the equilibrium of the solutions, for time-continuous equations as well as for time-discrete equations. Our method uses new types of discrete Poincare inequalities for a two-direction discretization of the derivative in velocity. For the inhomogeneous problem, we adapt hypocoercive methods to the discrete cases.

First-passage distributions for the one-dimensional Fokker-Planck equation

We present an analytical framework to study the first-passage (FP) and first-return (FR) distributions for the broad family of models described by the one-dimensional Fokker-Planck equation in finite domains, identifying general properties of these distributions for different classes of models. When in the Fokker-Planck equation the diffusion coefficient is positive (nonzero) and the drift term is bounded, as in the case of a Brownian walker, both distributions may exhibit a power-law decay with exponent -3/2 for intermediate times. We discuss how the influence of an absorbing state changes this exponent. The absorbing state is characterized by a vanishing diffusion coefficient and/or a diverging drift term. Remarkably, the exponent of the Brownian walker class of models is still found, as long as the departure and arrival regions are far enough from the absorbing state, but the range of times where the power law is observed narrows. Close enough to the absorbing point, though, a new exponent may appear. The particular value of the exponent depends on the behavior of the diffusion and the drift terms of the Fokker-Planck equation. We focus on the case of a diffusion term vanishing linearly at the absorbing point. In this case, the FP and FR distributions are similar to those of the voter model, characterized by a power law with exponent -2. As an illustration of the general theory, we compare it with exact analytical solutions and extensive numerical simulations of a two-parameter voter-like family models. We study the behavior of the FP and FR distributions by tuning the importance of the absorbing points throughout changes of the parameters. Finally, the possibility of inferring relevant information about the steady-sate probability distribution of a model from the FP and FR distributions is addressed.

One-Dimensional Fokker–Planck Equation with Quadratically Nonlinear Quasilocal Drift

The Fokker–Planck equation in one-dimensional spacetime with quadratically nonlinear nonlocal drift in the quasilocal approximation is reduced with the help of scaling of the coordinates and time to a partial differential equation with a third derivative in the spatial variable. Determining equations for the symmetries of the reduced equation are derived and the Lie symmetries are found. A group invariant solution having the form of a traveling wave is found. Within the framework of Adomian’s iterative method, the first iterations of an approximate solution of the Cauchy problem are obtained. Two illustrative examples of exact solutions are found.

Effective colloidal interactions in rotating magnetic fields.

Non-equilibrium, steady-state effective pair potentials of micron-sized superparamagnetic particles in rotating magnetic fields are obtained vs. field frequency and amplitude. Trajectories of center-to-center distance between particle pairs from Brownian dynamic simulations, which were previously matched to experimental measurements, are analyzed to obtain local drift and diffusion coefficients. These coefficients are used to obtain effective interaction potentials from solving a one-dimensional Fokker-Planck equation. Biased sampling of the effective energy landscape was implemented by intermittent switching between the field of interest and a repulsive field. Our findings show how the shape and attractive well-depth of pair interactions can be tuned by changing field frequency and amplitude.

Communication: On the diffusion tensor in macroscopic theory of cavitation.

The classical description of nucleation of cavities in a stretched fluid relies on a one-dimensional Fokker-Planck equation (FPE) in the space of their sizes r, with the diffusion coefficient D(r) constructed for all r from macroscopic hydrodynamics and thermodynamics, as shown by Zeldovich. When additional variables (e.g., vapor pressure) are required to describe the state of a bubble, a similar approach to construct a diffusion tensor D^ generally works only in the direct vicinity of the thermodynamic saddle point corresponding to the critical nucleus. It is shown, nevertheless, that "proper" kinetic variables to describe a cavity can be selected, allowing to introduce D^ in the entire domain of parameters. In this way, for the first time, complete FPE's are constructed for viscous volatile and inertial fluids. In the former case, the FPE with symmetric D^ is solved numerically. Alternatively, in the case of an inertial fluid, an equivalent Langevin equation is considered; results are compared with analytics. The suggested approach is quite general and can be applied beyond the cavitation problem.

Discrete Beckner inequalities via the Bochner–Bakry–Emery approach for Markov chains

Discrete convex Sobolev inequalities and Beckner inequalities are derived for time-continuous Markov chains on finite state spaces. Beckner inequalities interpolate between the modified logarithmic Sobolev inequality and the Poincaré inequality. Their proof is based on the Bakry–Emery approach and on discrete Bochner-type inequalities established by Caputo, Dai Pra and Posta and recently extended by Fathi and Maas for logarithmic entropies. The abstract result for convex entropies is applied to several Markov chains, including birth-death processes, zero-range processes, Bernoulli–Laplace models, and random transposition models, and to a finite-volume discretization of a one-dimensional Fokker–Planck equation, applying results by Mielke.

Convergence to equilibrium in the free Fokker–Planck equation with a double-well potential

We consider the one-dimensional free Fokker-Planck equation $\frac{\partial \mu_t}{\partial t} = \frac{\partial}{\partial x} \left[ \mu_t \left( \frac12 V' - H\mu_t \right) \right]$, where $H$ denotes the Hilbert transform and $V$ is a particular double-well quartic potential, namely $V(x) = \frac14 x^4 + \frac{c}{2} x^2$, with $-2 \le c < 0$. We prove that the solution $(\mu_t)_{t \ge 0}$ of this PDE converges to the equilibrium measure $\mu_V$ as $t$ goes to infinity, which provides a first result of convergence in a non-convex setting. The proof involves free probability and quadratic differentials techniques.

The mean shape of transition and first-passage paths

Based on the one-dimensional Fokker-Planck equation in an arbitrary free energy landscape including a general inhomogeneous diffusivity profile, we analytically calculate the mean shape of transition paths and first-passage paths, where the shape of a path is defined as the kinetic profile in the plane spanned by the mean time and the position. The transition path ensemble is the collection of all paths that do not revisit the start position x(A) and that terminate when first reaching the final position x(B). In contrast, a first-passage path can revisit its start position x(A) before it terminates at x(B). Our theoretical framework employs the forward and backward Fokker-Planck equations as well as first-passage, passage, last-passage, and transition-path time distributions, for which we derive the defining integral equations. We show that the mean shape of transition paths, in other words the mean time at which the transition path ensemble visits an intermediate position x, is equivalent to the mean first-passage time of reaching the position x(A) when starting from x without ever visiting x(B). The mean shape of first-passage paths is related to the mean shape of transition paths by a constant time shift. Since for a large barrier height U, the mean first-passage time scales exponentially in U, while the mean transition path time scales linearly inversely in U, the time shift between first-passage and transition path shapes is substantial. We present explicit examples of transition path shapes for linear and harmonic potentials and illustrate our findings by trajectories obtained from Brownian dynamics simulations.

Geodesic convexity of the relative entropy in reversible Markov chains

We consider finite-dimensional, time-continuous Markov chains satisfying the detailed balance condition as gradient systems with the relative entropy E as driving functional. The Riemannian metric is defined via its inverse matrix called the Onsager matrix K. We provide methods for establishing geodesic λ-convexity of the entropy and treat several examples including some discretizations of one-dimensional Fokker–Planck equations.

The nonlinear Fokker-Planck equation: comparison of the classical and quantum (boson and fermion) characteristics

A "free energy" functional F is studied for the one-dimensional Fokker-Planck equation (FPE). Using F, a notion of distance between a global Maxwellian function M, which is a stationary explicit solution of the FPE, and an arbitrary solution f (with the same total density ρ at the fixed moment t) is introduced. In this way we generalize the important Kullback-Leibler distance. A comparison between classical and quantum characteristics of the FPE solutions is made.

Classes of N-Dimensional Nonlinear Fokker-Planck Equations Associated to Tsallis Entropy

Several previous results valid for one-dimensional nonlinear Fokker-Planck equations are generalized to N-dimensions. A general nonlinear N-dimensional Fokker-Planck equation is derived directly from a master equation, by considering nonlinearitiesin the transition rates. Using nonlinear Fokker-Planck equations, the H-theorem is proved;for that, an important relation involving these equations and general entropic forms is introduced. It is shown that due to this relation, classes of nonlinear N-dimensional Fokker-Planck equations are connected to a single entropic form. A particular emphasis is given to the class of equations associated to Tsallis entropy, in both cases of the standard, and generalized definitions for the internal energy.

Variational supersymmetric approach to evaluate Fokker–Planck probability

In this work we introduce a method to determine the time dependent probability density for the one-dimensional Fokker–Planck equation. The treatment is based in an analysis of the Schrödinger equation through the variational method associated to the formalism of supersymmetric quantum mechanics (SQM). The approach uses an ansatz for the superpotential which allows us to obtain the trial functions of the variational method. The hierarchy of effective Hamiltonians permits us to determine the variational eigenfunctions and energies of the excited states to the evaluation of the probability. The symmetric bistable potential is used to illustrate the approach whose results are compared with results obtained by the state-dependent diagonalization method and by direct numerical calculation.

Equivalent single-junction model for two-junction quantum interferometers with small inductance in the presence of noise

We illustrate, in the limit of small values of the parameter β , a method to analyze the stochastic equations governing the time evolution of a two-junction quantum interferometer in the presence of noise. The analysis takes into account the different characteristic times of the two processes involved in the system, in such a way that the appropriate probability density function for each process is derived by means of one-dimensional Fokker–Planck equations. Therefore, the present analysis is equivalent to a generalization of the reduced single-junction model in the case of thermal fluctuations in the junctions. Voltage–current characteristics, transfer functions and circulating currents in 0-SQUIDs and π -SQUIDs are calculated by means of the proposed analytic approach.

On the solution of the Fokker–Planck equation using a high-order reduced basis approximation

The numerical solution of the one-dimensional Fokker–Planck equation for describing the evolution of the configuration probability density function associated with kinetic theory models in polymer dynamics is presented. The finitely extensible non-linear elastic (FENE) model is considered and the spectral element discretisation is applied using an adaptive reduced basis technique. This technique facilitates a significant reduction in the number of degrees of freedom required in the approximation through the determination of an optimal basis using the singular value decomposition (SVD). The basis functions are constructed dynamically so that the numerical approximation is optimal in the current finite-dimensional subspace of the solution space. This is achieved through basis enrichment and projection. The reduced basis method is extended to the high-dimensional Fokker–Planck equation, using the d-dumbbell FENE model, by consideration of a high-dimensional singular value decomposition. Some numerical results are presented to demonstrate the efficiency of the numerical scheme.