Backward Fokker-Planck Equation Research Articles

Existence and uniqueness of solutions for forward and backward nonlocal Fokker-Planck equations with time-dependent coefficients

Based on the Feynman-Kac formula, we develop a new approach to show the global existence and uniqueness of the solutions for the forward and backward time-dependent multidimensional nonlocal Fokker-Planck equations, which are associated with a class of time-dependent stochastic differential equations driven by symmetric (or asymmetric) non-Gaussian Lévy process. First, after deriving the forward time-dependent multidimensional nonlocal Fokker-Planck equation, the Feynman-Kac formula for the forward nonlocal Fokker-Planck equation is established by using Itô's formula and techniques for the backward nonlocal Fokker-Planck equation corresponding to the backward stochastic differential equation driven by Lévy process. Then, from the Feynman-Kac formula, we prove the global existence and uniqueness of the solutions for the forward and backward time-dependent multidimensional nonlocal Fokker-Planck equations by using techniques from stochastic analysis. Finally, the dynamic evolution of the probability density function corresponding to the stochastic model for the MeKS network (reflecting the interactions among the MecA complex, comK, comS) is investigated over an extended period by Feynman-Kac formula and Monte Carlo simulations. In particular, we reveal the influence of symmetric and asymmetric stable noises as well as tempered stable noise on the probability density function corresponding to the stochastic MeKS network model.

Brownian bridges for stochastic chemical processes-An approximation method based on the asymptotic behavior of the backward Fokker-Planck equation.

A Brownian bridge is a continuous random walk conditioned to end in a given region by adding an effective drift to guide paths toward the desired region of phase space. This idea has many applications in chemical science where one wants to control the endpoint of a stochastic process-e.g., polymer physics, chemical reaction pathways, heat/mass transfer, and Brownian dynamics simulations. Despite its broad applicability, the biggest limitation of the Brownian bridge technique is that it is often difficult to determine the effective drift as it comes from a solution of a Backward Fokker-Planck (BFP) equation that is infeasible to compute for complex or high-dimensional systems. This paper introduces a fast approximation method to generate a Brownian bridge process without solving the BFP equation explicitly. Specifically, this paper uses the asymptotic properties of the BFP equation to generate an approximate drift and determine ways to correct (i.e., re-weight) any errors incurred from this approximation. Because such a procedure avoids the solution of the BFP equation, we show that it drastically accelerates the generation of conditioned random walks. We also show that this approach offers reasonable improvement compared to other sampling approaches using simple bias potentials.

Making Birth–Death Processes from Backward Fokker–Planck Equations for Computing Expectations in Langevin Systems

A method to direct evaluation of expectations for Langevin systems (stochastic differential equations) is proposed. The method is based on a birth-death process which is derived using combinations of dummy variables and It{\^o} formula. As a pedagogical example, a double-well system and expectations for sigmoid-type functions are used. It is shown that the proposed method has some merits from computational point of view; only one time-integration for the birth-death process gives expectations for various initial conditions in the original Langevin systems. Furthermore, the same time-integration result is available for computing various center positions of the sigmoid-type functions.

Stochastically gated local and occupation times of a Brownian particle.

We generalize the Feynman-Kac formula to analyze the local and occupation times of a Brownian particle moving in a stochastically gated one-dimensional domain. (i) The gated local time is defined as the amount of time spent by the particle in the neighborhood of a point in space where there is some target that only receives resources from (or detects) the particle when the gate is open; the target does not interfere with the motion of the Brownian particle. (ii) The gated occupation time is defined as the amount of time spent by the particle in the positive half of the real line, given that it can only cross the origin when a gate placed at the origin is open; in the closed state the particle is reflected. In both scenarios, the gate randomly switches between the open and closed states according to a two-state Markov process. We derive a stochastic, backward Fokker-Planck equation (FPE) for the moment-generating function of the two types of gated Brownian functional, given a particular realization of the stochastic gate, and analyze the resulting stochastic FPE using a moments method recently developed for diffusion processes in randomly switching environments. In particular, we obtain dynamical equations for the moment-generating function, averaged with respect to realizations of the stochastic gate.

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.

First-passage-time statistics of a Brownian particle driven by an arbitrary unidimensional potential with a superimposed exponential time-dependent drift

In one-dimensional systems, the dynamics of a Brownian particle are governed by the force derived from a potential as well as by diffusion properties. In this work, we obtain the first-passage-time statistics of a Brownian particle driven by an arbitrary potential with an exponential temporally decaying superimposed field up to a prescribed threshold. The general system analyzed here describes the sub-threshold signal integration of integrate-and-fire neuron models, of any kind, supplemented by an adaptation-like current, whereas the first-passage-time corresponds to the declaration of a spike. Following our previous studies, we base our analysis on the backward Fokker–Planck equation and study the survival probability and the first-passage-time density function in the space of the initial condition. By proposing a series solution we obtain a system of recurrence equations, which given the specific structure of the exponential time-dependent drift, easily admit a simpler Laplace representation. Naturally, the present general derivation agrees with the explicit solution we found previously for the Wiener process in (2012 J. Phys. A: Math. Theor. 45 185001). However, to demonstrate the generality of the approach, we further explicitly evaluate the first-passage-time statistics of the underlying Ornstein–Uhlenbeck process. To test the validity of the series solution, we extensively compare theoretical expressions with the data obtained from numerical simulations in different regimes. As shown, agreement is precise whenever the series is truncated at an appropriate order. Beyond the fact that both the Wiener and Ornstein–Uhlenbeck processes have a direct interpretation in the context of neuronal models, given their ubiquity in different fields, our present results will be of interest in other settings where an additive state-independent temporal relaxation process is being developed as the particle diffuses.

Fluctuations of Time Averages for Langevin Dynamics in a Binding Force Field

We derive a simple formula for the fluctuations of the time average x(t) around the thermal mean <x>(eq) for overdamped brownian motion in a binding potential U(x). Using a backward Fokker-Planck equation, introduced by Szabo, Schulten, and Schulten in the context of reaction kinetics, we show that for ergodic processes these finite measurement time fluctuations are determined by the Boltzmann measure. For the widely applicable logarithmic potential, ergodicity is broken. We quantify the large nonergodic fluctuations and show how they are related to a superaging correlation function.

Rapidly driven nanoparticles: Mean first-passage times and relaxation of the magnetic moment

We present an analytical method of calculating the mean first-passage times (MFPTs) for the magnetic moment of a uniaxial nanoparticle which is driven by a rapidly rotating, circularly polarized magnetic field and interacts with a heat bath. The method is based on the solution of the equation for the MFPT derived from the two-dimensional backward Fokker-Planck equation in the rotating frame. We solve these equations in the high-frequency limit and perform precise, numerical simulations which verify the analytical findings. The results are used for the description of the rates of escape from the metastable domains which in turn determine the magnetic relaxation dynamics. A main finding is that the presence of a rotating field can cause a drastic decrease of the relaxation time and a strong magnetization of the nanoparticle system. The resulting stationary magnetization along the direction of the easy axis is compared with the mean magnetization following from the stationary solution of the Fokker-Planck equation.

Random walk methods for scalar transport problems subject to Dirichlet, Neumann and mixed boundary conditions

Three stochastic-based methods are proposed for solving unsteady scalar transport problems in bounded, single-phase domains. The first (Method I), a local solution appropriate to problems having Dirichlet conditions, adapts a well-known local stochastic solution of a backward Fokker–Planck equation to scalar transport. Method II, a local solution applicable to Dirichlet, Neumann and/or mixed initial boundary value problems (IBVPs), and representing a time-dependent extension of a recently reported heuristic steady solution, provides a straightforward addition to the limited collection of techniques available for Neumann and mixed problems. This approach is shown to be equivalent to a long-standing, rigorous low-order solution and, in addition, allows development of a probabilistic-based analytical solution to Neumann problems, stated in terms of an exit probability. Method III, a global solution, likewise suitable for Neumann and mixed IBVPs, follows by combined application of domain boundary Taylor expansions and Method I. This approach is shown to be computationally equivalent to a global version of Method II.

Dynamical and thermal effects in nanoparticle systems driven by a rotating magnetic field

We study dynamical and thermal effects that are induced in nanoparticle systems by a rotating magnetic field. Using the deterministic Landau-Lifshitz equation and appropriate rotating coordinate systems, we derive the equations that characterize the steady-state precession of the nanoparticle magnetic moments and study a stability criterion for this type of motion. On this basis, we describe (i) the influence of the rotating field on the stability of the small-angle precession, (ii) the dynamical magnetization of nanoparticle systems, and (iii) the switching of the magnetic moments under the action of the rotating field. Using the backward Fokker-Planck equation, which corresponds to the stochastic Landau-Lifshitz equation, we develop a method for calculating the mean residence times that the driven magnetic moments dwell in the up and down states. Within this framework, the features of the induced magnetization and magnetic relaxation are elucidated.

Residence Time Statistics for Normal and Fractional Diffusion in a Force Field

We investigate statistics of occupation times for an over-damped Brownian particle in an external force field. A backward Fokker-Planck equation introduced by Majumdar and Comtet describing the distribution of occupation times is solved. The solution gives a general relation between occupation time statistics and probability currents which are found from solutions of the corresponding problem of first passage time. This general relationship between occupation times and first passage times, is valid for normal Markovian diffusion and for non-Markovian sub-diffusion, the latter modeled using the fractional Fokker-Planck equation. For binding potential fields we find in the long time limit ergodic behavior for normal diffusion, while for the fractional framework weak ergodicity breaking is found, in agreement with previous results of Bel and Barkai on the continuous time random walk on a lattice. For non-binding potential rich physical behaviors are obtained, and classification of occupation time statistics is made possible according to whether or not the underlying random walk is recurrent and the averaged first return time to the origin is finite. Our work establishes a link between fractional calculus and ergodicity breaking.

Drowsy cheetah hunting antelopes: a diffusing predator seeking fleeing prey

We consider a system of three random walkers (a ‘cheetah’ surrounded by two‘antelopes’) diffusing in one dimension. The cheetah and the antelopes diffuse, butthe antelopes experience in addition a deterministic relative drift velocity, awayfrom the cheetah, proportional to their distance from the cheetah, such that theytend to move away from the cheetah with increasing time. Using the backwardFokker–Planck equation we calculate, as a function of their initial separations,the probability that the cheetah has caught neither antelope after infinite time.

Vicious walkers in a potential

We consider N vicious walkers moving in one dimension in a one-body potential v(x). Using the backward Fokker-Planck equation we derive exact results for the asymptotic form of the survival probability Q(x,t) of vicious walkers initially located at (x_1,...,x_N) = x, when v(x) is an arbitrary attractive potential. Explicit results are given for a square-well potential with absorbing or reflecting boundary conditions at the walls, and for a harmonic potential with an absorbing or reflecting boundary at the origin and the walkers starting on the positive half line. By mapping the problem of N vicious walkers in zero potential onto the harmonic potential problem, we rederive the results of Fisher [J. Stat. Phys. 34, 667 (1984)] and Krattenthaler et al. [J. Phys. A 33}, 8835 (2000)] respectively for vicious walkers on an infinite line and on a semi-infinite line with an absorbing wall at the origin. This mapping also gives a new result for vicious walkers on a semi-infinite line with a reflecting boundary at the origin: Q(x,t) \sim t^{-N(N-1)/2}.

Optimal paths and the calculation of state selection probabilities.

The addition of noise to a dynamical system means that initial states near points of instability may no longer decay to a unique stable state. A common example of this behavior occurs in a dynamical system with two degrees of freedom and with two or more stable states. If the initial state of the system is near the separatrices bounding the basins of attraction of these stable states, then the addition of noise to the system means that there is a nonzero probability that the stable state selected is in a different basin of attraction to that of the initial state. We discuss a method of calculating these state-selection probabilities based on a path-integral representation of the stochastic dynamics. The relationship of this approach to a method based on the solution of the backward Fokker-Planck equation is particularly stressed, since this was used in previous studies of problems of this type. However, while the method based on the backward Fokker-Planck equation is a powerful one for systems with one degree of freedom, in systems with more degrees of freedom it is much less useful. Since the standard method of solution in this case involves a series of mappings onto a deterministic dynamics which is simply the classical dynamics associated with the path-integral formulation, we argue that for systems with more than one degree of freedom, the path-integral method is a very natural way of calculating state-selection probabilities. We illustrate this on a simple example taken from population biology, and find that the state-selection probabilities are in excellent agreement with Monte Carlo simulations.

On the use of the backward Fokker–Planck equation to calculate range profiles

A new mathematical model for range calculations of implanted impurities – the backward Fokker–Planck equation – is derived from the backward Boltzmann transport equation using decomposition of the scattering integral for small energy transfers in nuclear collisions. The present model describes ion transport processes dominated by collisions with small energy transfer such as the diffusion-like motion of the angular direction. Furthermore, we extend the present model to take into account single collisions events in high-energy implantation.

First-passage-time transfer functions for groundwater tracer tests conducted in radially convergent flow

Forced-gradient groundwater tracer tests may be conducted using a variety of hydraulic schemes, so it is useful to have simple semi-analytic models available that can examine various injection/withdrawal scenarios. Models for radially convergent tracer tests are formulated here as transfer functions, which allow complex tracer test designs to be simulated by a series of simple mathematical expressions. These mathematical expressions are given in Laplace space, so that transfer functions may be placed in series by simple multiplication. Predicted breakthrough is found by numerically inverting the composite transfer function to the time-domain, using traditional computer programs or commercial mathematical software. Transport is assumed to be dictated by a radially convergent or uniform flow field, and is based upon an exact first-passage-time solution of the backward Fokker–Planck equation. These methods are demonstrated by simulating a weak-dipole tracer test conducted in a fractured granite formation, where mixing in the injection borehole is non-ideal.

The multivariate Langevin and Fokker–Planck equations

A novel derivation of the Langevin equation that was recently presented in this journal for a univariate continuous Markov process is generalized here to the more widely applicable multivariate case. The companion multivariate forward and backward Fokker–Planck equations are also derived. The derivations require just a few modest assumptions, and are driven by a self-consistency condition and some established theorems of random variable theory and ordinary calculus. The constructive nature of the derivations shows why a multivariate continuous Markov process must evolve according to equations of the canonical Langevin and Fokker–Planck forms, and also sheds new light on some uniqueness issues. The need for self-consistency in the time-evolution equations of both Markovian and non-Markovian stochastic processes is emphasized, and it is pointed out that for a great many non-Markovian processes self-consistency can be ensured most easily through the multivariate Markov theory.

Extension of eigenfunction-expansion solutions of a fokker-planck equation—II. Second order system

In an earlier paper the authors presented results for eigenfunction-expansion solutions to the forward Fokker-Planck equation associated with a specific, non-linear, first-order system subject to white noise excitation. This work is concerned with eigenfunction-expansion solutions to the forward and backward Fokker-Planck equations associated with a specific, non-linear, second-order system subject to white noise excitation. Expansion terms through the fourth-order have been generated using a digital computer. Using this new information, inverted Domb-Sykes plots revealed a pattern in the coefficients for certain values of the parameters. Through this pattern, Dingle's theory of terminants was used to recast the series into a more favorable computational form.

A stochastic analysis of the growth of competing microbial populations in a continuous biochemical reactor

The dynamics of a chemostat in which two microbial populations grow and compete for a common substrate is examined. It is shown that the two populations cannot coexist in a spatially uniform environment which is subject to time invariant external influences unless the dilution rate takes on one of a discrete set of special values. The dynamics of the same system are next considered in the stochastic environment created by random fluctuations of the dilution rate about a value that allows coexistence. The information needed for the description of the random process of the state of the chemostat is obtained from the transition probability density function. By modeling the system as a Markov process continuous in time and space, the transition probability density is obtained as solution of the Fokker-Planck equation. Analytical and numerical solutions of this equation show that extinction of either one population or the other will ultimately take place. The time required for extinction, the evolution of the mean composition with time, the steady states of the latter and the dependence of all the above on the intensity of the random noise are also calculated using constants appropriate to the competition of E. coli and Spirillum sp. The question of making predictions as to which population is the more likely to become extinct is treated finally, and the probabilities of extinction are calculated as solutions of the steady state version of the backward Fokker-Planck (Kolmogorov) equation.