Fokker-Planck Differential Equation Research Articles

MEASUREMENTS AGAINST NOISE BACKGROUND AND WITH FLUCTUATIONS

When performing non-contact and remote measurements of various parameters, additional obstacles arise due to the influence of the intermediate medium and the measuring path between different links of the measuring system. These obstacles include:
 
 Noises arising in communication paths, intermediate media, complexes of measuring and auxiliary equipment. These noises are superimposed on the useful signal and distort it.
 Fluctuations of the general and selective energy absorption of the useful signal in the intermediate medium and measuring communication path. In all these cases, it is necessary to isolate the useful signal and restore its original true value and nature of non-stationarity.
 
 The task of filtering in its general form has not been completely solved until now. In order to obtain maximum information about the useful signal, such an operator L is selected that the function Y(τ)=L[X(τ)] slightly differs from the useful signal U(τ), where X(τ)= U(τ)+ V(τ ); V(τ) is noise (interference). To determine the values of the function Y(τ) at a certain time τ_0, all previous values of the function X(τ) from the interval -∞≤τ≤τ_0 should be used. To distinguish the functions Y(τ) and U(τ) the variance of the difference of these functions is used. The problem of Wiener filtering has been solved by various authors, in particular the Kalman-Busy filter equation for linear Gaussian systems. Optimal (suboptimal) Bayesian filters are constructed for a nonlinear stochastic system disturbed by "white" Gaussian noise with continuous time and discrete measurements that minimizes the root-mean-square error of estimation. The exact solution of this problem in the interval between the moments of measurement results is subject to the Fokker-Planck differential equation with private derivatives.
 Considered examples: measuring the temperature and pressure of the environment under investigation, distorted by various random factors in the form of "white" noise, as well as tracking the coordinates and velocities of an aircraft performing a maneuver in the horizontal plane - as test tasks in the construction of a sigma-point Kalman filter - as a multi-cube Kalman filter.

Numerical approximation of the first-passage time distribution of time-varying diffusion decision models: A mesh-free approach

Sequential sampling theory is the dominant theoretical framework for explaining human decision making behavior. In this theory, the decision process is determined by a stochastic process in a bounded domain, with the likelihood function of the model being the first passage-time distribution of this process. However, many sequential sampling models have no analytic formulation for the first passage-time distribution, making these models difficult and computationally taxing to fit. Thus, developing an efficient and general approximation procedure for the first-passage time distribution of sequential sampling models is crucial for the field of mathematical psychology. Here, we have developed a numerical algorithm based on the mesh-free techniques for numerically solving the Fokker–Planck differential equation, which allows us to approximate the first passage-time distribution of some important classes of sequential sampling models, including the urgency signal model, Ornstein–Uhlenbeck model, and collapsing threshold model. Since the considered equation is a singular partial differential equation in a moving boundary domain, the proposed numerical method first transforms the equation to a regular domain, and then the solution is approximated based on the collocation method. However, since satisfying the boundary conditions in the collocation algorithms can be very tricky, we have imposed the boundary conditions in the basis functions. Furthermore, for the time-discrete and the full-discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method. The performance of the proposed method is illustrated by solving several examples.

Description of the Distribution Law and Non-Linear Dynamics of Growth of Comments Number in News and Blogs Based on the Fokker-Planck Equation

The article considers stationary and dynamic distributions of news by the number of comments. The processing of the observed data showed that static distribution of news by the number of comments relating to that news obeys a power law, and the dynamic distribution (the change in number of comments over time) in some cases has an S-shaped character, and in some cases a more complex two-stage character. This depends on the time interval between the appearance of a comment at the first level and a comment attached to that comment. The power law for the stationary probability density of news distribution by the number of comments can be obtained from the solution of the stationary Fokker-Planck equation, if a number of assumptions are made in its derivation. In particular, we assume that the drift coefficient μ(x) responsible in the Fokker-Planck equation for a purposeful change in the state of system x (x is the current number of comments on that piece of news) linearly depends on the state x, and the diffusion coefficient D(x) responsible for a random change depends quadratically on x. The solution of the unsteady Fokker-Planck differential equation with these assumptions made it possible to obtain an analytical equation for the probability density of transitions between the states of the system per unit of time, which is in good agreement with the observed data, considering the effect of the delay time between the appearance of the first-level comment and the comment on that comment.

Deterministic approach to skyrmionic dynamics at nonzero temperatures: Pinning sites and racetracks

The discovery of room-temperature skyrmions in some magnetic materials has boosted the investigation of their dynamics in view of future applications. We study the dynamics of skyrmions in the presence of defects or borders using a deterministic (finding and solving a deterministic Fokker-Planck differential equation) while probabilistic (the solution is the probability density for the presence of a skyrmion) approach. The probability that a skyrmion becomes trapped in a pinning center or the probability of the survival of a skyrmion along a racetrack are obtained as a function of temperature. The present work can be relevant in the design of skyrmionic devices where the probability of finding skyrmions at a given position and time is crucial for their feasibility.

Kolmogorov’s forward PIDE and forward transition rates in life insurance

We consider a doubly stochastic Markov chain, where the transition intensities are modelled as diffusion processes. Here we present a forward partial integro-differential equation (PIDE) for the transition probabilities. This is a generalisation of Kolmogorov’s forward differential equation. In this set-up, we define forward transition rates, generalising the concept of forward rates, e.g. the forward mortality rate. These models are applicable in e.g. life insurance mathematics, which is treated in the paper. The results presented follow from the general forward PIDE for stochastic processes, of which the Fokker–Planck differential equation and Kolmogorov’s forward differential equation are the two most known special cases. We end the paper by considering the semi-Markov case, which can also be considered a special case of a general forward partial integro-differential equation.

Measurement of vascular water transport in human subjects using time‐resolved pulsed arterial spin labelling

Most approaches to arterial spin labelling (ASL) data analysis aim to provide a quantitative measure of the cerebral blood flow (CBF). This study, however, focuses on the measurement of the transfer time of blood water through the capillaries to the parenchyma (referred to as the capillary transfer time, CTT) as an alternative parameter to characterise the haemodynamics of the system. The method employed is based on a non-compartmental model, and no measurements need to be added to a common time-resolved ASL experiment. Brownian motion of labelled spins in a potential was described by a one-dimensional general Langevin equation as the starting point, and as a Fokker-Planck differential equation for the averaged distribution of labelled spins at the end point, which takes into account the effects of flow and dispersion of labelled water by the pseudorandom nature of the microvasculature and the transcapillary permeability. Multi-inversion time (multi-TI) ASL data were acquired in 14 healthy subjects on two occasions in a test-retest design, using a pulsed ASL sequence and three-dimensional gradient and spin echo (3D-GRASE) readout. Based on an error analysis to predict the size of a region of interest (ROI) required to obtain reasonably precise parameter estimates, data were analysed in two relatively large ROIs, i.e. the occipital lobe (OC) and the insular cortex (IC). The average values of CTT in OC were 260 ± 60 ms in the first experiment and 270 ± 60 ms in the second experiment. The corresponding IC values were 460 ± 130 ms and 420 ± 139 ms, respectively. Information related to the water transfer time may be important for diagnostics and follow-up of cerebral conditions or diseases characterised by a disrupted blood-brain barrier or disturbed capillary blood flow.

A Model for Service Life Control of Selected Device Systems

Abstract This paper presents a way of determining distribution of limit state exceedence time by a diagnostic parameter which determines accuracy of maintaining zero state. For calculations it was assumed that the diagnostic parameter is deviation from nominal value (zero state). Change of deviation value occurs as a result of destructive processes which occur during service. For estimation of deviation increasing rate in probabilistic sense, was used a difference equation from which, after transformation, Fokker-Planck differential equation was obtained [4, 11]. A particular solution of the equation is deviation increasing rate density function which was used for determining exceedance probability of limit state. The so-determined probability was then used to determine density function of limit state exceedance time, by increasing deviation. Having at disposal the density function of limit state exceedance time one determined service life of a system of maladjustment. In the end, a numerical example based on operational data of selected aircraft [weapon] sights was presented. The elaborated method can be also applied to determining residual life of shipboard devices whose technical state is determined on the basis of analysis of values of diagnostic parameters.

Asymptotic and Finite User PER Analysis of Successive Interference Cancellation for DS-CDMA

An expression is derived for the average Packet Error Rate (PER) of a Successive Interference Canceller (SIC) for DS-CDMA when the number of users asymptotically tends to infinity. The asymptotic probability density function of the interference power is governed by a Fokker-Planck differential equation with drift and (asymptotically vanishing) diffusion depending on the PER function of the adopted forward error correcting code (FEC). In addition to the asymptotic solution for the PER, a particle-based algorithm is also developed for computing efficiently the PER in the finite user case.

On the Per-Sample Capacity of Nondispersive Optical Fibers

The capacity of the channel defined by the stochastic nonlinear Schr\"odinger equation, which includes the effects of the Kerr nonlinearity and amplified spontaneous emission noise, is considered in the case of zero dispersion. In the absence of dispersion, this channel behaves as a collection of parallel per-sample channels. The conditional probability density function of the nonlinear per-sample channels is derived using both a sum-product and a Fokker-Planck differential equation approach. It is shown that, for a fixed noise power, the per-sample capacity grows unboundedly with input signal. The channel can be partitioned into amplitude and phase subchannels, and it is shown that the contribution to the total capacity of the phase channel declines for large input powers. It is found that a two-dimensional distribution with a half-Gaussian profile on the amplitude and uniform phase provides a lower bound for the zero-dispersion optical fiber channel, which is simple and asymptotically capacity-achieving at high signal-to-noise ratios (SNRs). A lower bound on the capacity is also derived in the medium-SNR region. The exact capacity subject to peak and average power constraints is numerically quantified using dense multiple ring modulation formats. The differential model underlying the zero-dispersion channel is reduced to an algebraic model, which is more tractable for digital communication studies, and in particular it provides a relation between the zero-dispersion optical channel and a $2 \times 2$ multiple-input multiple-output Rician fading channel. It appears that the structure of the capacity-achieving input distribution resembles that of the Rician fading channel, i.e., it is discrete in amplitude with a finite number of mass points, while continuous and uniform in phase.

Generalized pricing formulas for stochastic volatility jump diffusion models applied to the exponential Vasicek model

Path integral techniques for the pricing of financial options are mostly based on models that can be recast in terms of a Fokker-Planck differential equation and that, consequently, neglect jumps and only describe drift and diffusion. We present a method to adapt formulas for both the path-integral propagators and the option prices themselves, so that jump processes are taken into account in conjunction with the usual drift and diffusion terms. In particular, we focus on stochastic volatility models, such as the exponential Vasicek model, and extend the pricing formulas and propagator of this model to incorporate jump diffusion with a given jump size distribution. This model is of importance to include non-Gaussian fluctuations beyond the Black-Scholes model, and moreover yields a lognormal distribution of the volatilities, in agreement with results from superstatistical analysis. The results obtained in the present formalism are checked with Monte Carlo simulations.

Non-linear differential equation for atomic mixing: 2. Numerical results

A discussion is given of a non-linear Fokker-Planck (diffusion-type) differential equation for atomic mixing, developed in an earlier paper by the authors. Some simplifications of the earlier theory are presented, together with a sign-correction. Numerical results are given, obtained by a recently-developed machine program, for the case of an initial uniform layer of silver on a silicon substrate, irradiated by a 45 keV beam of Ar + ions at normal incidence.

Effect of scale on solute dispersion in saturated porous media

Consider a saturated porous m edium in which water is flowing slowly with a steady velocity. Suppose at some space-time scale the concentration C(x, r) of a non-reactive dilute solute is governed by the following Fokker-Planck differential equation:

Internal conversion in the rhodamine Dye, fast acid violet 2R

The results of fluorescence lifetime measurements of Fast Acid Violet 2R in a series of alcohols and diols having a wide range of viscosity have been fitted to Kramers' equation, an approximate solution of the Fokker–Planck differential equation, applicable in the case of conformational change, hindered by viscous drag of a solvent. Three unknown parameters in Kramers' equation were determined by a least-squares method. One of these parameters, d, the effective radius of the part of the molecule undergoing conformational change, was found to be > 4A in normal alcohols indicating that solvent attachment takes place. The value of d in diols was anomalously low.

Calculation of the E.S.R. line shape in highly viscous media

A method for calculation of the E.S.R. line shape is developed for the case when the Kubo and Tomita theory is invalid. The method is based on the equivalence of functional integration over trajectories and solution of the Fokker-Planck differential equation with some potential. The adiabatic approximation is used in establishing the form of the solution.

Energy changes of cosmic rays in the solar system

The energy (in the fixed frame of reference) of a cosmic-ray particle with velocity w in the frame of the solar wind fluctuates between 1 2 M(w + v) 2 and 1 2 M(w − v) 2 as the particle random walks in the magnetic field, where v is the wind velocity. This energy change cannot be treated as a simple Markov process, together with the random walk and adiabatic deceleration, in a fixed frame of reference. To describe the particle motion by a Fokker-Planck differential equation, the energy variable must be the particle energy in a frame of reference moving with the wind. The appropriate transformations of both the solutions and boundary conditions are presented to order v / w . The corrections due to this effect are ∼10% for 1 MeV particles and correspondingly less for higher energy particles. The energy balance between the solar wind and cosmic rays is examined and is shown to depend in a fundamental manner on these energy transformations. Although the solar wind does net work on the cosmic-ray gas, no particle gains more than 2 Mvw kinetic energy and the few particles which penetrate deeply into the solar system suffer a net energy loss because of adiabatic deceleration. Энepгия (B cTaциoнapнoй cиcTeMe кoopдинaT B пpocTpaнcTBe) чacTицы кocMичecкич лyчeй, co cкopocTью w W B cиcTeMe coлнeчнoгo BeTpa, кoлeблeTcя Meждy 1 2 m(w + v) 2 и 1 2 m(w − v) 2 , B To BpeMя кaк чacTицa бpoдиT B MaпниTнoM пoлe, V яBляeTcя cкopocTью BeTpa. Taкyю пepeMeиy эиepгии нe cлeдyeT paccMaTpиBaTь кaк oбыкиoBeииый пpoцecc MapкoBa, BMecTe c бpoжeниeM и aдиaбaTныM зaMeдлeниeM cкopocTи, B cTaциoнapнoй иcTeMe кoopдинaT B пpocTpaнcTBe. чToбы oпиcaTь дBижeниe чacTицы пpи пoMoщи диффepeнциaльнoгo ypaBнeния qFoккepПлaикa, нeoбчoдиMo, чToбы пepeMeннaя энepгии яBянлacь чacTицeй энepгии B пpoдBигaющeйcя BMecTe c BeTpoM cиcTeMe кoopдинaT B пpocTpaнcTBe. BыcTaBляюTcя copaзMepныe пpeBpaщeния кaк peщeний, Taк и гpaничиычycлoBий, cooTBeTcTBeннo пopядкy v / w . ПpoиcTeкaющиe oT эToгo эффeкTa пoпpaBки cocTaBляюT ~10% для чacTиц B 1. Mev и cooTBeTcTBeнныM oбpaзoM Meньщe для чacTиц бoлee Bыcoкoй энepгии. ЙccлeдyeTcя paBнoBecиe зиepгии Meждy coлнeчиыM BeTpoM и кocMичecкиMи лyчaMи, кoTopoe, пoBидиMoMy, элeMeнTapныM oбpaзoM зaBиcиT oT эTич пpeoбpaзoBaний зиepгии. ЧoTя coлиeчный BeTep нe oпepиpyeT нa гaзe кocMичecкич лyчeй, ни oднa чacTицa нe ycилиBaeTcя нa бoлee чeM 2 mvw кинeTичecкoй энepгии и Te нemиoгиe чacTицы, кoTopыe глyбoкo нpoникaюT B coлнeчнyю cиcTey иoдBepгa. oTcн чиcToй иoTepe зиepгии из-зa aдиaбaTнoгo зaMeдлeния cкopocTи.

On the spectrum of the mean diffusion velocities in the ocean

By interpretation of the concentration s of a diffusing substance in the sea as probability density, a Fokker-Planck differential equation of the second order is derived from a Markov functional equation for s. For small time differences only the momenta of distance of the first and second order are taken into account. Scales of length r are determined as radii of the circles that enclose the areas bounded by isoconcentration lines. The area mean of the diffusion velocity, given by the momentum of the first order, yields a velocity of 2P(r). These results are compared with experimental observations, and they indicate a dependence on r similar to that assumed by Stommel for the velocity spectrum in the ocean.