Steady, one-dimensional Brownian motion with an absorbing boundary

Harris, S.

doi:10.1063/1.442406

Cited by 56 publications

(13 citation statements)

References 10 publications

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“…The boundary conditions for these equations must fulfill the requirement that the virtual layers are absorbing at a and b, in the sense that particles are counted out once they reach the boundary. While some difficulty might exist when dealing with the probability function P (z, t|z 0 ) in the SE [2] and even in the stochastic free diffusion equation [4,5,13] , the absorbing boundary condition for a survival or exit probability is straightforward, namely, G(a, t) = G(b, t) = 0, and for the MFPT, .…”

Section: Defining the Smoluchowski Operatormentioning

confidence: 99%

“…A serious difficulty found was that the time scale of the Langevin Dynamics and of the Molecular Dynamics did not match. In fact, it was discussed by Burschka et al [3], Harris [4], and Razi Naqvi et al [5] , that the conditional probability obtained as the stationary solution of the Fokker-Planck equation, associated to the Langevin equation for a system with absorbing boundaries conditions fails to vanish at those boundaries. For the same space sampling layers width L, the Langevin persistence probability is then known to be lower than the corresponding MD persistence probability.…”

Section: Introductionmentioning

confidence: 99%

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Scaling of Langevin and molecular dynamics persistence times of nonhomogeneous fluids

Olivares-Rivas

Colmenares

2012

Phys. Rev. E

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The existing solution for the Langevin equation of an anisotropic fluid allowed the evaluation of the position-dependent perpendicular and parallel diffusion coefficients, using molecular dynamics data. However, the time scale of the Langevin dynamics and molecular dynamics are different and an ansatz for the persistence probability relaxation time was needed. Here we show how the solution for the average persistence probability obtained from the backward Smoluchowski-Fokker-Planck equation (SE), associated to the Langevin dynamics, scales with the corresponding molecular dynamics quantity. Our SE perpendicular persistence time is evaluated in terms of simple integrals over the equilibrium local density. When properly scaled by the perpendicular diffusion coefficient, it gives a good match with that obtained from molecular dynamics.

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Section: Defining the Smoluchowski Operatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Scaling of Langevin and molecular dynamics persistence times of nonhomogeneous fluids

Olivares-Rivas

Colmenares

2012

Phys. Rev. E

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“…Employing the fact that the inner integrals can be written in terms of the error function, using the fundamental theorem of calculus 1 to differentiate P(γ < c) with respect to c, and exploiting the fact that the product of two Gaussians is proportional 1 The fundamental theorem of calculus as used here is: to another Gaussian 2 , after some manipulation we arrive at the pdf of γ, to first order in ∆t:…”

Section: First Constraint Encountermentioning

confidence: 99%

“…Particle absorption is the simplest scenario, characterizing the distribution of a population of particles as they move at constant speed near a parallel boundary, while diffusing in the lateral direction. Particle absorption focuses mainly on random walks, however, a canonical case that does not necessarily extend to more general linear systems [1][2][3][4]. Seminal work by Crandall and others [5][6][7] explored the statistics of first-excursion for a lightly-damped oscillator with Gaussian process noise.…”

Section: Introductionmentioning

confidence: 99%

A Single Differential Equation for First-Excursion Time in a Class of Linear Systems

Greytak

Hover

2011

Journal of Dynamic Systems, Measurement, and Control

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First-excursion times have been developed extensively in the literature for oscillators; one major application is structural dynamics of buildings. Using the fact that most closed-loop systems operate with a moderate to high damping ratio, we have derived a new procedure for calculating first-excursion times, for a class of linear continuous, time-varying systems. In several examples we show that the algorithm is both accurate and timeefficient. These are important attributes for real-time path planning in stochastic environments, and hence the work should be useful for autonomous robotic systems involving marine and air vehicles. Nomenclature x(t)State vector. y(t)System output, scalar.A(t), G(t), C System matrices. w(t)Unit Wiener noise process, vector.Lower and upper boundaries, scalar. Σ Σ Σ(t)State covariance matrix.Σ y (t) Output covariance, scalar.P a (t) Probability of no collision, scalar.P hit (t) Collision probability, 1 − P a (t), scalar.

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“…Several algorithms have been suggested (Ermak & Buckholtz, 1980;van Gunsteren & Berendsen, 1982;Allen, 1982), and the technique has found wide application, for example, in modelling atom-transfer reactions (Allen, 1980), motion in ionic solutions (Ermak, 1975), and protein dynamics (Dickinson, 1985). There has been little progress in obtaining analytical solutions for first passage time distributions in the case of Brownian dynamics (Harris, 1981). Consequently such problems are usually investigated by simulation.…”

Section: Brownian Dynamicsmentioning

confidence: 99%

Statistical Models of Chemical Kinetics in Liquids

Clifford

Green

Pilling

1987

Journal of the Royal Statistical Society Series B: Statistical Methodology

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Statistical methods are used increasingly in theoretical chemistry. Applications range from the use of stochastic relaxation techniques in determining minimum energy molecular configurations to dynamical Monte Carlo simulations of molecular motion in liquids. This paper focuses on diffusion-controlled reactions in radiation chemistry. Here the interest is in describing the evolution of isolated clusters, containing a few chemically active particles, resulting from the passage of ionising radiation through a liquid. The subsequent chemistry is determined by the rate at which these particles can encounter each other, pairwise, in the course of their random motion. Classically the trajectories of the particles are described as sample paths of a continuous stochastic process, which in many cases can be assumed to be a diffusion. We have devised an approximate theory based on the approximation that pair distances evolve independently. The effect of this geometric distortion has been studied for small systems by simulation. The extension of this work to a wider class of realistic chemical processes poses many problems. Analytical progress is difficult and methods of simulating large spatial systems are not well developed. Problems which remain to be solved relate to the random geometry of the cluster as its constituent particles diffuse, the development of good approximations to first passage time distributions for diffusions with inhomogeneous drift and methods for the analysis of random motion in a liquid on the basis of detailed non-Markovian models of molecular movement.

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Steady, one-dimensional Brownian motion with an absorbing boundary

Abstract: Onedimensional Brownian motion near an absorbing boundary: Solution to the steady state Fokker-Planck equation J. Chem. Phys. 79, 2302 (1983); 10.1063/1.446034 Comment on ''Steady, onedimensional Brownian motion with an absorbing boundary''

Cited by 56 publications

References 10 publications

Scaling of Langevin and molecular dynamics persistence times of nonhomogeneous fluids

Scaling of Langevin and molecular dynamics persistence times of nonhomogeneous fluids

A Single Differential Equation for First-Excursion Time in a Class of Linear Systems

Statistical Models of Chemical Kinetics in Liquids

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