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AIP conference proceedings

Evaluation of the Fokker-Planck probability by Asymptotic Taylor Expansion Method

Kenan Firat, Okan Özer

Published Jan 1, 2017• 0 citations
Publisher

Abstract

The one-dimensional Fokker-Planck equation is solved by the Asymptotic Taylor Expansion Method for the time-dependent probability density of a particle. Using an ansatz wave function, one obtains the series expansion of the solution for the Schrödinger and it allows one to find out the eigen functions and eigen energies of the states to the evaluation of the probability. The eigen energies of some certain kind of Bistable potentials are calculated for some certain potential parameters. The probability function is determined and graphed for potential parameters. The numerical results are compared with existing literature, and a conclusion about the advantages and disadvantages on the method is given.

DOI
10.1063/1.4976483

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The one-dimensional Fokker-Planck equation is solved by the Asymptotic Taylor Expansion Method for the time-dependent probability density of a particle. Using an ansatz wave function, one obtains the series expansion of the solution for the Schrödinger and it allows one to find out the eigen functions and eigen energies of the states to the evaluation of the probability. The eigen energies of some certain kind of Bistable potentials are calculated for some certain potential parameters. The probability function is determined and graphed for potential parameters. The numerical results are compared with existing literature, and a conclusion about the advantages and disadvantages on the method is given.
Abstract We discuss the systematic expansion of the solution of the Fokker–Planck equation with the help of the eigenfunctions of the time-dependent Fokker–Planck operator. The expansion parameter is the time derivative of the external parameter which controls the form of an external potential. Our expansion corresponds to the perturbative calculation of the adiabatic motion in quantum mechanics. With this method, we derive a new formula to calculate the irreversible work order by order, which is expressed as the expectation value with a pseudo density matrix. Applying this method to the case of the harmonic potential, we show that the first order term of the expansion gives the exact result. Because we do not need to solve the coupled differential equations of moments, our method simplifies the calculations of various functions such as the fluctuation of the irreversible work per unit time. We further investigate the exact optimized protocol to minimize the irreversible work by calculating its variation with respect to the control parameter itself.
A simple framework for accurate solution of a general class of one-dimensional Fokker-Planck and/or Schr\"odinger equations is presented. The main idea is representing the propagator in the form P(x,t∣x0)=P0(x,t∣x0)exp[W(x,t∣x0)]P(x, t|{x}_{0})={P}_{0}(x, t|{x}_{0})\mathrm{exp}[W(x, t|{x}_{0})]P(x,t∣x0​)=P0​(x,t∣x0​)exp[W(x,t∣x0​)] and expanding the exponent WWW in a power series in a given function of ttt, where P0{P}_{0}P0​ is the exact solution of a reference harmonic-oscillator problem. The expansion coefficients are analytically evaluated from recursive relations. This approach is shown to be a dramatic improvement over the standard Taylor series expansion for the propagator in that just a few terms of the present expansion are sufficient to attain a very accurate description in the whole time domain.
A newly developed discrete ordinate method of solution of differential equations is applied to the solution of the Fokker-Plank equation employed in the a nalysis is of optical bistability. The solution is written in terms of the eigenfunctions of the Fokker-PlancK operator and the time evolution of the probability density function is studied. The present analysis includes a study of the convergence of the eigenualues and eigen functions for different choices of the quadrature rule upon which the discrete ordinate method is based.
This chapter focuses on widely used approximate method: the numerical integration of the Schrodinger equation, restricting ourselves to one dimension for simplicity. There is a plethora of approaches that yield highly accurate numerical eigenvalues and eigenfunctions. In quantum mechanics one sometimes seeks after bound stationary states that satisfy given physical boundary conditions. Boundary conditions at infinity play a relevant role in quantum mechanics, in which case one looks for bound states that vanish asymptotically at the end points of coordinate interval. Solving the propagation matrix equation exactly for every coordinate pair in the physical interval is as difficult as solving the original differential equation. However, sometimes the former is preferable to develop approximate numerical algorithms for the accurate calculation of eigenvalues and eigenfunctions. The perturbation method based on the Taylor expansion only requires derivatives of the potential energy function, which one obtains easily in most cases and appears to be preferable for the systematic improvement of the approximate propagation matrix.
An approximate method is proposed to solve the position-dependent mass (PDM) Schrödinger equation. The procedure suggested here leads to the solution of the PDM Schrödinger equation without transforming the potential function to the mass space or vice versa. The method based on the asymptotic Taylor expansion of the function produces an approximate analytical expression for eigenfunction and numerical results for eigenvalues of the PDM Schrödinger equation. The results show that the PDM and constant mass Schrödinger equations are not isospectral. The calculations are carried out with the aid of a computer system of symbolic or numerical calculation by constructing a simple algorithm.
A Taylor-series expansion of the density functional F[n] is constructed from the given expansion for the energy E[v]. The formalism is used to investigate the noninteracting kinetic-energy functional for one-dimensional systems. Furthermore, a formal analysis of the Taylor expansion of the density functional for an interacting electron system at finite temperature is given and the relation between density-functional theory and the field-theoretical approach to the many-particle problem is enlightened.