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Spectral and Pseudospectral Methods of Solution of the Fokker-Planck and Schrödinger Equations

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Spectral Methods in Chemistry and Physics

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Abstract

Spectral and pseudospectral methods based on classical and nonclassical polynomial basis sets are used for the solution of the Fokker-Planck and Schrödinger equations. Fokker-Planck equations describe many different processes in chemistry and physics, and their study has attracted considerable attention by researchers in many different fields including astrophysics, finance and biology. Pseudospectral methods of solution of the Fokker-Planck equation are presented for several systems such as the Ornstein-Uhlenbeck model for Brownian motion, electron thermalization in atomic moderators, charged particle relaxation in plasmas and models for chemical reactions based on Kramers’ equation. A Fokker-Planck equation can be transformed to a Schrödinger equation with a potential that belongs to the class of potentials in supersymmetric quantum mechanics and expressed in terms of the superpotential. The quantum harmonic oscillator and the Morse potential belong to this class of Schrödinger equations. The pseudospectral methods developed for the solution of the Fokker-Planck equation based on nonclassical basis sets are also applied to a large number of the Schrödinger equations including the Henon-Heles potential. Fundamental aspects of different pseudospectral methods such as the Discrete Variable Representation, the Quadrature Discretization method, the Lagrange mesh method and Fourier grid methods are discussed.

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Notes

  1. 1.

    Adrian Fokker (1887–1972) was a Dutch physicist who made contributions to relativity and statistical mechanics in collaboration with Albert Einstein. The Fokker-Planck equation used to model numerous processes in physics, astrophysics, chemistry, finance and biology bears his name. He also made numerous contributions to music theory.

  2. 2.

    Max Planck (1858–1947) was a German physicist and the 1918 Nobel laureate for his contributions to the explanation of the photoelectric effect, energy quantization and the introduction of the Planck constant. The basis for this work was his doctoral work on thermodynamics as related to black body radiation at equilibrium. Planck and Fokker independently derived the Fokker-Planck equation of statistical physics.

  3. 3.

    Robert Brown (1773–1858) was a Scottish botanist who made important contributions to botany and statistical physics from his use of a microscope to observe the random motion of pollen grains which was later referred to as Brownian motion.

  4. 4.

    Paul Langevin (1872–1946) was a French physicist and doctoral student with J.J. Thompson at the Cavendish Laboratory and Pierre Curie in Paris. He worked extensively on paramagnetism and diamagnetism as well as in kinetic theory and theory of Brownian motion following on Einstein’s work.

  5. 5.

    Marian Smoluchowski (1872–1917) was a Polish physicist who was responsible for the development of fundamental concepts in statistical physics, kinetic theory and Brownian motion. His name is associated with integral equations for coagulation and a Fokker-Planck equation for chemical reactions.

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    Bernard Shizgal

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Shizgal, B. (2015). Spectral and Pseudospectral Methods of Solution of the Fokker-Planck and Schrödinger Equations. In: Spectral Methods in Chemistry and Physics. Scientific Computation. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9454-1_6

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