Chapter 1. Basic Concepts of the Probability Theory

  1. Prof. Dr. Emmanuil G. Sinaiski1 and
  2. Prof. Dr. Leonid I. Zaichik2

Published Online: 21 FEB 2008

DOI: 10.1002/9783527621804.ch1

Statistical Microhydrodynamics

Statistical Microhydrodynamics

How to Cite

Sinaiski, E. G. and Zaichik, L. I. (2008) Basic Concepts of the Probability Theory, in Statistical Microhydrodynamics, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, Germany. doi: 10.1002/9783527621804.ch1

Author Information

  1. 1

    An der Kotsche 12, 04207 Leipzig, Germany

  2. 2

    Nuclear Safety Institute, Russian Academy of Science, B. Tulskaya 52, 115191 Moscow, Russia

Publication History

  1. Published Online: 21 FEB 2008
  2. Published Print: 23 JAN 2008

ISBN Information

Print ISBN: 9783527406562

Online ISBN: 9783527621804

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Keywords:

  • probability theory;
  • random variables;
  • average value;
  • variance;
  • correlation functions

Summary

This chapter contains sections titled:

  • Events, Set of Events, and Probability

  • Random Variables, Probability Distribution Function, Average Value, and Variance

  • Generalized Functions

  • Methods of Averaging

  • Characteristic Functions

  • Moments and Cumulants of Random Variables

  • Correlation Functions

  • Bernoulli, Poisson, and Gaussian Distributions

  • Stationary Random Functions, Homogeneous Random Fields

  • Isotropic Random Fields. Spectral Representation

  • Stochastic Processes. Markovian Processes. The Chapman–Kolmogorov Integral Equation

  • The Chapman–Kolmogorov, Chapman–Feller, Fokker–Planck, and Liouville Differential Equations

    • Derivation of the Differential Chapman–Kolmogorov Equation

    • Discontinuous (“Jump”) Processes. The Kolmogorov–Feller Equation

    • Diffusion Processes. The Fokker–Planck Equation

    • Deterministic Processes. The Liouville Equation

  • Stochastic Differential Equations. The Langevin Equation

    • The Langevin Equation

    • The Diffusion Equation

      • The Diffusion Equation with Chemical Reactions Taken into Account

      • Brownian Motion of a Particle in a Hydrodynamic Medium

  • Variational (Functional) Derivatives

  • The Characteristic Functional