Advertisement

Fokker-Planck Equation

  • Roberto Mauri
Part of the Soft and Biological Matter book series (SOBIMA)

Abstract

The time evolution of the probability density function of a set of random variables is described by the Fokker-Planck equation, named after Adriaan Fokker and Max Planck. Originally, it was developed to describe the motion of Brownian particles and later was generalized to follow the evolution of a set of random variables with linear phenomenological constitutive relations. In this chapter, the Fokker-Planck equation is derived in the framework of Markov processes (Sects. 4.1 and 4.2), showing its most general solution (Sect. 4.3) and how the fluctuation-dissipation theorem follows from it (Sect. 4.4). Then, in Sect. 4.5, a counter example is illustrated, where we show that when the applied forces are non-conservative the fluctuation-dissipation theorem cannot be applied. Finally, in Sect. 4.6, we study the simplest case of Brownian motion, namely pure diffusion, also referred to as the Wiener process, named in honor of Norbert Wiener, stressing how the associated mathematical inconsistencies can be completely resolved only by applying the theory of stochastic differential equations (see Chap.  5).

References

  1. 1.
    Chandrasekhar, S.: Rev. Mod. Phys. 15, 1 (1943) MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. 2.
    Gardiner, C.W.: Handbook of Stochastic Methods. Springer, Berlin (1985) Google Scholar
  3. 3.
    Leporini, D., Mauri, R.: Fluctuations of non-conservative systems. J. Stat. Mech., P03002 (2007) Google Scholar
  4. 4.
    Mauri, R., Leporini, D.: Violation of the fluctuation-dissipation theorem in confined driven colloids. Europhys. Lett. 76, 1022–1028 (2006) ADSCrossRefGoogle Scholar
  5. 5.
    Uhlenbeck, G.E., Ornstein, L.S.: On the theory of the Brownian motion. Phys. Rev. 36, 823–841 (1930) ADSzbMATHCrossRefGoogle Scholar
  6. 6.
    Wax, N.: Noise and Stochastic Processes. Dover, New York (1954) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Roberto Mauri
    • 1
  1. 1.Department of Chemical Engineering, Industrial Chemistry and Material ScienceUniversity of PisaPisaItaly

Personalised recommendations