Chapter 2
The Fokker-Planck Equation
During the last years with the studies of stochastic processes: neurons networks,
molecular motors, dynamics models, anomalous diffusion, disordered media, etc,
several methods have evolved to apply the Focker-Planck equation (FPE) to these
phenomena. We present here the solution of the Fokker-Planck equation by the
Crank-Nicholson formalism. The von Neumann amplification factor, ξ (k), is
independent of dt, so the method is stable for any size dt. The method is suitable
for modeling molecular motors because the great amount of interactions in these
systems, vectors and matrices oriented methods are needed, suited to work with
Matlab.
In the Appendix of this chapter is given some notions of Stochastic Dynamics
2.1 The Methods
The method of Fractional Focker-Planck equation (FFPE) [1] was derived from a
generalized continuous time random walk, which includes space dependent jump
probabilities which are the result of an external field. In [2] was presented the
solution of the FFPE in terms of an integral transformation. In [3] a half order FFPE
was derived from the generalized scheme of random walks on the comlike structure.
In [4, 5] the FFPE was used to study and describe also the anomalous diffusion in
external fields. In [6] the FFPE was used to study ultraslow kinetics. In [7] was
introduced a heterogeneous FFPE involving external force fields describing systems
on heterogeneous fractal structure medium. In [8] was studied the dynamical
properties of bistable systems described by the one-dimensional sub-diffusive FFPE,
for the natural boundary conditions as well as the absorbing boundary conditions. In
[9] was shown that the subordinated Brownian process is a stochastic solution of the
FFPE. In [10] was proven that the asymptotic shape of the solution of the FFPE is a
stretched Gaussian and that its solution can be expressed in the form of a function of
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in
Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9_2
15
The Fokker-Planck Equation
During the last years with the studies of stochastic processes: neurons networks,
molecular motors, dynamics models, anomalous diffusion, disordered media, etc,
several methods have evolved to apply the Focker-Planck equation (FPE) to these
phenomena. We present here the solution of the Fokker-Planck equation by the
Crank-Nicholson formalism. The von Neumann amplification factor, ξ (k), is
independent of dt, so the method is stable for any size dt. The method is suitable
for modeling molecular motors because the great amount of interactions in these
systems, vectors and matrices oriented methods are needed, suited to work with
Matlab.
In the Appendix of this chapter is given some notions of Stochastic Dynamics
2.1 The Methods
The method of Fractional Focker-Planck equation (FFPE) [1] was derived from a
generalized continuous time random walk, which includes space dependent jump
probabilities which are the result of an external field. In [2] was presented the
solution of the FFPE in terms of an integral transformation. In [3] a half order FFPE
was derived from the generalized scheme of random walks on the comlike structure.
In [4, 5] the FFPE was used to study and describe also the anomalous diffusion in
external fields. In [6] the FFPE was used to study ultraslow kinetics. In [7] was
introduced a heterogeneous FFPE involving external force fields describing systems
on heterogeneous fractal structure medium. In [8] was studied the dynamical
properties of bistable systems described by the one-dimensional sub-diffusive FFPE,
for the natural boundary conditions as well as the absorbing boundary conditions. In
[9] was shown that the subordinated Brownian process is a stochastic solution of the
FFPE. In [10] was proven that the asymptotic shape of the solution of the FFPE is a
stretched Gaussian and that its solution can be expressed in the form of a function of
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in
Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9_2
15