The Fokker-Planck Equation
Methods of Solution and Applications
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26 Result(s)
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Professor Dr. Hannes Risken
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Book
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Chapter
Probability Theory
In this chapter we recapitulate some of the basic ideas and conceptions of probability theory needed to unterstand the other chapters. Though there are many text books on probability theory [2.1 – 6], a select...
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Chapter
Reduction of the Number of Variables
Usually, the difficulty of solving the Fokker-Planck equation like any other partial differential equation increases with increasing number of independent variables. It is therefore advisable to eliminate as m...
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Chapter
Introduction
A Fokker-Planck equation was first used by Fokker [1.1] and Planck [1.2] to describe the Brownian motion of particles. To become familiar with this equation we first discuss the Brownian motion of particles in it...
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Chapter
Linear Response and Correlation Functions
We consider a system in a stable steady state or in equilibrium. If we disturb the system by applying some external fields or by changing some parameter the system will be driven away from its former steady st...
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Chapter
Fokker-Planck Equation
As shown in Sects. 3.1, 2 we can immediately obtain expectation values for processes described by the linear Langevin equations (3.1, 31). For nonlinear Langevin equations (3.67, 110) expectation values are mu...
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Chapter
Fokker-Planck Equation for Several Variables; Methods of Solution
In this chapter we discuss methods of solution for the Fokker-Planck equation (4.94a, 95) for time-independent drift and diffusion coefficients, i.e., for
6.1 -
Chapter
Brownian Motion in Periodic Potentials
In this chapter we apply some of the methods discussed in Chap. 10 for solving the Kramers equation for the problem of Brownian motion in a periodic potential. As discussed below, this problem arises in severa...
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Chapter
Langevin Equations
We first investigate the solution of the Langevin equation for Brownian motion. In Sect. 3.2 we treat a system of linear Langevin equations, followed in Sects. 3.3, 4 by general nonlinear Langevin equations.
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Chapter
Fokker-Planck Equation for One Variable; Methods of Solution
We now want to discuss methods for solving the one-variable Fokker-Planck equation (4.44, 45) with time-independent drift and diffusion coefficients, assuming D (2)(x) > 0
5.1 ... -
Chapter
Solutions of the Kramers Equation
The Kramers equation is a special Fokker-Planck equation describing the Brownian motion in a potential. For a one-dimensional problem it is an equation for the distribution function in position and velocity sp...
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Chapter
Statistical Properties of Laser Light
The Fokker-Planck equation has become a very useful tool for treating noise in quantum optics. In this chapter we investigate noise in a laser, which is the most important device in quantum optics. This subjec...
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Chapter
Solutions of Tridiagonal Recurrence Relations, Application to Ordinary and Partial Differential Equations
As shown in the next chapter, the Fokker-Planck equation describing the Brownian motion in arbitrary potentials, i.e., the Kramers equation, can be cast into a tridiagonal vector recurrence relation by suitabl...
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Book
-
Chapter
Linear Response and Correlation Functions
We consider a system in a stable steady state or in equilibrium. If we disturb the system by applying some external fields or by changing some parameter the system will be driven away from its former steady st...
-
Chapter
Probability Theory
In this chapter we recapitulate some of the basic ideas and conceptions of probability theory needed to unterstand the other chapters. Though there are many text books on probability theory [2.1 – 6], a select...
-
Chapter
Reduction of the Number of Variables
Usually, the difficulty of solving the Fokker-Planck equation like any other partial differential equation increases with increasing number of independent variables. It is therefore advisable to eliminate as m...
-
Chapter
Introduction
A Fokker-Planck equation was first used by Fokker [1.1] and Planck [1.2] to describe the Brownian motion of particles. To become familiar with this equation we first discuss the Brownian motion of particles in it...
-
Chapter
Langevin Equations
We first investigate the solution of the Langevin equation for Brownian motion. In Sect. 3.2 we treat a system of linear Langevin equations, followed in Sects. 3.3, 4 by general nonlinear Langevin equations.
-
Chapter
Fokker-Planck Equation for One Variable; Methods of Solution
We now want to discuss methods for solving the one-variable Fokker-Planck equation (4.44, 45) with time-independent drift and diffusion coefficients, assuming D (2)(x) > 0
(5.1) ...
