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## Über dieses Buch

One of the central problems synergetics is concerned with consists in the study of macroscopic qualitative changes of systems belonging to various disciplines such as physics, chemistry, or electrical engineering. When such transitions from one state to another take place, fluctuations, i.e., random processes, may play an im­ portant role. Over the past decades it has turned out that the Fokker-Planck equation pro­ vides a powerful tool with which the effects of fluctuations close to transition points can be adequately treated and that the approaches based on the Fokker­ Planck equation are superior to other approaches, e.g., based on Langevin equa­ tions. Quite generally, the Fokker-Planck equation plays an important role in problems which involve noise, e.g., in electrical circuits. For these reasons I am sure that this book will find a broad audience. It pro­ vides the reader with a sound basis for the study of the Fokker-Planck equation and gives an excellent survey of the methods of its solution. The author of this book, Hannes Risken, has made substantial contributions to the development and application of such methods, e.g., to laser physics, diffusion in periodic potentials, and other problems. Therefore this book is written by an experienced practitioner, who has had in mind explicit applications to important problems in the natural sciences and electrical engineering.

## Inhaltsverzeichnis

### 1. Introduction

Abstract
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 its simplest form.
Hannes Risken

### 2. Probability Theory

Abstract
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 selection of basic ideas and concepts of probability theory in a simplified form may be in order for the reader not very familiar with probability theory.
Hannes Risken

### 3. Langevin Equations

Abstract
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.
Hannes Risken

### 4. Fokker-Planck Equation

Abstract
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 much more difficult to obtain, so here we first try to derive an equation for the distribution function. As mentioned already in the introduction, a differential equation for the distribution function describing Brownian motion was first derived by Fokker [1.1] and Planck [1.2]: many review articles and books on the Fokker-Planck equation now exist [1.5 – 15].
Hannes Risken

### 5. Fokker-Planck Equation for One Variable; Methods of Solution

Abstract
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
$$\partial W(x,t)/\partial t = {L_{FP}}W(x,t) = - (\partial /\partial x)S(x,t),$$
(5.1)
$${L_{FP}}(x) = - \frac{\partial }{{\partial x}}{D^{(1)}}(x) + \frac{{{\partial ^2}}}{{\partial {x^2}}}{D^{(2)}}(x).$$
(5.2)
Hannes Risken

### 6. Fokker-Planck Equation for Several Variables; Methods of Solution

Abstract
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
$$\partial W/\partial t = {L_{FP}}W = \partial {S_i}/\partial {x_i},$$
(6.1)
$${L_{FP}} = - \frac{\partial }{{\partial {x_i}}}{D_i}(\{ x\} ) + \frac{{{\partial ^2}}}{{\partial {x_i}\partial {x_j}}}{D_{ij}}(\{ x\} ).$$
(6.2)
(With the exception of Sect. 6.6.5 the summation convention for Latin indices is used in this chapter.)
Hannes Risken

### 7. Linear Response and Correlation Functions

Abstract
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 state. The external fields or the changes of the parameters are usually small. Then we only need to take into account those deviations from the steady state which are linear in the external fields (linear response). The deviations of expectation values from their steady-state values also depend linearly on the fields. This dependence can be described by a response function. If the external fields are switched off, the deviations from the steady state decay or dissipate (in the physical literature the word ‘dissipate’ is usually used for the decay of energy).
Hannes Risken

### 8. Reduction of the Number of Variables

Abstract
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 many variables as possible, so we discuss below three cases where the number of independent variables can be reduced.
Hannes Risken

### 9. Solutions of Tridiagonal Recurrence Relations, Application to Ordinary and Partial Differential Equations

Abstract
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 suitable expansion of the distribution function. In this chapter we shall investigate the solutions of tridiagonal vector recurrence relations. As it turns out, the Laplace transform of these solutions as well as the eigenvalues and eigenfunctions can be obtained in terms of matrix continued fractions. Therefore, the corresponding solutions of the Kramers equation can also be given in terms of matrix continued fractions. This method has the advantage that no detailed balance condition is needed for its application. This matrix continued-fraction method is especially suitable for numerical calculations and for some problems it seems to be the most accurate and fastest method, as will be discussed in other chapters. Besides its advantage for numerical purposes, the matrix continued-fraction solutions are also very useful for analytical evaluations. By a proper Taylor series expansion of the matrix continued fractions we obtain, for instance, in Sect. 10.4 the high-friction limit solutions of the Kramers equation.
Hannes Risken

### 10. Solutions of the Kramers Equation

Abstract
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 space. This Kramers equation was derived and used by Kramers [1.17] to describe reaction kinetics. Later on it turned out that it had more general applicability, e.g., to such different fields as superionic conductors, Josephson tunneling junction, relaxation of dipoles, second-order phase-locked loops. These applications will be discussed in Chap. 11. For large damping constants the Kramers equation reduces to the Smoluchowski equation which is a special Fokker-Planck equation for the distribution function for the position coordinate only. In this chapter some of the well-known solutions for linear forces are presented. Next we shall derive a general solution of the Kramers equation in terms of matrix continued fractions for arbitrary forces. Expansion of these matrix continued-fraction solutions for large damping constants into powers of the inverse friction constant gives the Smoluchowski equation and its different correction terms. Whereas the position will become a slow variable and the velocity a fast variable in the high-friction limit, the energy will become a slow variable and the position (or velocity) a fast variable in the low-friction limit (see Sect. 8.3 for a discussion of slow and fast variables). In the low-friction limit the procedure depends on the topology of the energy surface in phase space, which in turn depends on the specific form of the potential.
Hannes Risken

### 11. Brownian Motion in Periodic Potentials

Abstract
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 several fields of science, for instance in physics, chemical physics and communication theory. Restricting ourselves to the one-dimensional case, we deal with particles which are kicked around by the Langevin forces and move in a one-dimensional periodic potential (Fig. 11.1). Because of the excitation due to the Langevin forces the particles may leave the well and go either to the neighboring left or right well or they may move in the course of time to other wells which are further away. For long enough times the particles will thus diffuse in both directions of the x axis. As shown in Sect. 11.7 this diffusion can be described by a diffusion constant D, if we wait long enough. Thus the mean-square displacement is given by
$$\left\langle {{{[x(t) - x(0)]}^2}} \right\rangle = 2Dt$$
(11.1)
for large times t. (The particles are then distributed over many potential wells.)
Hannes Risken

### 12. Statistical Properties of Laser Light

Abstract
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 subject together with other applications of the Fokker-Planck equation in quantum optics are already treated in a number of handbooks, books and review articles [12.1 – 13, 1.28, 4.8]. The main purpose of this chapter is to demonstrate how some of the methods of Chaps. 2 – 9 can be applied to a simple laser model (one mode, adiabatic elimination of all variables with the exception of the laser field, threshold region). The following two points make it difficult but also interesting to investigate the statistical properties of laser light.
Hannes Risken

### Backmatter

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