*Melvin Lax, Wei Cai, and Min Xu*

- Published in print:
- 2006
- Published Online:
- January 2010
- ISBN:
- 9780198567769
- eISBN:
- 9780191718359
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567769.003.0008
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Thermal noise and Gaussian noise sources combine to create a category of Markovian processes known as Fokker–Planck processes. This chapter discusses Fokker–Planck processes such as ...
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Thermal noise and Gaussian noise sources combine to create a category of Markovian processes known as Fokker–Planck processes. This chapter discusses Fokker–Planck processes such as generation-recombination processes, linearly damped processes, Doob's theorem, and multivariable processes. The determination of the behaviour of a Markovian random variable is reduced to the solution of a partial differential equation for the probability that such variable will assume the value at a certain time. In this way, a problem of stochastic processes has been reduced to a more conventional mathematical problem, the solution of a partial differential equation. This chapter also examines drift vectors and diffusion coefficients, the average motion of a general random variable, generalised Fokker–Planck equation, characteristic function, path integral average, linear damping and homogeneous noise, backward equation, extension of the Fokker–Planck equation to many variables, and time reversal in the linear case.Less

Thermal noise and Gaussian noise sources combine to create a category of Markovian processes known as Fokker–Planck processes. This chapter discusses Fokker–Planck processes such as generation-recombination processes, linearly damped processes, Doob's theorem, and multivariable processes. The determination of the behaviour of a Markovian random variable is reduced to the solution of a partial differential equation for the probability that such variable will assume the value at a certain time. In this way, a problem of stochastic processes has been reduced to a more conventional mathematical problem, the solution of a partial differential equation. This chapter also examines drift vectors and diffusion coefficients, the average motion of a general random variable, generalised Fokker–Planck equation, characteristic function, path integral average, linear damping and homogeneous noise, backward equation, extension of the Fokker–Planck equation to many variables, and time reversal in the linear case.

*Melvin Lax, Wei Cai, and Min Xu*

- Published in print:
- 2006
- Published Online:
- January 2010
- ISBN:
- 9780198567769
- eISBN:
- 9780191718359
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567769.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This book discusses random processes in physics, from basic probability theory to the Fokker-Planck process and Langevin equation for the Fokker Planck process and diffusion. It includes diverse ...
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This book discusses random processes in physics, from basic probability theory to the Fokker-Planck process and Langevin equation for the Fokker Planck process and diffusion. It includes diverse applications, such as explanations of very narrow laser width, analytical solutions of the elastic Boltzmann transport equation, and a critical viewpoint of mathematics currently used in the world of finance. The name ‘econophysics’ has been used to denote the use of the mathematical techniques developed for the study of random processes in physical systems to applications in the economic and financial worlds. Since a substantial number of physicists are now employed in the financial arena or are doing research in this area, it is appropriate to give a course that emphasises and relates physical applications to financial applications. The course and text in this book differ from mathematical texts by emphasising the origins of noise, as opposed to an analysis of its transformation by linear and nonlinear devices. Among other topics, this book discusses thermal noise, shot noise, fluctuation-dissipation theorem, and delta functions.Less

This book discusses random processes in physics, from basic probability theory to the Fokker-Planck process and Langevin equation for the Fokker Planck process and diffusion. It includes diverse applications, such as explanations of very narrow laser width, analytical solutions of the elastic Boltzmann transport equation, and a critical viewpoint of mathematics currently used in the world of finance. The name ‘econophysics’ has been used to denote the use of the mathematical techniques developed for the study of random processes in physical systems to applications in the economic and financial worlds. Since a substantial number of physicists are now employed in the financial arena or are doing research in this area, it is appropriate to give a course that emphasises and relates physical applications to financial applications. The course and text in this book differ from mathematical texts by emphasising the origins of noise, as opposed to an analysis of its transformation by linear and nonlinear devices. Among other topics, this book discusses thermal noise, shot noise, fluctuation-dissipation theorem, and delta functions.

*Melvin Lax, Wei Cai, and Min Xu*

- Published in print:
- 2006
- Published Online:
- January 2010
- ISBN:
- 9780198567769
- eISBN:
- 9780191718359
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567769.003.0010
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter discusses the Langevin treatment of the Fokker–Planck process and diffusion. The form of Langevin equation used is different from the stochastic differential equation using Ito's ...
More

This chapter discusses the Langevin treatment of the Fokker–Planck process and diffusion. The form of Langevin equation used is different from the stochastic differential equation using Ito's calculus lemma. The transform of the Langevin equation obeys the ordinary calculus rule, hence can be easily performed and some misleadings can be avoided. The origin of the difference between this approach and that using Ito's lemma comes from the different definitions of the stochastic integral. This chapter also discusses drift velocity, an example with an exact solution, use of Langevin equation for a general random variable, extension of this equation to the multiple dimensional case, and means of products of random variables and noise source.Less

This chapter discusses the Langevin treatment of the Fokker–Planck process and diffusion. The form of Langevin equation used is different from the stochastic differential equation using Ito's calculus lemma. The transform of the Langevin equation obeys the ordinary calculus rule, hence can be easily performed and some misleadings can be avoided. The origin of the difference between this approach and that using Ito's lemma comes from the different definitions of the stochastic integral. This chapter also discusses drift velocity, an example with an exact solution, use of Langevin equation for a general random variable, extension of this equation to the multiple dimensional case, and means of products of random variables and noise source.