Kramers–Moyal expansion: Difference between revisions
Content deleted Content added
No edit summary |
Citation bot (talk | contribs) Alter: url, template type. URLs might have been internationalized/anonymized. Add: isbn, date, bibcode. Removed parameters. | You can use this bot yourself. Report bugs here. | Suggested by AManWithNoPlan | All pages linked from cached copy of User:AManWithNoPlan/sandbox2 | via #UCB_webform_linked |
||
| Line 1: | Line 1: | ||
In [[stochastic processes]], '''Kramers–Moyal expansion''' refers to a [[Taylor series]] expansion of the [[master equation]], named after [[Hans Kramers]] and [[José Enrique Moyal]].<ref>{{cite journal |last=Kramers |first=H. A. |year=1940 |title=Brownian motion in a field of force and the diffusion model of chemical reactions |journal=Physica |volume=7 |issue=4 |pages=284–304 |doi=10.1016/S0031-8914(40)90098-2 }}</ref><ref>{{cite journal |last=Moyal |first=J. E. |year=1949 |title=Stochastic processes and statistical physics |journal=[[Journal of the Royal Statistical Society]] |series=Series B (Methodological) |volume=11 |issue=2 |pages=150–210 |jstor=2984076 }}</ref> This expansion transforms the [[integro-differential equation|integro-differential]] [[master equation]] |
In [[stochastic processes]], '''Kramers–Moyal expansion''' refers to a [[Taylor series]] expansion of the [[master equation]], named after [[Hans Kramers]] and [[José Enrique Moyal]].<ref>{{cite journal |last=Kramers |first=H. A. |year=1940 |title=Brownian motion in a field of force and the diffusion model of chemical reactions |journal=Physica |volume=7 |issue=4 |pages=284–304 |doi=10.1016/S0031-8914(40)90098-2 |bibcode=1940Phy.....7..284K }}</ref><ref>{{cite journal |last=Moyal |first=J. E. |year=1949 |title=Stochastic processes and statistical physics |journal=[[Journal of the Royal Statistical Society]] |series=Series B (Methodological) |volume=11 |issue=2 |pages=150–210 |jstor=2984076 }}</ref> This expansion transforms the [[integro-differential equation|integro-differential]] [[master equation]] |
||
:<math>\frac{\partial p(x,t)}{\partial t} =\int dx'[W(x|x')p(x',t)-W(x'|x)p(x,t)]</math> |
:<math>\frac{\partial p(x,t)}{\partial t} =\int dx'[W(x|x')p(x',t)-W(x'|x)p(x,t)]</math> |
||
where <math>p(x,t|x_0,t_0)</math> (for brevity, this probability is denoted by <math>p(x,t)</math>) is the transition probability density, to an infinite order [[partial differential equation]]<ref>{{cite book |last=Gardiner |first=C. |year=2009 |title=Stochastic Methods |edition=4th |location=Berlin |publisher=Springer |isbn=978-3-642-08962-6 }}</ref><ref>{{cite book |last=Van Kampen |first=N. G. |year=1992 |title=Stochastic Processes in Physics and Chemistry |location= |publisher=Elsevier |isbn=0-444-89349-0 }}</ref><ref>{{cite book |last=Risken |first=H. |year=1996 |
where <math>p(x,t|x_0,t_0)</math> (for brevity, this probability is denoted by <math>p(x,t)</math>) is the transition probability density, to an infinite order [[partial differential equation]]<ref>{{cite book |last=Gardiner |first=C. |year=2009 |title=Stochastic Methods |edition=4th |location=Berlin |publisher=Springer |isbn=978-3-642-08962-6 }}</ref><ref>{{cite book |last=Van Kampen |first=N. G. |year=1992 |title=Stochastic Processes in Physics and Chemistry |location= |publisher=Elsevier |isbn=0-444-89349-0 }}</ref><ref>{{cite book |last=Risken |first=H. |year=1996 |title=The Fokker–Planck Equation |pages=63–95 |publisher=Springer |location=Berlin, Heidelberg |isbn=3-540-61530-X }}</ref> |
||
:<math>\frac{\partial p(x,t)}{\partial t} = \sum_{n=1}^\infty \frac{(-1)^n}{n!} \frac{\partial^n}{\partial x^n}[\alpha_n(x) p(x,t)]</math> |
:<math>\frac{\partial p(x,t)}{\partial t} = \sum_{n=1}^\infty \frac{(-1)^n}{n!} \frac{\partial^n}{\partial x^n}[\alpha_n(x) p(x,t)]</math> |
||
| Line 14: | Line 14: | ||
==Pawula theorem== |
==Pawula theorem== |
||
The Pawula theorem states that the expansion either stops after the first term or the second term<ref>R. F. Pawula, "Generalizations and extensions of the Fokker- Planck-Kolmogorov equations," in IEEE Transactions on Information Theory, vol. 13, no. 1, pp. 33-41, January 1967, doi: 10.1109/TIT.1967.1053955.</ref><ref>Pawula, R. F. (1967). Approximation of the linear Boltzmann equation by the Fokker-Planck equation. Physical review, 162(1), 186.</ref>. If the expansion continues past the second term it must contain an infinite number of terms, in order that the solution to the equation be interpretable as a probability density function.<ref>{{cite |
The Pawula theorem states that the expansion either stops after the first term or the second term<ref>R. F. Pawula, "Generalizations and extensions of the Fokker- Planck-Kolmogorov equations," in IEEE Transactions on Information Theory, vol. 13, no. 1, pp. 33-41, January 1967, doi: 10.1109/TIT.1967.1053955.</ref><ref>Pawula, R. F. (1967). Approximation of the linear Boltzmann equation by the Fokker-Planck equation. Physical review, 162(1), 186.</ref>. If the expansion continues past the second term it must contain an infinite number of terms, in order that the solution to the equation be interpretable as a probability density function.<ref>{{cite book|last1=Risken|first1=Hannes|title=The Fokker-Planck Equation: Methods of Solution and Applications|date=6 December 2012|isbn=9783642968075|url=https://books.google.com/books?id=dXvpCAAAQBAJ&q=Pawula-Theorem&pg=PA70}}</ref> |
||
==Implementations== |
==Implementations== |
||
* Implementation in Python: https://github.com/LRydin/KramersMoyal <ref>{{cite journal |last1=Rydin Gorjão |first1=L. |last2=Meirinhos | first2=F. |year=2019 |title=kramersmoyal: Kramers--Moyal coefficients for stochastic processes |journal=[[Journal of Open Source Software]] |volume=4 |issue=44 |pages=1693 |doi=10.21105/joss.01693|doi-access=free }}</ref> |
* Implementation in Python: https://github.com/LRydin/KramersMoyal <ref>{{cite journal |last1=Rydin Gorjão |first1=L. |last2=Meirinhos | first2=F. |year=2019 |title=kramersmoyal: Kramers--Moyal coefficients for stochastic processes |journal=[[Journal of Open Source Software]] |volume=4 |issue=44 |pages=1693 |doi=10.21105/joss.01693|bibcode=2019JOSS....4.1693G |doi-access=free }}</ref> |
||
==References== |
==References== |
||
Revision as of 19:44, 2 October 2020
In stochastic processes, Kramers–Moyal expansion refers to a Taylor series expansion of the master equation, named after Hans Kramers and José Enrique Moyal.[1][2] This expansion transforms the integro-differential master equation
![{\displaystyle {\frac {\partial p(x,t)}{\partial t}}=\int dx'[W(x|x')p(x',t)-W(x'|x)p(x,t)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52cd1ff057fad02a4d8973266fa48180ce68bf55)
where (for brevity, this probability is denoted by ) is the transition probability density, to an infinite order partial differential equation[3][4][5]
![{\displaystyle {\frac {\partial p(x,t)}{\partial t}}=\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n!}}{\frac {\partial ^{n}}{\partial x^{n}}}[\alpha _{n}(x)p(x,t)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2ebeb240f22abdcec80ed08908598ee76af8a2a)
where
Here is the transition probability rate. Fokker–Planck equation is obtained by keeping only the first two terms of the series in which is the drift and is the diffusion coefficient.
Pawula theorem
The Pawula theorem states that the expansion either stops after the first term or the second term[6][7]. If the expansion continues past the second term it must contain an infinite number of terms, in order that the solution to the equation be interpretable as a probability density function.[8]
Implementations
- Implementation in Python: https://github.com/LRydin/KramersMoyal [9]
References
- ^ Kramers, H. A. (1940). "Brownian motion in a field of force and the diffusion model of chemical reactions". Physica. 7 (4): 284–304. Bibcode:1940Phy.....7..284K. doi:10.1016/S0031-8914(40)90098-2.
- ^ Moyal, J. E. (1949). "Stochastic processes and statistical physics". Journal of the Royal Statistical Society. Series B (Methodological). 11 (2): 150–210. JSTOR 2984076.
- ^ Gardiner, C. (2009). Stochastic Methods (4th ed.). Berlin: Springer. ISBN 978-3-642-08962-6.
- ^ Van Kampen, N. G. (1992). Stochastic Processes in Physics and Chemistry. Elsevier. ISBN 0-444-89349-0.
- ^ Risken, H. (1996). The Fokker–Planck Equation. Berlin, Heidelberg: Springer. pp. 63–95. ISBN 3-540-61530-X.
- ^ R. F. Pawula, "Generalizations and extensions of the Fokker- Planck-Kolmogorov equations," in IEEE Transactions on Information Theory, vol. 13, no. 1, pp. 33-41, January 1967, doi: 10.1109/TIT.1967.1053955.
- ^ Pawula, R. F. (1967). Approximation of the linear Boltzmann equation by the Fokker-Planck equation. Physical review, 162(1), 186.
- ^ Risken, Hannes (6 December 2012). The Fokker-Planck Equation: Methods of Solution and Applications. ISBN 9783642968075.
- ^ Rydin Gorjão, L.; Meirinhos, F. (2019). "kramersmoyal: Kramers--Moyal coefficients for stochastic processes". Journal of Open Source Software. 4 (44): 1693. Bibcode:2019JOSS....4.1693G. doi:10.21105/joss.01693.