Kramers–Moyal expansion: Difference between revisions
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The [[Fokker–Planck equation]] is obtained by keeping only the first two terms of the series in which <math>\alpha_1</math> is the [[drift velocity|drift]] and <math>\alpha_2</math> is the diffusion coefficient.<ref>{{cite book |first=Wolfgang |last=Paul |first2=Jörg |last2=Baschnagel |chapter=A Brief Survey of the Mathematics of Probability Theory |title=Stochastic Processes |pages=17–61 [esp. 33–35] |publisher=Springer |year=2013 |isbn= |doi=10.1007/978-3-319-00327-6_2 }}</ref> |
The [[Fokker–Planck equation]] is obtained by keeping only the first two terms of the series in which <math>\alpha_1</math> is the [[drift velocity|drift]] and <math>\alpha_2</math> is the diffusion coefficient.<ref>{{cite book |first=Wolfgang |last=Paul |first2=Jörg |last2=Baschnagel |chapter=A Brief Survey of the Mathematics of Probability Theory |title=Stochastic Processes |pages=17–61 [esp. 33–35] |publisher=Springer |year=2013 |isbn= |doi=10.1007/978-3-319-00327-6_2 }}</ref> |
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Also, the moments, assuming they exist, evolves as<ref>{{Citation |last=Tabar |first=M. Reza Rahimi |title=Kramers–Moyal Expansion and Fokker–Planck Equation |date=2019 |url=https://doi.org/10.1007/978-3-030-18472-8_3 |work=Analysis and Data-Based Reconstruction of Complex Nonlinear Dynamical Systems: Using the Methods of Stochastic Processes |pages=19–29 |editor-last=Rahimi Tabar |editor-first=M. Reza |access-date=2023-06-09 |series=Understanding Complex Systems |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-030-18472-8_3 |isbn=978-3-030-18472-8}}</ref> |
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<math display="block">\frac{\partial}{\partial t}\left\langle x^n\right\rangle=\sum_{k=1}^n \frac{n !}{(n-k) !}\left\langle x^{n-k} D^{(k)}(x, t)\right\rangle</math>where angled brackets mean taking the expectation: <math>\left\langle f\right\rangle = \int f(x) p(x, t)dx</math>. |
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=== n-dimensional version === |
=== n-dimensional version === |
Revision as of 09:10, 9 June 2023
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]
Statement
Start with the integro-differential master equation
where is the transition probability function, and is the probability density at time .
The Kramers–Moyal expansion transforms the above to an infinite order partial differential equation[3][4][5]
and also
where are the Kramers–Moyal coefficients, defined by
The 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.[6]
Also, the moments, assuming they exist, evolves as[7]
n-dimensional version
The above version is the one-dimensional version. It generalizes to n-dimensions. (Section 4.7 [5])
Proof
In usual probability, where the probability density does not change, the moments of a probability density function determines the probability density itself by a Fourier transform (details may be found at the characteristic function page):
The same computation with gives the other equation.
Forward and backward equations
The equation can be recast into a linear operator form, using the idea of infinitesimal generator.
Define the linear operator
is the Hermitian adjoint of .
Computing the Kramers–Moyal coefficients
By definition,
Since the even central moments are nonnegative, we have for all .
When the stochastic process is the Markov process , we can directly solve for as approximated by a normal distribution with mean and variance . This then allows us to compute the central moments, and so
Pawula theorem
Let the operator be defined such . The probability density evolves by . Different order of gives different level of approximation.
- : the probability density does not evolve
- : it evolves by deterministic drift only.
- : it evolves by drift and Brownian motion (Fokker-Planck equation)
- : the fully exact equation.
Pawula theorem states that for any other choice of , there exists a probability density function that can become negative during its evolution (and thus fail to be a probability density function).[8][9][10]
However, this doesn't mean that Kramers-Moyal expansions truncated at other choices of is useless. Though the solution must have negative values at least for sufficiently small times, the resulting approximation probability density may still be better than the approximation.
Proof
By Cauchy–Schwarz inequality, the central moment functions satisfy . So, taking the limit, we have .
If some for some , then . In particular, . So the existence of any nonzero coefficient of order implies the existence of nonzero coefficients of arbitrarily large order.
Also, if , then . So the existence of any nonzero coefficient of order implies all coefficients of order are positive.
Thus, either the sequence becomes zero at the third term, or all its even terms are positive.
Applications
In many textbooks, the expansion is used only to derive the Fokker–Planck equation, and never used again.
For a real stochastic process, one can compute its central moment functions from experimental data on the process, from which one can then compute its Kramers–Moyal coefficients, and thus empirically measure its Kolmogorov forward and backward equations. This is implemented as a python package [11]
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. S2CID 33337019.
- ^ 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.
- ^ a b Risken, H. (1996). The Fokker–Planck Equation. Berlin, Heidelberg: Springer. pp. 63–95. ISBN 3-540-61530-X.
- ^ Paul, Wolfgang; Baschnagel, Jörg (2013). "A Brief Survey of the Mathematics of Probability Theory". Stochastic Processes. Springer. pp. 17–61 [esp. 33–35]. doi:10.1007/978-3-319-00327-6_2.
- ^ Tabar, M. Reza Rahimi (2019), Rahimi Tabar, M. Reza (ed.), "Kramers–Moyal Expansion and Fokker–Planck Equation", Analysis and Data-Based Reconstruction of Complex Nonlinear Dynamical Systems: Using the Methods of Stochastic Processes, Understanding Complex Systems, Cham: Springer International Publishing, pp. 19–29, doi:10.1007/978-3-030-18472-8_3, ISBN 978-3-030-18472-8, retrieved 2023-06-09
- ^ Pawula, R. F. (1967). "Generalizations and extensions of the Fokker–Planck–Kolmogorov equations" (PDF). IEEE Transactions on Information Theory. 13 (1): 33–41. 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): 186–188. Bibcode:1967PhRv..162..186P. doi:10.1103/PhysRev.162.186.
- ^ 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. arXiv:1912.09737. Bibcode:2019JOSS....4.1693G. doi:10.21105/joss.01693.