scholarly journals Using some results about the Lie evolution of differential operators to obtain the Fokker-Planck equation for non-Hamiltonian dynamical systems of interest

2018 ◽  
Vol 59 (5) ◽  
pp. 053303 ◽  
Author(s):  
Marco Bianucci
Keyword(s):  
Dynamical Systems ◽  
Differential Operators ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
Hamiltonian Dynamical Systems ◽  
Fokker Planck

scholarly journals Beating the curse of dimension with accurate statistics for the Fokker–Planck equation in complex turbulent systems

2017 ◽  
Vol 114 (49) ◽  
pp. 12864-12869 ◽  
Author(s):  
Nan Chen ◽  
Andrew J. Majda
Keyword(s):  
Dynamical Systems ◽  
Planck Equation ◽  
Dimensional Subspace ◽  
Gaussian Kernel ◽  
High Dimensional ◽  
Block Decomposition ◽  
Curse Of Dimension ◽  
Fokker Planck Equation ◽  
Non Gaussian ◽  
Fokker Planck

Solving the Fokker–Planck equation for high-dimensional complex dynamical systems is an important issue. Recently, the authors developed efficient statistically accurate algorithms for solving the Fokker–Planck equations associated with high-dimensional nonlinear turbulent dynamical systems with conditional Gaussian structures, which contain many strong non-Gaussian features such as intermittency and fat-tailed probability density functions (PDFs). The algorithms involve a hybrid strategy with a small number of samples L, where a conditional Gaussian mixture in a high-dimensional subspace via an extremely efficient parametric method is combined with a judicious Gaussian kernel density estimation in the remaining low-dimensional subspace. In this article, two effective strategies are developed and incorporated into these algorithms. The first strategy involves a judicious block decomposition of the conditional covariance matrix such that the evolutions of different blocks have no interactions, which allows an extremely efficient parallel computation due to the small size of each individual block. The second strategy exploits statistical symmetry for a further reduction of L. The resulting algorithms can efficiently solve the Fokker–Planck equation with strongly non-Gaussian PDFs in much higher dimensions even with orders in the millions and thus beat the curse of dimension. The algorithms are applied to a 1,000-dimensional stochastic coupled FitzHugh–Nagumo model for excitable media. An accurate recovery of both the transient and equilibrium non-Gaussian PDFs requires only L=1 samples! In addition, the block decomposition facilitates the algorithms to efficiently capture the distinct non-Gaussian features at different locations in a 240-dimensional two-layer inhomogeneous Lorenz 96 model, using only L=500 samples.

AN APPLICATION OF FOKKER–PLANCK EQUATION TO JOSEPHSON JUNCTION

1989 ◽  
Vol 9 (1) ◽  
pp. 109-120
Author(s):  
G. Liao ◽  
A.F. Lawrence ◽  
A.T. Abawi
Keyword(s):  
Josephson Junction ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
Fokker Planck
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