scholarly journals Smoluchowski approach to nonlinear Vlasov-Fokker-Planck equations: Stability analysis of beam dynamics and Haïssinski theory

2006 ◽  
Vol 9 (8) ◽  
Author(s):  
T. D. Frank
Keyword(s):  
Stability Analysis ◽  
Beam Dynamics ◽  
Fokker Planck Equations ◽  
Fokker Planck

scholarly journals Fokker–Planck representations of non-Markov Langevin equations: application to delayed systems

2019 ◽  
Vol 377 (2153) ◽  
pp. 20180131 ◽  
Author(s):  
Luca Giuggioli ◽  
Zohar Neu
Keyword(s):  
Langevin Equation ◽  
Complex Dynamics ◽  
Probability Distributions ◽  
Delay Systems ◽  
Langevin Equations ◽  
Delayed Systems ◽  
Temporal Features ◽  
Fokker Planck Equations ◽  
Random Fluctuations ◽  
Fokker Planck

Noise and time delays, or history-dependent processes, play an integral part in many natural and man-made systems. The resulting interplay between random fluctuations and time non-locality are essential features of the emerging complex dynamics in non-Markov systems. While stochastic differential equations in the form of Langevin equations with additive noise for such systems exist, the corresponding probabilistic formalism is yet to be developed. Here we introduce such a framework via an infinite hierarchy of coupled Fokker–Planck equations for the n -time probability distribution. When the non-Markov Langevin equation is linear, we show how the hierarchy can be truncated at n  = 2 by converting the time non-local Langevin equation to a time-local one with additive coloured noise. We compare the resulting Fokker–Planck equations to an earlier version, solve them analytically and analyse the temporal features of the probability distributions that would allow to distinguish between Markov and non-Markov features. This article is part of the theme issue ‘Nonlinear dynamics of delay systems’.

Modified Fokker-Planck equations

1978 ◽  
Vol 36 (1) ◽  
pp. 65-78 ◽  
Author(s):  
Glenn T. Evans
Keyword(s):  
Fokker Planck Equations ◽  
Fokker Planck
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