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Nanoscale Fluid Dynamics
Published in Klaus D. Sattler, 21st Century Nanoscience – A Handbook, 2020
Ravi Radhakrishnan, N. Ramakrishnan, David M. Eckmann, Portonovo S. Ayyaswamy
where ξ(t) represents a unit-normalized white noise process. The SDE encodes for the Brownian dynamics (BD) of the particle in the limit of zero inertia. The corresponding equation when inertia of the particle is added is often referred to as the Langevin equation. In summary, Brownian or thermal effects are described within the hydrodynamics framework using either the FHD approach or the BD/Langevin equation approach.
Quantum Signals at Microwave Devices
Published in Maged Marghany, Automatic Detection Algorithms of Oil Spill in Radar Images, 2019
From the point of view of physics, Langevin equation is named after Paul Langevin, which is a stochastic differential equation demonstrating the time growth of a subset of the degree of freedom. These quantities of freedom usually are combined variables differing only gradually in contrast to the other variables of the system. The fast microwave photon particles are responsible for the stochastic nature of the Langevin equation.
Ulam stability for nonlinear-Langevin fractional differential equations involving two fractional orders in the ψ-Caputo sense
Published in Applicable Analysis, 2022
Zidane Baitiche, Choukri Derbazi, Mohammed M. Matar
On the other hand, the Langevin equation was formulated by Paul Langevin in 1908 to describe the evolution of physical phenomena in fluctuating environments such as Brownian motion [43]. After that, various generalizations of the Langevin equation were proposed and studied by many scholars we mention here some works [44–53]. Recently, many researchers have investigated sufficient conditions for the existence, uniqueness, and stability of solutions for the nonlinear fractional Langevin equations involving various types of fractional derivatives and by using different types of methods such as standard fixed point theorems, Leray–Schauder theory, variational methods, or monotone iterative technique combined with the method of upper and lower solutions, etc. For more details see [54–66]. However, to the best of our knowledge, few results can be found on the existence and the Ulam–Hyers stability of solutions for fractional Langevin equations with the ψ-Caputo fractional derivative except that of [67].
Investigation of mode coupling in low and high NA step index plastic optical fibres using the Langevin equation
Published in Journal of Modern Optics, 2020
Svetislav Savović, Alexandar Djordjevich
The Fokker–Planck equation (3) can be transformed into the Langevin equation [8] in the form: where gΓ(z) is a Gaussian distributed random Langevin force with the strength g. This equation is also called stochastic differential equation because it contains the random force. The is assumed to satisfy the following equations [8]: where the brackets denote an average over the probability distribution function. Following the Ito rule [8], one obtains V = h and D = g2. Thus the Langevin equation can be expressed in terms of the drift and diffusion coefficients of the Fokker–Planck equation in the following form: