Asymptotic behavior of velocity process in the Smoluchowski–Kramers approximation for stochastic differential equations

1991 ◽  
Vol 23 (02) ◽  
pp. 317-326
Author(s):  
Kiyomasa Narita
Keyword(s):  
Mean Field ◽  
Planck Equation ◽  
Equation Of Motion ◽  
Distribution Functions ◽  
Two Dimensional ◽  
Large Parameter ◽  
Uhlenbeck Process ◽  
One Dimensional ◽  
Kramers Equation ◽  
Non Linear Oscillator

Here a response of a non-linear oscillator of the Liénard type with a large parameter α ≥ 0 is formulated as a solution of a two-dimensional stochastic differential equation with mean-field of the McKean type. This solution is governed by a special form of the Fokker–Planck equation such as the Smoluchowski–Kramers equation, which is an equation of motion for distribution functions in position and velocity space describing the Brownian motion of particles in an external field. By a change of time and displacement we find that the velocity process converges to a one-dimensional Ornstein–Uhlenbeck process as α →∞.

Asymptotic behavior of velocity process in the Smoluchowski–Kramers approximation for stochastic differential equations

1991 ◽  
Vol 23 (2) ◽  
pp. 317-326 ◽  
Author(s):  
Kiyomasa Narita
Keyword(s):  
Mean Field ◽  
Planck Equation ◽  
Equation Of Motion ◽  
Distribution Functions ◽  
Two Dimensional ◽  
Large Parameter ◽  
Uhlenbeck Process ◽  
One Dimensional ◽  
Kramers Equation ◽  
Non Linear Oscillator

Here a response of a non-linear oscillator of the Liénard type with a large parameter α ≥ 0 is formulated as a solution of a two-dimensional stochastic differential equation with mean-field of the McKean type. This solution is governed by a special form of the Fokker–Planck equation such as the Smoluchowski–Kramers equation, which is an equation of motion for distribution functions in position and velocity space describing the Brownian motion of particles in an external field. By a change of time and displacement we find that the velocity process converges to a one-dimensional Ornstein–Uhlenbeck process as α →∞.

The Smoluchowski–Kramers approximation for the stochastic Liénard equation by mean-field

1991 ◽  
Vol 23 (2) ◽  
pp. 303-316 ◽  
Author(s):  
Kiyomasa Narita
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Diffusion Process ◽  
Langevin Equation ◽  
Mean Field ◽  
Two Dimensional ◽  
Large Parameter ◽  
Rigorous Evaluation ◽  
One Dimensional ◽  
Limit Diffusion

The oscillator of the Liénard type with mean-field containing a large parameter α < 0 is considered. The solution of the two-dimensional stochastic differential equation with mean-field of the McKean type is taken as the response of the oscillator. By a rigorous evaluation of the upper bound of the displacement process depending on the parameter α, a one-dimensional limit diffusion process as α → ∞is derived and identified. Then our result extends the Smoluchowski–Kramers approximation for the Langevin equation without mean-field to the McKean equation with mean-field.

The Smoluchowski–Kramers approximation for the stochastic Liénard equation by mean-field

1991 ◽  
Vol 23 (02) ◽  
pp. 303-316 ◽  
Author(s):  
Kiyomasa Narita
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Diffusion Process ◽  
Langevin Equation ◽  
Mean Field ◽  
Two Dimensional ◽  
Large Parameter ◽  
Rigorous Evaluation ◽  
One Dimensional ◽  
Limit Diffusion

The oscillator of the Liénard type with mean-field containing a large parameter α &lt; 0 is considered. The solution of the two-dimensional stochastic differential equation with mean-field of the McKean type is taken as the response of the oscillator. By a rigorous evaluation of the upper bound of the displacement process depending on the parameter α, a one-dimensional limit diffusion process as α → ∞is derived and identified. Then our result extends the Smoluchowski–Kramers approximation for the Langevin equation without mean-field to the McKean equation with mean-field.

Comparison Between Theoretical Values and Simulation Results of Transport Coefficients for the Dissipative Particle Dynamics Method

2004 ◽  
Author(s):  
Akira Satoh
Keyword(s):  
Transport Coefficients ◽  
Dissipative Particle Dynamics ◽  
Planck Equation ◽  
Equation Of Motion ◽  
Particle Dynamics ◽  
Momentum Conservation ◽  
Distribution Functions ◽  
Fokker Planck Equation ◽  
Dynamics Method ◽  
Fokker Planck

In the present study, we have derived an expression for transport coefficients such as viscosity, from the equation of motion of dissipative particles. In the concrete, we have shown the Fokker-Planck equation in phase space, and macroscopic conservation equations such as the equation of continuity and the equation of momentum conservation. The basic equations of the single-particle and pair distribution functions have been derived using the Fokker-Planck equation. The solutions of these distribution functions have approximately been solved by the perturbation method under the assumption of molecular chaos. The expression of the viscosity due to dissipative forces has been obtained using the approximate solutions of the distribution functions. Also, we have conducted non-equilibrium dynamics simulations to investigate the influence of the parameters, which have appeared in defining the equation of motion in the dissipative particle dynamics method.

Stochastic Properties of a One-Dimensional Discrete Ginzburg-Landau Field

1981 ◽  
Vol 36 (1) ◽  
pp. 1-9
Author(s):  
M. Jaspers ◽  
W. Schattke
Keyword(s):  
Mean Field ◽  
Planck Equation ◽  
Field Approximation ◽  
Mean Field Approximation ◽  
Ginzburg Landau ◽  
Mean Values ◽  
One Dimensional ◽  
Slowing Down ◽  
Set Up ◽  
Stochastic Properties

Starting from a master equation for a discrete order parameter a dynamical model is set up via mean-field approximation in the Fokker-Planck equation. The time evolution of some mean values is calculated numerically, showing two transitions with characteristic slowing down of the relaxation time

scholarly journals Research on two dimensional Wiener stochastic degradation model based on the wear model

2018 ◽  
Vol 169 ◽  
pp. 01037
Author(s):  
Hui Zhang ◽  
Jun Yao ◽  
Yan-Lin Zhao
Keyword(s):  
Planck Equation ◽  
Degradation Process ◽  
Analytical Form ◽  
Wear Volume ◽  
Two Dimensional ◽  
Wear Model ◽  
Degradation Model ◽  
One Dimensional ◽  
Wear Process ◽  
Degradation Characteristic

One dimensional Wiener degradation process is often used to describe the degradation of product performance. However one dimensional Wiener degradation process doesn’t sufficiently consider the relevance of multiple degradation factors and wear process, which lead to inaccurate results. To overcome these problems, a new two dimensional Wiener stochastic degradation model is proposed, which applys to the products on wear process and stochastic degradation process. Combining wear model and two dimensional Wiener stochastic degradation model, a new reliability analytical form is obtained by constructing the Fokker-Planck equation. Then using the relation among wear volume, degradation characteristic lifetime and drift parameter, parameters of two dimensional Wiener degradation model on the basic of wear model can be estimated. Compared with the existing approaches, the proposed method can effectively improve accuracy. Finally, a case study is illustrated the application and advantages of the proposed method.

GLOBAL DYNAMICS IN NETWORKS OF 1D ELEMENTS WITH TIME DELAYED MEAN FIELD COUPLING SUBJECT TO NOISE: SYNCHRONIZATION THRESHOLD, MULTISTABILITY AND HYSTERESIS

2010 ◽  
Vol 20 (06) ◽  
pp. 1825-1836 ◽  
Author(s):  
A. POTOTSKY ◽  
N. JANSON
Keyword(s):  
Boundary Value Problem ◽  
Mean Field ◽  
Planck Equation ◽  
Global Dynamics ◽  
Polynomial Potential ◽  
One Dimensional ◽  
Field Coupling ◽  
Exact Location ◽  
Continuation Technique ◽  
Fokker Planck

We determine the boundary of the synchronization domain of a large number of one-dimensional continuous stochastic elements with time delayed nonhomogeneous mean-field coupling. The exact location of the synchronization threshold is shown to be a solution of the boundary value problem (BVP) which was derived from the linearized Fokker–Planck equation. Here the synchronization threshold is found by solving this BVP using the continuation technique (AUTO). Approximate analytics is obtained using expansion into eigenfunctions of the stationary Fokker–Planck operator. Multistability and hysteresis are demonstrated for the case of bistable elements with a polynomial potential.

The simulation of plasma double-layer structures in two dimensions

1983 ◽  
Vol 29 (1) ◽  
pp. 45-84 ◽  
Author(s):  
Joseph E. Borovsky ◽  
Glenn Joyce
Keyword(s):  
Magnetic Field ◽  
Electron Beam ◽  
Double Layer ◽  
Ion Beam ◽  
Two Dimensions ◽  
Distribution Functions ◽  
Double Layers ◽  
Two Dimensional ◽  
One Dimensional ◽  
Electron Cyclotron Waves

Electrostatic plasma double layers are numerically simulated by means of a magnetized 2½-dimensional particle-in-cell method, periodic in one direction and bounded by reservoirs of Maxwellian plasma in the other. The investigation of planar double layers indicates that these one-dimensional potential structures are susceptible to periodic disruption by plasma instabilities. A slight increase in the double-layer thickness with an increase in its obliqueness to the magnetic field is observed. It is noted that weak magnetization results in the double-layer electric-field alignment of particles accelerated by these potential structures and that strong magnetization results in their magnetic-field alignment. Electron-beam-excited electrostatic electron cyclotron waves and ion-beam-driven electrostatic turbulence are present in the plasmas adjacent to the double layers. The numerical simulations of spatially periodic two-dimensional double layers also exhibit cyclical instability. A morphological invariance in two-dimensional double layers with respect to the degree of magnetization implies that the potential structures scale with Debye lengths rather than with gyroradii. Ion-beam-driven electrostatic turbulence and electron-beam-driven plasma waves are again detected. A simplified one-dimensional model of oblique plasma double layers, using water-bag velocity distribution functions, is presented in an appendix.

A one-dimensional self-gravitating stellar gas

1966 ◽  
Vol 25 ◽  
pp. 46-48 ◽  
Author(s):  
M. Lecar
Keyword(s):  
Gravitational Field ◽  
Phase Space ◽  
Motion Picture ◽  
Space Distribution ◽  
Two Dimensional ◽  
One Dimensional ◽  
Phase Space Distribution ◽  
Dimensional Phase Space ◽  
The Mean

“Dynamical mixing”, i.e. relaxation of a stellar phase space distribution through interaction with the mean gravitational field, is numerically investigated for a one-dimensional self-gravitating stellar gas. Qualitative results are presented in the form of a motion picture of the flow of phase points (representing homogeneous slabs of stars) in two-dimensional phase space.

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