Brownian motion
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The term Brownian motion (in honor of the botanist Robert Brown) refers to either '''' 1. The physical phenomenon that minute particles, immersed in a fluid, move about randomly; or '''' 2. The mathematical models used to describe those random movements. '''' [...] '''' Mathematically, Brownian motion is a Wiener process in which the conditional probability distribution of the particle's position at time t + dt, given that its position at time t is p, is a normal distribution with a mean of p + µ dt and a variance of κ^2> dt; the parameter µ is the drift velocity, and the parameter κ^2> is the power of the noise. '''' |
(Wikipedia)
Thus brownian motion along one coordinate, without drift velocity, can be simulated by adding to the particle's position, at each time, random displacements that are normally distributed with mean zero.
In the Micro Rotation Workbench we simulate brownian motion along each space coordinate with the aid of a growing array of normally distributed values B(t_i) with mean µ = 0 and standard deviation σ = D dt
1/2>, where t_i is the time frame being simulated, t_0 = 0, t_i+1 = t_i + dt. We can think of D = κ
The regular brownian motion makes the particle to move a total distance r(t_i) from the origin that is the sumation of the elements of the array from t_0 to t_i.
A limited diffusion (due to the effect of the electrical fields) can be roughly simulated by restricting this sumation inside a limited time window only, from t_i- Δt to t_i. When Δt → 0 we recover a completely random "white noise" around the origin, with no effective diffusion.
- Introduction and simulation
- Sh. Kogan, Electronic Noise and Fluctuations in Solids (ISBN 0521460344) Cambridge University Press, 1996.
See also the Huygens' Object Stabilizer.
Other sources
- Andrei N Borodin, Paavo Salminen, Handbook of Brownian Motion (ISBN 3764367059) Birkhäuser, 2002.
- Ronald B Guenther, Partial Differential Equations of Mathematical Physics and Integral Equations (ISBN 0486688895) Courier Dover Publications, 1996.
White noise and Brownian motion
- Introduction
- D Kannan, V Lakshmikantham, Kannan Kannan, Handbook of Stochastic Analysis and Applications chap. 3 (ISBN 0824706609) Marcel Dekker, 2001.
- Kuo Kuo, Hui-Hsiung Kuo, White Noise Distribution Theory (ISBN 0849380774) CRC Press, 1996.
- more details: Robert Hermann, Cartanian Geometry Nonlinear Waves/Interdisciplinary Mathematics Series ; Vol Xxi (ISBN 0915692295) Math Science Pr, 1980.