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Scientific Volume Imaging B.V.

Scientific Volume Imaging

Deconvolution - Visualization - Analysis


Brownian motion

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
2> as some kind of diffusion coefficient: it is the average displacement of many particles along this coordinate per unit time.

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.

See also the Huygens' Object Stabilizer.

Other sources

White noise and Brownian motion