Brownian Motion
General
- Overview article by A. Wipf, University of Jena
- Brownian Motion by Peter Mörters and Yuval Peres
"The Brownian process describes the disordered motion of small particles suspended in a liquid. It is believed that Brown studied pollen particles floating in water under the microscope. He observed minute particles executing a jittery motion. The theory of this motion has been invented by Albert Einstein and Marian Smoludchowski. The mathematically rigorous construction of the corresponding stochastic process has been developed by Norbert Wiener." (Extraced from: A. Wipf "Path Integrals" (2019), p. 60, Brownian Motion, University of Jena)
#brownianMotion #multipleParticle #forwardModeling
Analysis of the densities
In the following you have the possibility to have a closer look at the distributions in certain timesteps.
Explanations:
- \(t\): integration time in time units (default: 1000)
- \(n\): stepsize (default: 0.5)
- \(N\): number of steps (default: \(t/n = 2000\))
- \(N_{particles}\): number of the particles (default: \(N_{particles} = 1000\))
- \(h\): matrix with the shape \((N, 40)\) where \(40\) is the number of equally spaced categories in which the values are grouped. Thus in a histogram up to 40 bars would be possible.
- \(C_a\): factor of random deviation
Figure a) represents the distribution of the values between the steps \(N/2\) and \(N\).
Figure b) how often does which value occur?
Figure c) shows the distribution of the values in the first step (R: h[1,]; Python: h[0]). Here it is to be noted that with fixed starting value in the first step always the same distribution occurs. However, if the start is chosen randomly, variations can be seen here as well.
Figure d), e) and f) show the distributions of \(N_{particles}\) in the second, fourth and eighth step, respectively.
Figure g) shows the distribution of \(N_{particles}\) at step \(N/2\).
Figure h) shows the average of the distributions of the last and penultimate step.
Driving function
Following is the diving function. It is the backbone of the above results and can clarify its behavior.
- Driving function:
\( \begin{align} f_{(x)} &= dx^3+cx^2+bx-a \\ \\ F(x) = \int f(x) dx &= \frac{d}{4}x^4+\frac{c}{3}x^3+\frac{b}{2}x^2-x \end{align} \)