Brownian motion simulation
Simulation of the Brownian motion of a large (red) particle with a radius of 0.7 m and mass 2 kg, surrounded by 124 (blue) particles with radii of 0.2 m and mass 0.9 kg, released at (x,y) = (0,0) at time t = 0 s. A blue line traces the movement of the large red particle. Initial velocity components of the blue particles are uniformly distributed random values between -2.0 m/s and 2.0 m/s.
Simulation is performed using a script written in Matlab. Moderate numbers of simulations indicate the mean squared displacement (relative to initial coordinates) of the red particle at a given time is proportional with the time the particle traveled. From this it follows that the probability density distribution for the large particle can be approximated by:
ρ(r,t)=N/(4πDt) exp(-r^2/(4Dt))
Where ρ is the local probability density (or concentration), r the radial coordinate (m), t the time (s), N the number of (simulated) particles (in this case 1) and D the diffusion coefficient (m^2/s). The distribution is the solution to the unsteady state diffusion equation describing the time evolution of a Gaussian pulse:
∂ρ/∂t=D/r ∂/∂r(r∂ρ/∂r)
With boundary conditions: 2π∫ρ(r=0,t=0)rdr = 1, ρ(r ≠ 0,0) = 0,
∂ρ(r = inf, t ≠ 0)/∂r = 0, ∂ρ(r=0, t ≠ 0)/∂r = 0.
Particle collisions were taken into account and velocities were updated using the hard-sphere method described by Wang (1992).
Refs:
Wang, Y. and Mason, M.T., (1992). Two-dimensional rigid-body collisions with friction. J. Appl. Mech. 59, 635.
The Matlab script can be found at:
https://github.com/derekharrison/Brow...
Update:
Simulation of Brownian motion is revisited in • Brownian Motion Simulation II