Brownian Motion Simulation II
Simulation of the Brownian motion of a large (blue) particle with a radius of 0.3 m and mass 1 kg, surrounded by 149 (green) particles with radii of 0.1 m and mass 0.4 kg, released at (x,y) = (0,0) at time t = 0 s. A blue line traces the movement of the large blue particle. Initial velocity components of the particles are normally distributed random values. The simulation is performed using a script written in Python. Large numbers of simulations show the mean squared displacement (relative to initial coordinates) of the blue 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 is given 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. The scripts used for simulation and analysis can be found at: https://github.com/derekharrison/BrownianMotionRev
Simulation of the Brownian motion of a large (blue) particle with a radius of 0.3 m and mass 1 kg, surrounded by 149 (green) particles with radii of 0.1 m and mass 0.4 kg, released at (x,y) = (0,0) at time t = 0 s. A blue line traces the movement of the large blue particle. Initial velocity components of the particles are normally distributed random values. The simulation is performed using a script written in Python. Large numbers of simulations show the mean squared displacement (relative to initial coordinates) of the blue 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 is given 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. The scripts used for simulation and analysis can be found at: https://github.com/derekharrison/BrownianMotionRev