Brownian Motion Animation Using Hard Sphere Scattering

Brownian motion in a gas occurs when a large (very massive) particle is embedded in a gas. At thermal equilibrium the average energy of the large particle is the same as the average energy of each gas particle. However, the speed ratio V/v of the large particle/gas particle is reduced by a factor V/v=sqrt(m/M) where m is the mass of a gas particle and M is the mass of the large particle. On the other hand, the momentum ratio is P/p=sqrt(M/m) and this means that the large particle is the "elephant in the room" and can go where it pleases until slowed down by large groups of gas particles. A typical practical scenario would be a 5 micrometer diameter water droplet immersed in air. This will have a mass of about 1e-13 kilograms while the air molecules each have a mass of about 5e-26 kilograms. The speed ratio is then about 9e-7. Since the speed of the large particle is so slow, it "walks" through the gas slowly, scattering when it encounters a gas particle. In this animation, the mass ratio and the radius ratio of the large particle to the gas particle is adjustable but not to the extent of this scenario. In addition, the number of time increments per walk is adjustable. In order to collect statistical data on the large particle movement, another adjustment is the number of walks. After each walk the final x and y positions are used to increment (fill) the corrsponding frequency bin for that x and y. A color plot of the current filling of the frequency bins is then made. Since the large particle displacement during a walk are small, the displayed positions of the bins are "zoomed" by an adjustable factor. In this animation, the starting kinetic energy for the large particle has been set to zero instead of being equal the kinetic energy of the gas particles as would be the case in thermal equilibrium. This is necessary because, for equal kinetic energy, the momentum ratio, P/p, is so large that the initial large particle momentum would cause it to go to large displacements in the steps that we use for collecting the end walk (x,y) data. This results in a doughnut shaped color frequency distribution plot. I have chosen to provide three different ways of showing the animation: "Animate All Displacements" shows the displaements of all the particles (large and gas) during a walk as well as the final displacement. "Animate Final Displacements" shows just the final (x,y) displacement and plots the distibution of the displacements. Of course, Animate All Displacements is a lot slower because of the time it takes to draw the gas particles at each step. In addition to the 2D color plot of the final displacements, I have plotted a histogram of the radial distribution of the bins. Since Brownian Motion is essentially a diffusion process, I have included a red gaussian fit to the radial plot.

The third animation button shows how the large particle (starting from zero) very slowly gains kinetic energy, This energy gain has similar statistics to the displacement of random walk. The gas particles can both increase or decrease the large particle energy. Initially the large particle has negligble velocity. If the gas particle velocity is parallel or antiparallel to the large particle velocity, they will increase the large particle energy by approximately 4m/M times their energy. After the large particle gains some momentum, collisions with the gas particles can reduce its energy. Since many random walks take a long time to calculate, the color plot of the distribution is stored in an array that may be played back much faster than the calculation time (Start Movie button). As always, the learner is welcome to examine the Javascript code when using the Chrome browser by pressing F12 in Windows.