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Brownian motion may not seem very important but, if you read the Wikipedia reference, you can see that it was at the forefront of physics in 1905 when Einstein and others wrote papers describing a theory for it. It gave a basis for the particles in Boltzmann's statistical theory for particles in nature. It also provided a means to experimentally determine both Avogadro's number of molecules in a gram molecular weight and the size of molecules.

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 `mum` diameter water droplet immersed in air. This will have a
mass of about `10^(-13)` kilograms while the air molecules each have a mass of about `5*10^(-26)` kilograms.
The speed ratio is then about `9*10^(-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 energy gain of the large particle 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 an example of the results see 2500 Walks
As always, the learner is welcome to examine the Javascript code when using the Chrome browser by pressing F12 in Windows.

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