This program simulates Browninan motion (random walk)
as the basis of diffusion and of Boltzman probability distribution of energy
This animation illustrates the random walk
of a particle (a gas molecule or a molecule of solute) according to the theory
of Browninan motion, described by Einstein equation, that predicts that
the probability distribuition of the position of the particle along any axis
described by a Gaussian with variance equal to \(D\cdot t\), where
\(D\) is the “diffusion coefficient” (\({\mu}m^2\)) and \(t\)
is the time lapsed since the moment the movement started:
\(p(x) =\) \(\frac{\exp(-x^2/2Dt)}{\sqrt{2\pi Dt}}\)
with average distance \(\bar{x}=\sqrt{Dt}\)
The position of each particle results at each moment from the sum of the steps
in random directions. Similarly, its velocity would result at each moment from the sum
of the hits it received in random directions.
Thus, the probability distribution of the velocities will be
\(p(v) =\) \(\frac{\exp(-v^2/2C)}{\sqrt{2\pi C}}\),
with average velocity \(\bar{v}=\sqrt{C}\)
and, since the kinetic energy of a particle is \(E=m\cdot{v}^2\), the probability
distribution of the energies will be
\(p(E) =\) \(\frac{\exp(-E/2RT)}{\sqrt{2\pi RT}}\)
with average energy \(\bar{E}=RT\).
The fraction of the particles expected to have enough energy to overcome a barrier
(activation energy) \(E_A\) will be \(p(E>=E_A)=\exp(-E_A/RT)\)