Brownian Motion
Particle diffusion may also be described mathematically (Einstein,
1926). Note from your observations of the simulation that the
distance a particle moves in any given direction is not a
simple linear function of the time elapsed, as it would be, for
example, if you wanted to describe the movement of a snowball
you'd tossed at a professor. Rather, the particle (like the drunk)
exhibits what is called a random walk, where the average
displacement in a given direction (bar delta Y) is a linear
function of the square root of the time period (t) during
which its movement is measured. More commonly, the relationship is
expressed as:
___
(deltaY)2 = Dt
and D - the Diffusion Coefficient - is defined
by the Boltzmann's constant (k) and two variables: the
absolute temperature (T) and the frictional coefficient
(f) of the particle:
D = kT/f
Thus, diffusion is faster at higher temperatures and slower for
larger, irregularly shaped particles. For spherical particles,
the frictional coefficient is defined, in turn, by Stokes Law:
f = 6(pi)(nu)r
where (nu) is the viscosity of the medium surrounding the
particle and (r) is its radius. Combining these expressions,
the Diffusion Coefficient of a spherical particle
in a given direction is:
D = kT/[6(pi)(nu)r]
which is often referred to as the Stokes-Einstein equation.