PHY452H1S Basic Statistical Mechanics. Lecture 3: Random walk. Taught by Prof. Arun Paramekanti
Posted by peeterjoot on January 16, 2013
Peeter’s lecture notes from class. May not be entirely coherent.
Spatial Random walk
One dimensional case
With a time interval , our total time is
We form a random variable for the total distance moved
In the above, I assume that the cross terms were killed with an assumption of uncorrelation.
Two dimensional case
The two dimensional case can be used for either spatial generalization of the above, or a one dimensional problem with time evolution as illustrated in (Fig 3)
Since we have a small correction , this is approximately
We see that random microscopics lead to a well defined macroscopic equation.
Continuity equation and Ficks law
Requirements for well defined diffusion results
- Particle number conservation (local)Imagine that we put a drop of ink in water. We’ll see the ink gradually spread out and appear to disappear. But the particles are still there. We require particle number concentration for well defined diffusion results.
This is expressed as a continuity equation and illustrated in (Fig 4).
Here is the probability current density, and is the probability density.
- Phenomenological law, or Fick’s law.
The probability current is linearly proportional to the gradient of the probability density.
Combining the above we have
So for constant diffusion rates we also arrive at the random walk diffusion result
Random walk in velocity space
We are imagining that we are following one particle (particle ) in a gas, initially propagating without interaction at some velocity . After one collision we have
After two collisions
Where is the number of collisions. We expect
Here the mean is expected to be zero because the individual collisions are thought to be uncorrelated.
We’ll see that there is something wrong with this, and will figure out how to fix this in the next lecture.
In particular, the kinetic energy of any given particle is
and the sum of this is fixed for the complete system.
However, if we sum the kinetic energy squares above 1.0.17b shows that it is growing with the number of collisions. Fixing this we will arrive at the Maxwell Boltzmann distribution.