[Click here for a PDF of this post with nicer formatting]
Peeter’s lecture notes from class. May not be entirely coherent.
Review: Lead up to Maxwell distribution for gases
For the random walk, after a number of collisions , we found that a particle (labeled the th) will have a velocity
We argued that the probability distribution for finding a velocity was as in
Fig1: Velocity distribution found without considering kinetic energy conservation
What went wrong?
However, we know that this must be wrong, since we require
Where our argument went wrong is that when the particle has a greater than average velocity, the effect of a collision will be to slow it down. We have to account for
- Fluctuations “random walk”
- Dissipation “slowing down”
There were two ingredients to diffusion (the random walk), these were
- Conservation of particles
We can also think about a conservation of a particles in a velocity space
where is a probability current in this velocity space.
- Fick’s law in velocity space takes the form
The diffusion results in an “attempt” to flatten the distribution of the concentration as in (Fig 2).
Fig2: A friction like term is require to oppose the diffusion pressure
We’d like to add to the diffusion current an extra frictional like term
We want something directed opposite to the velocity and the concentration
Can we find a steady state solution to this equation when ? For such a steady state we have
Integrating once we have
supposing that at , integrating once more we have
This is the Maxwell-Boltzmann distribution, illustrated in (Fig3).
Fig3: Maxwell-Boltzmann distribution
The concentration has a probability distribution.
Calculating from this distribution we can identify the factor.
This also happens to be the energy in terms of temperature (we can view this as a definition of the temperature for now), writing
Equilibrium steady states
This is a specific example of the more general Fluctuation-Dissipation theorem.
Generalizing to 3D
Fick’s law and the continuity equation in 3D are respectively
As above we have for the steady state
Integrating once over all space
This is three sets of equations, one for each component of
So that our steady state equation is
Computing the average 3D squared velocity for this distribution, we have
For each component the normalization kills off all the contributions for the other components, leaving us with the usual ideal gas law kinetic energy.
Now let’s switch directions a bit and look at how to examine a more general system described by the phase space of generalized coordinates
- , where is the physical space dimension.
The motion in phase space will be governed by the knowledge of how each of these coordinates change for all the particles. Assuming a Hamiltonian and recalling that , gives us
we have for the following set of equations describing the entire system
Example, 1D SHO
This has phase space trajectories as in (Fig4).
Fig4: Classical SHO phase space trajectories