2024-09-23 8.333 - Statistical Mechanics 1

Summary

Today was a second step towards the kinetic theory of gases (in particular, deriving the boltzmann equation).

Mainly, it was motivating a method for coarse graining our system.
There are three physically relevant scales (in order from smallest to largest):
1. Duration and length scale of particle collisions
2. Duration and distance between particle collisions
3. The free transport hamiltonian (basically just the distance of the walls)
Since each scale is well separate from the next, our approach will be to successively coarse grain so that, in analyzing one level, we can have this abstracted away/simplified view of the density/dynamics.

I really like this coarse graining way of doing things. Its almost as if one system is multiple independent systems operating at different temporal and spatial scales. They can kind of influence each other, but only via very simple interfaces.
I think what's going to happen is this TAP equation type two variable recurrence like EM
- Where we compute how things equilibriate at the micro level, then compute, given the parameters that micro level equilibriates at, what happens at the macro scale, and back and forth.
One thing that feels weird is that (2) and (3) seem like they're talking about the same kind of thing, while (1) seems like a different kind of thing
- In that in 1, you're localizing to a particular collision, while the others, your sort of talking about how often these collisions occur...
- And formally, they are treated differently. (1) is only used to justify a scale at which to spatially coarsen the system, while my guess is that then (2) is going to be used to coarsen the dynamics (i.e. temporally coarsen)
What if we did all of this stuff from the heisenberg picture? That is, looking at how observables evolve?
- What would that look like? We'd still get a liouvilles equation type thing, and we could still define similar things like the one body density
  - [?] What's the one body density in this case?
    - Marginalization doesn't make sense... since operators live in $L_\infty$ space so you can't average over the whole thing...