I have two related questions:

How Boltzmann equation can be written in a covariant (using differential forms, connections, and etc.) way in a classical (not quantum) but curved system?

How does the Liouville's theorem change in a curved background?

Any answers, suggestions, clarifications, or references are welcome.

  • $\begingroup$ The derivation is in Straumann. $\endgroup$ – Slereah Dec 16 '17 at 20:17
  • $\begingroup$ @Slereah I didn't find it! Can you tell me which chapter? $\endgroup$ – Kiarash Dec 17 '17 at 8:53

Liouville's Theorem is pretty easy because it's the same in curved spacetime as in flat spacetime. MTW gives "Liouville's theorem in curved spacetime" as

The volume $\mathscr{V}$ occupied by a given swarm of $N$ particles is independent of location along the world line of the swarm.

See section 22.6 for the discussion. Basically, the only things that we might consider changed are how we measure volume and how the world line evolves.

The Boltzmann equation is more complicated. For example, if we just have a relativistic gas of particles with the same mass (and we choose units where that mass is 1 unit), and the gas at some point has momentum $p^\alpha$, then Boltzmann's equation for the distribution function $f$ looks like \begin{equation} p^\alpha \frac{\partial f}{\partial x^\alpha} - \Gamma^\gamma_{\alpha \beta} p^\alpha p^\beta \frac{\partial f}{\partial p^\gamma} = Q(f, f), \end{equation} where $Q$ is a complicated collision operator and $\Gamma$ are Christoffel's symbols. An original reference for this is Bichteler's paper, but a more modern discussion can be found here, for example.

  • $\begingroup$ I assume this is true so long as the trajectories are not allowed to diverge/split? $\endgroup$ – honeste_vivere Dec 21 '17 at 15:07

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