# Can statistical mechanics be formulated generally in terms of phase space?

In many statistical mechanics books, notably Landau and Lifschitz' volume in the course on theoretical physics, the quantities central to statistical mechanics such as entropy are defined in terms of phase space densities. This is because (classically) particles in the real world can be described in terms of $n$ coordinates and their corresponding momenta. Also, in order to derive the Gibbs distribution, L&L argue that a mechanical system has only seven independent additive conserved quantities.

However, many statistical mechanics models (such as spin models) are not formulated in terms of phase space and Hamiltonian mechanics at all. One still wants to be able to talk about entropy, probability distributions, etc. when discussing these models. Can the phase space formulation be generalized to systems that are not 'particles in a box' but can have arbitrary configurations and energies?

• Have you heard of the 'microcanonical', 'canonical' and 'grand canonical' ensembles? Landau ch. 1 develops the microcanonical ensemble, then the others in Ch. 3. Here odessa.phy.sdsmt.edu/~andre/PHYS743 are some video lectures closely following L&L's Stat Mech. – bolbteppa Jun 14 '14 at 13:38
• @bolbteppa The slides in your link developing the canonical and grand canonical ensembles (odessa.phy.sdsmt.edu/~andre/PHYS743/Lecture10.pdf) explicitly mention phase space coordinates $p,q$. – user111187 Jun 14 '14 at 13:59
• Do you understand the difference between classical and quantum statistical mechanics? – bolbteppa Jun 14 '14 at 14:36
• @bolbteppa Yes, but I do not see how that is relevant to the question, so maybe I am missing something. – user111187 Jun 14 '14 at 14:56
• You are beginning with states in a phase space, which are classical statistical mechanical states, and then asking about states like spin states which are not classical statistical mechanical states in a phase space but are instead quantum statistical mechanical states in a Hilbert space, so I would imagine you are thinking of statistical mechanics as purely classical statistical mechanics and not quantum stat mech, c.f. Sec 5 & 6 in Landau chapter 1. – bolbteppa Jun 14 '14 at 16:12