Hamiltonian thermodynamics of black holes in generic 2D dilaton gravity

We consider the Hamiltonian mechanics and thermodynamics of an eternal black hole in a box of fixed radius and temperature in generic 2D dilaton gravity. Imposing boundary conditions analogous to those used by Louko and Whiting for spherically symmetric gravity, we find that the reduced Hamiltonian generically takes the form $H(M,{\phi }_{+})={\sigma }_{0}E(M,{\phi }_{+})-{(N}_0/2\mathrm{}\pi \left)S\right(M),$ where $E(M,{\phi }_{+})$ is the quasilocal energy of a black hole of mass $M$ inside a static box (surface of fixed dilaton field ${\phi }_{+}$) and $S\left(M\right)\mathrm{}$ is the associated classical thermodynamical entropy. ${\sigma }_0$ and $N_0$ determine time evolution along the world line of the box and boosts at the bifurcation point, respectively. An ansatz for the quantum partition function is obtained by fixing ${\sigma }_0$ and $N_0$ and then tracing the operator $e^{-\beta H}$ over mass eigenstates. We analyze this partition function in some detail both generically and for the class of dilaton gravity theories that is obtained by dimensional reduction of Einstein gravity in $n+2\mathrm{}$ dimensions with $S^n$ spherical symmetry.