  A framework for statistical-mechanical analysis of quantum Hamiltonians is
introduced. The approach is based upon a gradient flow equation in the space of
Hamiltonians such that the eigenvectors of the initial Hamiltonian evolve
toward those of the reference Hamiltonian. The nonlinear double-bracket
equation governing the flow is such that the eigenvalues of the initial
Hamiltonian remain unperturbed. The space of Hamiltonians is foliated by
compact invariant subspaces, which permits the construction of statistical
distributions over the Hamiltonians. In two dimensions, an explicit dynamical
model is introduced, wherein the density function on the space of Hamiltonians
approaches an equilibrium state characterised by the canonical ensemble. This
is used to compute quenched and annealed averages of quantum observables.
