Statistical mechanics with homogeneous first-degree Lagrangians

Abstract

The equilibrium statistical mechanics is investigated of any system whose LagrangianL ₀(v, q) is a convex homogeneous function of generalized velocitiesv, with coordinatesq in a bounded setD. A member of a canonical ensemble, the system has a conjugate HamiltonianH ₀(p, q) that vanishes identically in some subsetC×D of its phase space. The subsetC may also be specified, in some systems with a finite functionf(p, q), convex inp≡∂L ₀/∂ν, and thenL ₀ is also convex and homogeneous inv. In either case, ifC is bounded and convex, thenC or the convex functionf constitutes the fundamental constraint on the system. Under this fundamental constraint, it is shown that the so-called partition function becomes a phase-space volumeG (classical) or a numberW of microstates (quantum) from which follows the thermodynamic fundamental relation, entropyS≡k InG (ork lnW).