Microcanonical Entropy and Dynamical Measure of Temperature for Systems with Two First Integrals

Abstract

We consider a generic classical many particle system described by an autonomous Hamiltonian H(x 1 , . . . , x N +2 ) which, in addition, has a conserved quantity V (x 1 , . . . , x N +2 ) = v, so that the Poisson bracket {H, V } vanishes. We derive in detail the microcanonical expressions for entropy and temperature. We show that both of these quantities depend on multidimensional integrals over sub-manifolds given by the intersection of the constant energy hyper-surfaces with those defined by V (x 1 , . . . , x N +2 ) = v. We show that temperature and higher order derivatives of entropy are microcanonical observable that, under the hypothesis of ergodicity, can be calculated as time averages of suitable functions. We derive the explicit expression of the function that gives the temperature.

On the dispute between Boltzmann and Gibbs entropy

“…This fact is proven by Rugh [49] in the case of many-particle systems for which the Hamiltonian is the only conserved quantity, and in Ref. [50] and Ref. [51] for the case of two and k ∈ N conserved quantities, respectively.…”

“…In fact, in Ref. [50] it has been considered the case k = 2 by studying a general classical autonomous many-body Hamiltonian system, whose coordinates and canonical momenta are indicated with x ∈ R L , and for which V (x) is a further conserved quantity in involution with H. For such a system, the motion takes place on…”

“…Furthermore, note that we have made use of the generalization [50] of the co-area formula [52] which is of very general validity and holds also for…”

Nonequilibrium discrete nonlinear Schrödinger equation

“…Thus, it is necessary to introduce suitable operative definitions of kinetic temperature T and chemical potential µ, to measure such quantities in actual simulations. In the following, we make use of a recent definition of the microcanonical temperature [24] and extend it for the estimate of the chemical potential.…”

Thermal Transport in Low Dimensions

“…As shown in [36,50], the partial derivative ∂ S /∂C i (i = 1, 2, with C 1 = H and C 2 = A) can be computed by exploiting the fact that C i is a conserved quantity,…”