A Micro-Thermodynamic Formalism

We consider the micro-canonical ensemble of a classical Hamiltonian dynamical system, the Hamiltonian being parameter dependent and in the possible presence of other first integrals. We describe a thermodynamic formalism in which a 1st law of thermodynamics, or fundamental relation, is based upon the bulk-entropy, SΩ. Under an ergodic hypothesis, SΩ is shown to be an adiabatic invariant. Expressions for derivatives and thermodynamic relations are derived within the micro-canonical ensemble itself. Equilibrium properties of an isolated Hamiltonian dynamical system with many, say 10, degrees of freedom is probably best described using a thermodynamic formalism for a canonical ensemble at a fixed temperature, even though this means introducing fluctuations in an otherwise conserved quantity, the energy. For a more moderate number of degrees of freedom, say 10− 10, numerical simulations become feasible, and it is desirable to obtain a description in terms of the micro-canonical ensemble itself where the values of the first integrals are fixed quantities. In such an approach, geometrical properties of the level surfaces reflect thermodynamic relations and, by invoking the ergodic hypothesis, also dynamical properties of the underlying system. In particular, when energy is the only first integral, measurements may then be done by time averaging (cf. [10, 11]). The purpose of this article is to develop such a microthermodynamic formalism further, taking into account parameter dependency and the presence of other first integrals. Within this framework we will also (section 2) discuss a natural formulation of a 1st law of thermodynamics, or fundamental relation, based upon the bulk-entropy, SΩ. We refer to Abraham and Marsden [1] as well as Landau and Lifshitz [6] for a general introduction to thermodynamic ensembles, to Lebowitz et al. [7] for an illustrative example of some differences between the ensembles and to Evans and Morriss [3] for practical calculations carried out in the micro canonical ensemble. We also refer to Jepps et al. [4] where the presence of other (approximative) first integrals is of relevance and to Otter [8] who studied reaction events using mixed ensemble averages. 1 Micro-canonical ensembles For simplicity, we consider a Euclidean phase space, Ω = R, d ≥ 1, and a Hamiltonian function, H : Ω → R, bounded from below and of sufficient rapid growth at infinity. The dynamics preserves the Liouville measure, here the Lebesgue measure, m= dxdp. There may be other first integrals, denoted F = F1, . . . , Fm, m ≥ 0 Note that in section 3 we shall write F0 = H for the Hamiltonian which is then considered at the same footing as the other first integrals. All first integrals are assumed to be in involution. We also assume that all functions are known analytically and that the Hamiltonian depends smoothly on some external real parameters, denoted Λ = Λ1, . . . ,Λn. By contrast we do not allow the other first integrals to depend on Λ. This is for technical reasons (cf. below), though in some cases such a condition could be relaxed. For fixed values of parameters, Λ=λ, of first integrals, F = I, and of the energy, Hλ =E, the subspace, A=A[E, I, λ] = {ξ ∈ R 2d : Hλ(ξ) =E,F (ξ) = I}, is invariant under the dynamics of Hλ. We will assume that values are chosen so that the differentials, dH , dF1,. . . ,dFm, are all independent on A[E, I, λ]. This in particular implies that A is a smooth co-dimension m+1 sub-manifold of our Euclidean space. In the literature one will find (at least) two definitions (denoted bulk and surface) of a micro-canonical entropy and temperature. It turns out that we shall need both. Thus we define eΩ ≡ ∫ m Θ(E−Hλ) δ (I−F ) (1) where Θ denotes the Heaviside function and eμ ≡ ∫ m δ (E−Hλ, I−F ) . (2) The bulkand the surface-temperature are then given by: 1 TΩ = ∂SΩ ∂E , 1 Tμ = ∂Sμ ∂E . (3) Derivatives with respect to other first integrals are considered in Section (3). We also have generalized bulkand surface-pressures: piΩ = TΩ ∂SΩ ∂λi , piμ = Tμ ∂Sμ ∂λi , i = 1, . . . , n. (4) Taking an average in the micro-canonical ensemble will here mean taking the surface-average, i.e. 〈φ〉μ ≡ 〈φ|E, I, λ〉 = ∫ m δ (E−Hλ, I−F )φ ∫ m δ (E−Hλ, I−F ) . (5) In practice, 〈φ〉μ, is often calculated by time-averaging (assuming ergodicity), thus giving a dynamical preference to the surface-average relative to other ensemble-averages. The calculation of either of the two entropies may be difficult or even impossible when the number of degrees of freedom in the system is large. On the other hand, the associated temperatures and generalized pressures may be probed using time-averaging (cf. Section 3). A bulk-pressure, pΩ, may be calculated as follows: piΩ = −〈 ∂Hλ ∂λi 〉μ = −〈 ∂Hλ ∂λi |E, I, λ〉. (6) To see this we note that the derivative of a Heavyside-function yields a delta-function. It follows that 1/TΩ = ∂SΩ/∂E = e μ/eΩ , and therefore, piΩ = TΩ ∂SΩ ∂λi = − ∫ m δ(E−Hλ, I−F ) ∂Hλ ∂λi ∫ m δ(E−Hλ, I−F ) . (7) By the very definition, there is always a 1st law of thermodynamics for the bulk-entropy: