Clusters are treated in the microcanonical ensemble. Two classical choices for the partition function are used in Boltzmann’s assumption for entropy: Phase space volume and phase space density. For both definitions, exact statistical analogs are derived for energy derivatives of entropy to arbitrary order. The analogs are used in rigorous microcanonical Monte Carlo simulation to measure entropy derivatives up to order three for systems of three and four atoms. All simulation results are confirmed by direct numerical integration of the statistical analogs.

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The form of the six-fold integral, Eq. (14a), permits pre-tabulation of the integrand. The trapezoid rule is used to avoid decisions on even/odd interval numbers. The inner ϕ-integration, as opposed to a r34-integration, avoids a Jacobian which may become singular near ϕ = 0, π. Figure 1 shows that the numerical effort very strongly depends on E. Symmetry conditions of the problem also depend on E. For E<U4(1) I use for the upper integration bounds r12 = rmax, r13 = r12, r23 = r13, r14 = r23, r24 = r14. For EU4(1) I use r12 = rmax, r13 = r12, r23 = r12, r14 = rmax, r24 = r14. The step size Δϕ = 2Δr/(rmaxrmin) is crudely adjusted to Δr for consistency. A step size of Δr = 0.01 at high E is expected to yield an accuracy to within 4 significant digits. Step sizes of Δr = 0.001 at low E are expected to yield an accuracy of 5 significant digits. More accurate algorithms may be found.

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