Reply to the Comment by F. Calvo et al.

Abstract

Calvo et al. [1] suggest that the heat capacity of a nanocluster may be negative when determined in the microcanonical ensemble, while it is strictly positive when determined in the canonical ensemble. Our letter [2] argues that there is but one thermodynamic equilibrium state, and therefore there is only one caloric curve corresponding to a system in thermodynamic equilibrium. While a canonical ensemble of nanoclusters can reach thermodynamic equilibrium, a microcanonical ensemble of nanoclusters cannot due to trapping, although it may reach a local thermal equilibrium. Thermodynamic formalism, including the Gibbs equation, is only valid for a system in thermodynamic equilibrium. Calvo et al. report on new “converged” Monte Carlo simulations of the cluster Na147. However, this new work is irrelevant to the issue since, contrary to their claim, we do not “suggest that previous simulations were not converged”. What we do suggest is that convergence is a necessary but not sufficient condition for thermodynamic equilibrium. In the microcanonical ensemble, convergence may imply a metastable (or frozen [3]) state, but this is not generally the true thermodynamic equilibrium state. In the canonical ensemble, convergence over the finite time of experiment or simulation also does not imply thermodynamic equilibrium. High energy barriers between isomers of different symmetry (see, for example, [4]) can lead to time local convergence in the canonical ensemble, but true thermodynamic equilibrium, demonstrating no dependence of any macroscopic measurement on initial conditions, may require impractically large simulation or experimental equilibration times. Our rough calculations based on transition state theory and the Gupta potential