Systems with negative specific heat in thermal contact: violation of the zeroth law

Using both numerical simulations and exact expressions for the free energy and microcanonical entropy for a modified Hamiltonian mean-field (HMF) model, we show that when two similar systems with the same intensive parameters but with negative specific heat are weakly coupled, they undergo a process in which the total entropy increases irreversibly. We find that the final equilibrium is such that two phases appear at a temperature (equal in both systems) that is generally different from the initial temperature. We corroborate our results using two different kinds of couplings between the HMF systems. We confirm that our results hold also for the Ising model with long- and short-range interactions, which also has a parameter region with negative specific heat in the microcanonical ensemble. Further, we show that we can couple each system having negative specific heat to a third system that can be used as a thermometer, as long as this thermometer is small enough not to drive the system out of the microcanonical ensemble. Therefore, we show an instance of violation of the zeroth law of thermodynamics.