Violation of the Zeroth Law of Thermodynamics in Systems with Negative Specific Heat

We show that systems with negative specific heat can violate the zeroth law of thermodynamics. By both numerical simulations and by using exact expressions for free energy and microcanonical entropy it is shown that if two systems with the same intensive parameters but with negative specific heat are thermally coupled, they undergo a process in which the total entropy increases irreversibly. The final equilibrium is such that two phases appear, that is, the subsystems have different magnetizations and internal energies at temperatures which are equal in both systems, but that can be different from the initial temperature.

A trivial observation on time reversal in random matrix theory

It is commonly thought that a state-dependent quantity, after being averaged over a classical ensemble of random Hamiltonians, will always become independent of the state. We point out that this is in general incorrect: if the ensemble of Hamiltonians is time reversal invariant, and the quantity involves the state in higher than bilinear order, then we show that the quantity is only a constant over the orbits of the invariance group on the Hilbert space. Examples include fidelity and decoherence in appropriate models.