Violation of the temperature-signifies-heat-flow rule in systems with long-range interactions

For a long-range interacting spin chain model, the microcanonical ensemble predicts a region of negative specific heat and a temperature jump at the transition energy. After two similar long-range interacting subsystems of different size at different temperatures are weakly coupled, they exchange energy and the total microcanonical entropy of the full system increases irreversibly. The hot subsystem could spontaneously absorb heat from the cold subsystem via the thermal contact and the final equilibrium temperature could be lower than the initial temperatures of the cold subsystem. This result is confirmed by numerical simulations using the microcanonical Monte Carlo algorithm.