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We study two small quantum systems coupled to the same reservoir which is inthermal equilibrium. By studying the particle density and the energy density inthe two systems before and after they contact each other, we find that the twosystems are not in thermal equilibrium with each other. Our result shows thatthe zeroth law of thermodynamics is broken in small quantum systems at lowtemperatures. Therefore, the traditional way of defining temperature fails dueto the breakdown of the transitive relation of thermal equilibrium. Then weshow a different way of defining temperature by attaching an auxiliary site,which plays the role of a thermometer, to the small quantum system.

The breakdown of the zeroth law of thermodynamics and the definition of temperature in small quantum systems