On the definition of temperature and its fluctuations in small systems

An analysis of the limits of applicability of the thermodynamic definition of temperature to small systems is given. It is shown that the classical thermodynamic definition, (dS/dU)=1/T (S being the entropy, U the energy, and T the absolute temperature), is not applicable to small systems. It results in an uncertainty in the definition of temperature of the order O(1/N), where N is the number of particles in the system. An alternative definition of temperature is proposed based on the statistical-mechanical description of ensembles of particles. Applying this definition to perfect gases, a rigorous expression for the distribution of temperatures is obtained valid also for small systems and even in the limit N→1. In contrast to alternative approaches based on the thermodynamic definition of temperature, this distribution retains the thermodynamic equilibrium conditions with respect to temperature (equality of average temperature of the small system and temperature of the thermostat) also for small systems resolving in this way a widely discussed in the past problem between thermodynamics and its statistical-mechanical interpretation. Further, a generalization of this distribution to nonideal systems of interacting particles is developed. The results are applied to an interpretation of recent molecular dynamics simulations of argon condensation. Some further consequences and different possible definitions of temperature for macroscopic systems are discussed briefly as well.