Understanding the thermodynamics of small systems has a well-established history [1]. The non-intuitive behavior of the intensive properties of small systems comprised of less than a few hundred particles, as first derived by Hill [1], has important implications in many areas of engineering and materials science. Nevertheless, many open questions about the thermophysical properties of systems not in the thermodynamic limit remain unanswered.

In this work, we go beyond the analysis of Hill by exploring the consequences of the coupling of two small subsystems, the combination of which make up a larger isolated system. Our results are obtained for two separate test cases: 1) two closed and rigid subsystems are initially separated by an adiabatic partition until a moment in time when the partition facilitates the exchange of energy between the subsystems, and 2) two closed subsystems are initially separated by a fixed diathermal partition until a moment in time when the partition of a given mass is allowed to move. The results of these two test cases are studied through the use of statistical mechanics and molecular dynamics simulations of Lennard-Jones particles in two dimensions.

There are two definitions of temperature (that are equivalent in the macro limit but different in the small system limit) that we use to discuss our findings. The first is the thermodynamic temperature, which is obtained directly from taking the derivative of the entropy with respect to the energy at fixed particle number and volume/area. The second is the kinetic temperature, which is obtained from the equipartition theorem. There are also two definitions of the pressure (which are again the same in the macro limit but differ in the small system limit). The thermodynamic pressure is obtained directly from the partition function, and the mechanical pressure is obtained from the virial theorem.

Results for the first test case demonstrate that when the two subsystems reach equilibrium their average inverse thermodynamic temperatures are always the same. The average thermodynamic temperature of the two subsystems, however, can be different, and this difference is related to the number of particles in each of the subsystems. The average kinetic temperatures are always equal when the systems are in equilibrium, but the average inverse kinetic temperatures can be different depending on the particle numbers. We show that the driving force for the system to reach equilibrium is through the equipartition of the total energy of the combined system. Throughout all of our theoretical calculations and simulations we find that the second law is always satisfied. We also discuss the implications of our results to the zeroth law of thermodynamics in small systems, as the temperature is not the indicator of the direction of energy transfer as it is for macroscopic systems.

The results from the second test case show that once equilibrium is attained, the average inverse thermodynamic temperature and the average thermodynamic pressure multiplied by the inverse thermodynamic temperature are equal in both subsystems. The average thermodynamic temperature and the average thermodynamic pressure of both subsystems can be different, with the relative difference again being based on the number of particles making up the two subsystems. The kinetic temperature and the mechanical pressure are always equal in equilibrium. We also provide theoretical and simulation proof that the chemical potential of the two subsystems do not need to be the same if the temperature and pressure of the two subsystems are the same, as would be the case in the macroscopic limit. The reason for this is that the Gibbs-Duhem equation relevant to small systems has an extra term, thereby allowing the chemical potential to not be directly coupled to the other two intensive properties (temperature and pressure).

**References**

[1] T. L. Hill, “Thermodynamics of Small Systems”, Dover Publications, Mineola, NY, 2002. Originally published in 1964.

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