Microcanonical Thermodynamics of Small Ideal Gas Systems

Abstract

We consider the thermal, mechanical, and chemical contact of two subsystems composed of ideal gases, both of which are not in the thermodynamic limit. After contact, the combined system is isolated, and the entropy is determined through the use of its standard connection to the phase space density (PSD), where only those microstates at a given energy value are counted. The various intensive properties of these small systems that follow from a derivative of the PSD, such as the temperature, pressure, and chemical potential (evaluated via a backward difference), while equal when the two subsystems are in equilibrium are nevertheless found not to behave in accordance with what is expected from macroscopic thermodynamics. Instead, it is the entropy, defined from its connection to the PSD, that still controls the behavior of these small (nonextensive) systems. We also analyze the contact of these two subsystems utilizing an alternative entropy definition, through its proposed connection to the phase space volume (PSV), where all microstates at or below a given energy value are counted. We show that certain key properties of these small systems obtained with the PSV either do not become equal or do not consistently describe the two subsystems when in contact, suggesting that the PSV should not be used for analyzing the behavior of small isolated systems.

Figure 1

Combined isolated system comprised of two subsystems (1 and 2, or L and R) that are in “contact” via a partition. Depending upon the properties of the partition, the subsystems are in (1) thermal contact (via a fixed and diathermal partition), or (2) mechanical and thermal contact (via a movable partition), or (3) chemical and thermal contact (via a permeable partition). For Cases 1 and 2, the number of particles in both subsystems are fixed at N₁ and N₂. For Case 3, the number of particles in both subsystems are allowed to vary, with “instantaneous” values given by N_L and N_R. The combined system has a fixed total number of particles, a fixed total volume, and a fixed total energy.