Hard Parallel Squares and Cubes

Hard Parallel Squares and Cubes are a simple particle model used in statistical mechanics. They were introduced by B. T. Geilikman in the Proceedings of the Academy of Science of the USSR 70, 25 (1950). The virial equations of state (pressure as a power series in the density) were studied by Zwanzig, Temperley, Hoover, and De Rocco. The latter two authors computed seven-term series for the models in the Journal of Chemical Physics 36, 3141 (1962). Both the sixth and seventh terms in the hard-cube series are negative, a counterintuitive result for repulsive interactions. In 1998 E. A. Jagla [ http://arxiv.org/pdf/cond-mat/9807032 ] investigated the melting transition for both parallel and rotating cube models, finding a qualitative difference in the nature of the transition for the two models. In that same year Martinez-Raton and Cuesta described cubes and mixtures of cubes [ http://arxiv.org/pdf/cond-mat/9809376 ].

Hard Parallel Squares and Cubes have another use, beyond providing a simple model for which seven terms in the Mayers' virial series can be evaluated. In 2009 the Hoovers pointed out [ http://arxiv.org/abs/0811.1807 ] that these models can be used as "ideal gas thermometers" capable of measuring the tensor temperature components $\{T_{xx},T_{yy},T_{zz}\}$. Kinetic theory shows that particles colliding with a hard-cube ideal gas at temperature T will lose or gain energy according to whether the particle kinetic temperature exceeds T or not. The independence of the temperature components for the hard parallel cubes (or square in two dimensions) allows them to serve as gedanken-experiment thermometers for all three temperature components.