Parallel hard cubes

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Parallel hard cubes are a simple particle model used in statistical mechanics. They were introduced by B. T. Geilikman [1] in 1950. The virial equation of state (pressure as a power series in the density) was studied by Zwanzig, Temperley, Hoover, and De Rocco [2][3]. The latter two authors computed seven-term series for the models [3]. Both the sixth and seventh terms in the hard-cube series are negative, a counter-intuitive result for repulsive interactions. In 1998 E. A. Jagla [4] 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 (See Mixtures [3]).

Usefulness of the Model[edit]

Parallel hard 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 [5] 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 Maxwell-Boltzmann 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 squares in two dimensions) allows them to serve as gedanken-experiment thermometers for all three temperature components.

Phase behavior[edit]

The phase diagram of parallel hard cubes shows a second-order phase transition from a fluid to a simple cubic crystal [6], which contains a large number of vacancies [7].


[8] [9] [10]


  1. B. T. Geilikman "", Proceedings of the Academy of Science of the USSR 70 pp. 25- (1950)
  2. Robert W. Zwanzig "Virial Coefficients of "Parallel Square" and "Parallel Cube" Gases", Journal of Chemical Physics 24 pp. 855-856 (1956)
  3. 3.0 3.1 3.2 William G. Hoover and Andrew G. De Rocco, "Sixth and Seventh Virial Coefficients for the Parallel Hard-Cube Model", Journal of Chemical Physics 36 pp. 3141- (1962) Cite error: Invalid <ref> tag; name "6and7" defined multiple times with different content Cite error: Invalid <ref> tag; name "6and7" defined multiple times with different content
  4. E. A. Jagla "Melting of hard cubes", Physical Review E 58 pp. 4701-4705 (1998)
  5. Wm. G. Hoover and C. G. Hoover "Nonlinear stresses and temperatures in transient adiabatic and shear flows via nonequilibrium molecular dynamics: Three definitions of temperature", Physical Review E 79 046705 (2009)
  6. B. Groh and B. Mulder, "A closer look at crystallization of parallel hard cubes", J. Chem. Phys. 114 pp. 3653 (2001)
  7. M. Marechal, U. Zimmermann and H. Loewen, "Freezing of parallel hard cubes with rounded edges", J. Chem. Phys. 136 pp. 144506-144506 (2012)
  8. José A. Cuesta "Fluid Mixtures of Parallel Hard Cubes", Physical Review Letters 76 pp. 3742-3745 (1996)
  9. José A. Cuesta and Yuri Martínez-Ratón "Fundamental measure theory for mixtures of parallel hard cubes. I. General formalism", Journal of Chemical Physics 107 pp. 6379- (1997)
  10. Yuri Martínez-Ratón and José A. Cuesta "Fundamental measure theory for mixtures of parallel hard cubes. II. Phase behavior of the one-component fluid and of the binary mixture", Journal of Chemical Physics 111 pp. 317- (1999)
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