Parallel hard cubes
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
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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ T_{xx},T_{yy},T_{zz}\}} . Kinetic theory shows that particles colliding with a hard-cube Maxwell-Boltzmann ideal gas at temperature Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} will lose or gain energy according to whether the particle kinetic temperature exceeds Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.
Mixtures
References
- ↑ B. T. Geilikman "", Proceedings of the Academy of Science of the USSR 70 pp. 25- (1950)
- ↑ Robert W. Zwanzig "Virial Coefficients of "Parallel Square" and "Parallel Cube" Gases", Journal of Chemical Physics 24 pp. 855-856 (1956)
- ↑ 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)
- ↑ E. A. Jagla "Melting of hard cubes", Physical Review E 58 pp. 4701-4705 (1998)
- ↑ 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)
- ↑ José A. Cuesta "Fluid Mixtures of Parallel Hard Cubes", Physical Review Letters 76 pp. 3742-3745 (1996)
- ↑ 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)
- ↑ 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)
- Related reading
- William G. Hoover, "High-Density Equation of State of Hard Parallel Squares and Cubes", Journal of Chemical Physics 40 pp. 937- (1964)
- T. R. Kirkpatrick "Ordering in the parallel hard hypercube gas", Journal of Chemical Physics 85 pp. 3515-3519 (1986)
- José A. Cuesta and Yuri Martínez-Ratón "Dimensional Crossover of the Fundamental-Measure Functional for Parallel Hard Cubes", Physical Review Letters 78 pp. 3681-3684 (1997)
- S. Belli, M. Dijkstra, and R. van Roij "Free minimization of the fundamental measure theory functional: Freezing of parallel hard squares and cubes", Journal of Chemical Physics 137 124506 (2012)