Parallel hard cubes: Difference between revisions

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'''Parallel hard cubes''' are a simple particle [[Models |model]] used in [[statistical mechanics]].  They were introduced by B. T. Geilikman <ref>B. T. Geilikman "", Proceedings of the Academy of Science of the USSR '''70''' pp. 25- (1950)</ref> in 1950.  The [[virial equation of state]] ([[pressure]] as a power series in the density) was studied by Zwanzig, Temperley, Hoover, and De Rocco <ref>[http://dx.doi.org/10.1063/1.1742621 Robert W. Zwanzig "Virial Coefficients of "Parallel Square" and "Parallel Cube" Gases", Journal of Chemical Physics  '''24''' pp. 855-856 (1956)]</ref><ref name="6and7">[http://dx.doi.org/10.1063/1.1732443  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)]</ref>.  The latter two authors computed seven-term series for the models <ref name="6and7"> </ref>.  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 <ref>[http://dx.doi.org/10.1103/PhysRevE.58.4701 E. A. Jagla "Melting of hard cubes", Physical Review E '''58''' pp. 4701-4705 (1998)]</ref> 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 [[Parallel hard cubes#Mixtures | Mixtures]] <ref name="6and7"> </ref>).
'''Parallel hard cubes'''
====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 <ref>[http://dx.doi.org/10.1103/PhysRevE.79.046705 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)]</ref> that these models can be used as "ideal gas thermometers" capable of measuring the tensor [[temperature]] components <math>\{ T_{xx},T_{yy},T_{zz}\}</math>.  Kinetic theory shows that particles colliding with a hard-cube [[Maxwell velocity distribution |Maxwell-Boltzmann]] [[ideal gas]] at temperature <math>T</math> will lose or gain energy according to whether the particle kinetic temperature exceeds <math>T</math> 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==
The phase diagram of parallel hard cubes shows a second-order phase transition from a fluid to a simple cubic crystal
<ref>[http://dx.doi.org/10.1063/1.1342816 B. Groh and B. Mulder, "A closer look at crystallization of parallel hard cubes", J. Chem. Phys. '''114''' pp. 3653 (2001)]</ref>,
which contains a large number of vacancies
<ref>[http://dx.doi.org/10.1063/1.3699086 M. Marechal, U. Zimmermann and H. Loewen, "Freezing of parallel hard cubes with rounded edges", J. Chem. Phys. '''136''' pp. 144506-144506 (2012)]</ref>.
 
 
==Mixtures==
==Mixtures==
#[http://dx.doi.org/10.1103/PhysRevLett.76.3742  José A. Cuesta "Fluid Mixtures of Parallel Hard Cubes", Physical Review Letters '''76''' pp. 3742-3745 (1996)]
<ref>[http://dx.doi.org/10.1103/PhysRevLett.76.3742  José A. Cuesta "Fluid Mixtures of Parallel Hard Cubes", Physical Review Letters '''76''' pp. 3742-3745 (1996)]</ref>
#[http://dx.doi.org/10.1063/1.474298 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)]
<ref>[http://dx.doi.org/10.1063/1.474298 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)]</ref>
#[http://dx.doi.org/10.1063/1.479273 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)]
<ref>[http://dx.doi.org/10.1063/1.479273 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)]</ref> 
==References==
==References==
#[http://dx.doi.org/10.1063/1.1742621 Robert W. Zwanzig "Virial Coefficients of "Parallel Square" and "Parallel Cube" Gases", Journal of Chemical Physics  '''24''' pp. 855-856 (1956)]
<references/>
#[http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCPSA6000036000012003141000001&idtype=cvips&gifs=Yes
;Related reading
  William G. Hoover and Andrew G. De Rocco, "Sixth and Seventh Virial Coefficients for the Parallel Hard-Cube Model", Journal of Chemical Physics '''36''', 3141 (1962).]
*[http://dx.doi.org/10.1063/1.1725285 William G. Hoover, "High-Density Equation of State of Hard Parallel Squares and Cubes", Journal of Chemical Physics '''40''' pp. 937- (1964)]
#[http://link.aip.org/llink/?JCPSA6/40/937/1 William G. Hoover, "High-Density Equation of State of Hard Parallel Squares and Cubes", Journal of Chemical Physics '''40''', 937 (1964).]
*[http://dx.doi.org/10.1063/1.450974 T. R. Kirkpatrick "Ordering in the parallel hard hypercube gas", Journal of Chemical Physics  '''85''' pp. 3515-3519 (1986)]
#[http://dx.doi.org/10.1063/1.450974 T. R. Kirkpatrick "Ordering in the parallel hard hypercube gas", Journal of Chemical Physics  '''85''' pp. 3515-3519 (1986)]
*[http://dx.doi.org/10.1103/PhysRevLett.78.3681 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)]
#[http://dx.doi.org/10.1103/PhysRevLett.78.3681 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)]
*[http://dx.doi.org/10.1063/1.4754836  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)]
 
 
[[category: models]]
[[category: models]]

Latest revision as of 15:34, 28 April 2013

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 {Txx,Tyy,Tzz}. 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].


Mixtures[edit]

[8] [9] [10]

References[edit]

  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)
  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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