http://www.sklogwiki.org/SklogWiki/api.php?action=feedcontributions&feedformat=atom&user=70.41.222.228SklogWiki - User contributions [en]2026-05-01T16:29:53ZUser contributionsMediaWiki 1.41.0http://www.sklogwiki.org/SklogWiki/index.php?title=Parallel_hard_cubes&diff=7671Parallel hard cubes2009-02-01T06:10:31Z<p>70.41.222.228: </p>
<hr />
<div>{{stub-general}}<br />
'''Parallel hard cubes'''<br />
<br />
==Usefulness of the Model==<br />
Parallel hard cubes provide a simple statistical mechanical model, amenable to analytic and theoretical work as well as computer simulation.<br />
Beginning with Geilikman's work in Russia in 1950 a succession of studies have clarified [1] the low-density virial expansion (pressure as<br />
a power series in the density) {2} the ordering phase transition at moderate density, [3] the properties of mixtures of cubes of different<br />
sizes, and [4] the conceptual role of the model as an ideal-gas thermometer capable of measuring separately the tensor components of<br />
temperature, <math>\{ T_{xx}, T_{yy}, T_{zz} \}</math>. <br />
==Mixtures==<br />
#[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)]<br />
#[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)]<br />
#[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)]<br />
==References==<br />
#[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)]<br />
#[http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCPSA6000036000012003141000001&idtype=cvips&gifs=Yes 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).]<br />
#[http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCPSA6000040000004000937000001&idtype=cvips&gifs=Yes William G. Hoover, "High-Density Equation of State of Hard Parallel Squares and Cubes", Journal of Chemical Physics '''40''', 937 (1964).]<br />
#[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)]<br />
#[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)]<br />
[[category: models]]</div>70.41.222.228http://www.sklogwiki.org/SklogWiki/index.php?title=Parallel_hard_cubes&diff=7670Parallel hard cubes2009-02-01T05:52:49Z<p>70.41.222.228: /* References */</p>
<hr />
<div>{{stub-general}}<br />
'''Parallel hard cubes'''<br />
==Mixtures==<br />
#[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)]<br />
#[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)]<br />
#[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)]<br />
==References==<br />
#[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)]<br />
#[http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCPSA6000036000012003141000001&idtype=cvips&gifs=Yes 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).]<br />
#[http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCPSA6000040000004000937000001&idtype=cvips&gifs=Yes William G. Hoover, "High-Density Equation of State of Hard Parallel Squares and Cubes", Journal of Chemical Physics '''40''', 937 (1964).]<br />
#[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)]<br />
#[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)]<br />
[[category: models]]</div>70.41.222.228http://www.sklogwiki.org/SklogWiki/index.php?title=Parallel_hard_cubes&diff=7669Parallel hard cubes2009-02-01T05:50:47Z<p>70.41.222.228: /* References */</p>
<hr />
<div>{{stub-general}}<br />
'''Parallel hard cubes'''<br />
==Mixtures==<br />
#[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)]<br />
#[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)]<br />
#[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)]<br />
==References==<br />
#[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)]<br />
#[http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCPSA6000036000012003141000001&idtype=cvips&gifs=Yes<br />
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).]<br />
#[http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCPSA6000040000004000937000001&idtype=cvips&gifs=Yes<br />
William G. Hoover, "High-Density Equation of State of Hard Parallel Squares and Cubes", Journal of Chemical Physics '''40''', 937 (1964).]<br />
#[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)]<br />
#[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)]<br />
[[category: models]]</div>70.41.222.228http://www.sklogwiki.org/SklogWiki/index.php?title=Parallel_hard_cubes&diff=7668Parallel hard cubes2009-02-01T05:49:05Z<p>70.41.222.228: /* References */</p>
<hr />
<div>{{stub-general}}<br />
'''Parallel hard cubes'''<br />
==Mixtures==<br />
#[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)]<br />
#[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)]<br />
#[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)]<br />
==References==<br />
#[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)]<br />
#[http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCPSA6000036000012003141000001&idtype=cvips&gifs=Yes]<br />
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).<br />
#[http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCPSA6000040000004000937000001&idtype=cvips&gifs=Yes]<br />
William G. Hoover, "High-Density Equation of State of Hard Parallel Squares and Cubes", Journal of Chemical Physics '''40''', 937 (1964).]<br />
#[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)]<br />
#[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)]<br />
[[category: models]]</div>70.41.222.228http://www.sklogwiki.org/SklogWiki/index.php?title=Parallel_hard_cubes&diff=7667Parallel hard cubes2009-02-01T05:45:11Z<p>70.41.222.228: /* References */</p>
<hr />
<div>{{stub-general}}<br />
'''Parallel hard cubes'''<br />
==Mixtures==<br />
#[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)]<br />
#[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)]<br />
#[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)]<br />
==References==<br />
#[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)]<br />
#[http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCPSA6000036000012003141000001&idtype=cvips&gifs=Yes<br />
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).]<br />
#[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).]<br />
#[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)]<br />
#[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)]<br />
[[category: models]]</div>70.41.222.228http://www.sklogwiki.org/SklogWiki/index.php?title=Parallel_hard_cubes&diff=7666Parallel hard cubes2009-02-01T05:39:17Z<p>70.41.222.228: /* References */</p>
<hr />
<div>{{stub-general}}<br />
'''Parallel hard cubes'''<br />
==Mixtures==<br />
#[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)]<br />
#[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)]<br />
#[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)]<br />
==References==<br />
#[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)]<br />
#[http://link.aip.org/llink/?JCPSA6/36/3141/1 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).]<br />
#[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).]<br />
#[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)]<br />
#[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)]<br />
[[category: models]]</div>70.41.222.228http://www.sklogwiki.org/SklogWiki/index.php?title=Parallel_hard_cubes&diff=7665Parallel hard cubes2009-02-01T05:27:56Z<p>70.41.222.228: /* References */</p>
<hr />
<div>{{stub-general}}<br />
'''Parallel hard cubes'''<br />
==Mixtures==<br />
#[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)]<br />
#[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)]<br />
#[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)]<br />
==References==<br />
#[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)]<br />
#[William G. Hoover, "High-Density Equation of State of Hard Parallel Squares and Cubes", Journal of Chemical Physics '''40''', 937 (1964).]<br />
#[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).]<br />
#[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)]<br />
#[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)]<br />
[[category: models]]</div>70.41.222.228http://www.sklogwiki.org/SklogWiki/index.php?title=Parallel_hard_cubes&diff=7664Parallel hard cubes2009-02-01T05:26:00Z<p>70.41.222.228: /* References */</p>
<hr />
<div>{{stub-general}}<br />
'''Parallel hard cubes'''<br />
==Mixtures==<br />
#[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)]<br />
#[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)]<br />
#[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)]<br />
==References==<br />
#[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)]<br />
#[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)]<br />
#[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)]<br />
[[category: models]]<br />
#[William G. Hoover, "High-Density Equation of State of Hard Parallel Squares and Cubes", Journal of Chemical Physics '''40''', 937 (1964).]<br />
#[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).]</div>70.41.222.228http://www.sklogwiki.org/SklogWiki/index.php?title=Parallel_hard_cubes&diff=7663Parallel hard cubes2009-02-01T05:23:59Z<p>70.41.222.228: /* References */</p>
<hr />
<div>{{stub-general}}<br />
'''Parallel hard cubes'''<br />
==Mixtures==<br />
#[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)]<br />
#[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)]<br />
#[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)]<br />
==References==<br />
#[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)]<br />
#[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)]<br />
#[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)]<br />
[[category: models]]<br />
William G. Hoover, "High-Density Equation of State of Hard Parallel Squares and Cubes", Journal of Chemical Physics '''40''', 937 (1964).<br />
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).</div>70.41.222.228http://www.sklogwiki.org/SklogWiki/index.php?title=Hard_Parallel_Squares_and_Cubes&diff=7662Hard Parallel Squares and Cubes2009-01-31T23:44:11Z<p>70.41.222.228: </p>
<hr />
<div>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 ].<br />
<br />
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 <math>\{ T_{xx},T_{yy},T_{zz}\}</math>. 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.</div>70.41.222.228http://www.sklogwiki.org/SklogWiki/index.php?title=Hard_Parallel_Squares_and_Cubes&diff=7661Hard Parallel Squares and Cubes2009-01-31T22:17:12Z<p>70.41.222.228: </p>
<hr />
<div>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 ].<br />
<br />
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 <math>\{ T_{xx},T_{yy},T_{zz}\}</math>. 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 squares in two dimensions) allows them to serve as gedanken-experiment thermometers for all three temperature components.</div>70.41.222.228http://www.sklogwiki.org/SklogWiki/index.php?title=Hard_Parallel_Squares_and_Cubes&diff=7660Hard Parallel Squares and Cubes2009-01-31T22:16:13Z<p>70.41.222.228: </p>
<hr />
<div>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 ].<br />
<br />
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 <math>\{ T_{xx},T_{yy},T_{zz}\}</math>. 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.</div>70.41.222.228http://www.sklogwiki.org/SklogWiki/index.php?title=Hard_Parallel_Squares_and_Cubes&diff=7659Hard Parallel Squares and Cubes2009-01-31T21:54:44Z<p>70.41.222.228: </p>
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<div>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 ]. 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 <math>\{ T_{xx},T_{yy},T_{zz}\}</math>. 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.</div>70.41.222.228http://www.sklogwiki.org/SklogWiki/index.php?title=Hard_Parallel_Squares_and_Cubes&diff=7658Hard Parallel Squares and Cubes2009-01-31T21:47:38Z<p>70.41.222.228: </p>
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<div>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 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 <math>\{ T_{xx},T_{yy},T_{zz}\}</math>. 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.</div>70.41.222.228http://www.sklogwiki.org/SklogWiki/index.php?title=Hard_Parallel_Squares_and_Cubes&diff=7657Hard Parallel Squares and Cubes2009-01-31T21:36:03Z<p>70.41.222.228: New page: 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...</p>
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<div>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 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 <math>\{ T_{xx},T_{yy},T_{zz}\}</math>. 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.</div>70.41.222.228http://www.sklogwiki.org/SklogWiki/index.php?title=Smooth_Particle_methods&diff=7656Smooth Particle methods2009-01-31T21:16:08Z<p>70.41.222.228: </p>
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<div>Smooth Particle Applied Mechanics [SPAM] and Smooth Particle Hydrodynamics [sph] are numerical methods for solving the equations of continuum mechanics (the continuity equation, the equation of motion, and the energy equation) with particles. This approach was originated by Lucy and Monaghan in 1977 for astrophysical applications, and has since been applied to many challenging problems in fluid and solid mechanics. The main advantage of smooth-particle methods is that the partial differential equations (continuity, motion, energy) are replaced by ordinary differential equations (like molecular dynamics) describing the motion of particles. The particles can be of any size, from microscopic to astophysical, and can obey any chosen constitutive equation. The main disadvantages (or research opportunities!) are the difficulties in treating sharp surfaces or interfaces with discrete particles and in avoiding the instabilities that can result for materials under tension, For a recent text see Wm. G. Hoover's "Smooth Particle Applied Mechanics -- The State of the Art" (World Scientific, Singapore, 2006).</div>70.41.222.228http://www.sklogwiki.org/SklogWiki/index.php?title=Smooth_Particle_methods&diff=7655Smooth Particle methods2009-01-31T21:15:05Z<p>70.41.222.228: SPAM and sph are particle methods for solving the continuum motion and energy equations.</p>
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<div>Smooth Particle Applied Mechanics {SPAM] and Smooth Particle Hydrodynamics [sph] are numerical methods for solving the equations of continuum mechanics (the continuity equation, the equation of motion, and the energy equation) with particles. This approach was originated by Lucy and Monaghan in 1977 for astrophysical applications, and has since been applied to many challenging problems in fluid and solid mechanics. The main advantage of smooth-particle methods is that the partial differential equations (continuity, motion, energy) are replaced by ordinary differential equations (like molecular dynamics) describing the motion of particles. The particles can be of any size, from microscopic to astophysical, and can obey any chosen constitutive equation. The main disadvantages (or research opportunities!) are the difficulties in treating sharp surfaces or interfaces with discrete particles and in avoiding the instabilities that can result for materials under tension, For a recent text see Wm. G. Hoover's "Smooth Particle Applied Mechanics -- The State of the Art (World Scientific, Singapore, 2006).</div>70.41.222.228http://www.sklogwiki.org/SklogWiki/index.php?title=Computer_simulation_techniques&diff=7654Computer simulation techniques2009-01-31T20:56:10Z<p>70.41.222.228: </p>
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<div>*[[Molecular dynamics]]<br />
*[[Monte Carlo]]<br />
==Material in common==<br />
*[[Boundary conditions]]<br />
*[[Coarse graining]]<br />
*[[Computation of phase equilibria]]<br />
*[[Configuration analysis]]<br />
*[[Dissipative particle dynamics]]<br />
*[[Electrostatics]]<br />
*[[Ergodic hypothesis]]<br />
*[[Finite size effects]]<br />
*[[Force fields]]<br />
*[[Gibbs-Duhem integration]]<br />
*[[Materials modelling and computer simulation codes]]<br />
*[[Models]]<br />
*[[Self-referential method]]<br />
*[[Smooth Particle methods]]<br />
*[[Tempering methods]]<br />
*[[Test area method]]<br />
*[[Test volume method]]<br />
*[[Verlet neighbour list]]<br />
*[[Widom test-particle method]]<br />
==Interesting reading==<br />
* W. W. Wood "Early history of computer simulations in statistical mechanics" in "Molecular-dynamics simulation of statistical-mechanical systems" (Eds. Giovanni Ciccotti and William G. Hoover) pp. 3-14 Società Italiana di Fisica (1986)<br />
*[http://physicsworldarchive.iop.org/full/pwa-pdf/9/4/phwv9i4a24.pdf Daan Frenkel and Jean-Pierre Hansen "Understanding liquids: a computer game?", Physics World '''9''' pp. 35–42 (April 1996)]<br />
*[http://arxiv.org/abs/0812.2086 Wm. G. Hoover "50 Years of Computer Simulation -- A Personal View (2008)]<br />
[[category: Computer simulation techniques]]</div>70.41.222.228http://www.sklogwiki.org/SklogWiki/index.php?title=Computer_simulation_techniques&diff=7653Computer simulation techniques2009-01-31T20:48:28Z<p>70.41.222.228: </p>
<hr />
<div>*[[Molecular dynamics]]<br />
*[[Monte Carlo]]<br />
==Material in common==<br />
*[[Boundary conditions]]<br />
*[[Coarse graining]]<br />
*[[Computation of phase equilibria]]<br />
*[[Configuration analysis]]<br />
*[[Dissipative particle dynamics]]<br />
*[[Electrostatics]]<br />
*[[Ergodic hypothesis]]<br />
*[[Finite size effects]]<br />
*[[Force fields]]<br />
*[[Gibbs-Duhem integration]]<br />
*[[Materials modelling and computer simulation codes]]<br />
*[[Models]]<br />
*[[Self-referential method]]<br />
*[[Smooth Particle methods]]<br />
*[[Tempering methods]]<br />
*[[Test area method]]<br />
*[[Test volume method]]<br />
*[[Verlet neighbour list]]<br />
*[[Widom test-particle method]]<br />
==Interesting reading==<br />
* W. W. Wood "Early history of computer simulations in statistical mechanics" in "Molecular-dynamics simulation of statistical-mechanical systems" (Eds. Giovanni Ciccotti and William G. Hoover) pp. 3-14 Società Italiana di Fisica (1986)<br />
*[http://physicsworldarchive.iop.org/full/pwa-pdf/9/4/phwv9i4a24.pdf Daan Frenkel and Jean-Pierre Hansen "Understanding liquids: a computer game?", Physics World '''9''' pp. 35–42 (April 1996)]<br />
[[category: Computer simulation techniques]]</div>70.41.222.228