Parallel hard cubes - Revision history

FSmallenburg: Added information and references on phase transition

2013-04-28T13:34:09Z

Added information and references on phase transition

← Older revision		Revision as of 15:34, 28 April 2013
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	====Usefulness of the Model====		====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.		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==

Carl McBride: Added a recent publication

2012-10-02T10:52:53Z

Added a recent publication

← Older revision		Revision as of 12:52, 2 October 2012
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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. 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 (~~Refs~~. 2 and 3). The latter two authors computed seven-term series for the models ~~(Ref. 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 (~~Ref. 7~~) 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. 3~~).		'''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>).
	====Usefulness of the Model====		====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. 8~~) 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.		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.

	==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==
	~~#B. T. Geilikman "", Proceedings of the Academy of Science of the USSR '''70''' pp. 25- (1950)~~		<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)]~~		;Related reading
	#[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)]		*[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://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://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)]
	#[http://dx.doi.org/10.~~1103~~/~~PhysRevE~~.58.~~4701 E~~. A. ~~Jagla~~ "~~Melting~~ of hard cubes", ~~Physical Review E~~ '''58''' ~~pp. 4701-4705~~ (~~1998~~)]
	#[http://arxiv.org/abs/0811.1807 Wm. G. Hoover, C. G. Hoover "Nonlinear Stresses and Temperatures in Transient Adiabatic and Shear Flows via Nonequilibrium Molecular Dynamics -- Three Definitions of Temperature", arXiv:0811.1807v3 (2009)]
	[[category: models]]		[[category: models]]

Nice and Tidy: /* Usefulness of the Model */ Changed inital wording.

2009-02-01T18:20:42Z

Usefulness of the Model: Changed inital wording.

← Older revision		Revision as of 20:20, 1 February 2009
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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. 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 (Refs. 2 and 3). The latter two authors computed seven-term series for the models (Ref. 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 (Ref. 7) 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. 3).		'''Parallel hard cubes''' are a simple particle [[Models \|model]] used in [[statistical mechanics]]. They were introduced by B. T. Geilikman (Ref. 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 (Refs. 2 and 3). The latter two authors computed seven-term series for the models (Ref. 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 (Ref. 7) 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. 3).
	====Usefulness of the Model====		====Usefulness of the Model====
	~~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 (Ref. 8) 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.		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. 8) 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.

	==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)]		#[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)]

Nice and Tidy: /* References */ Replaced links with DOI.

2009-02-01T16:05:31Z

References: Replaced links with DOI.

← Older revision		Revision as of 18:05, 1 February 2009
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	#B. T. Geilikman "", Proceedings of the Academy of Science of the USSR '''70''' pp. 25- (1950)		#B. T. Geilikman "", Proceedings of the Academy of Science of the USSR '''70''' pp. 25- (1950)
	#[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)]		#[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)]
	#[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).]		#[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)]
	#[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).]		#[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://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)]

Carl McBride: Transferred the content from the new page Hard Parallel Squares and Cubes.

2009-02-01T15:46:53Z

Transferred the content from the new page Hard Parallel Squares and Cubes.

← Older revision		Revision as of 17:46, 1 February 2009
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	~~{{stub-general}}~~		'''Parallel hard cubes''' are a simple particle [[Models \|model]] used in [[statistical mechanics]]. They were introduced by B. T. Geilikman (Ref. 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 (Refs. 2 and 3). The latter two authors computed seven-term series for the models (Ref. 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 (Ref. 7) 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. 3).
	'''Parallel hard cubes'''		====Usefulness of the Model====
			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 (Ref. 8) 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.
	~~==Usefulness of the Model==~~
	~~Parallel hard cubes provide~~ a simple statistical ~~mechanical model, amenable to analytic and theoretical work as well as computer simulation~~.
	~~Beginning with~~ Geilikman~~'s work in Russia~~ in 1950 ~~a succession~~ of ~~studies have clarified [1~~] ~~the low-density virial expansion~~ (pressure as
	a power series in the density) {2} the ~~ordering phase~~ transition ~~at moderate density~~, ~~[3]~~ the ~~properties~~ of mixtures of cubes of ~~different~~
	~~sizes~~, ~~and [4]~~ the ~~conceptual role of~~ the ~~model~~ as an ideal-gas ~~thermometer~~ capable of measuring ~~separately~~ the tensor components of
	~~temperature,~~ <math>\{ T_{xx}, T_{yy}, T_{zz} \}</math>.
	==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)]		#[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)]
	#[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)]		#[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)]
	#[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)]		#[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)]
	==References==		==References==
			#B. T. Geilikman "", Proceedings of the Academy of Science of the USSR '''70''' pp. 25- (1950)
	#[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)]		#[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)]
	#[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).]		#[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).]
Line 18:		Line 13:
	#[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/10.1103/PhysRevE.58.4701 E. A. Jagla "Melting of hard cubes", Physical Review E '''58''' pp. 4701-4705 (1998)]
			#[http://arxiv.org/abs/0811.1807 Wm. G. Hoover, C. G. Hoover "Nonlinear Stresses and Temperatures in Transient Adiabatic and Shear Flows via Nonequilibrium Molecular Dynamics -- Three Definitions of Temperature", arXiv:0811.1807v3 (2009)]
	[[category: models]]		[[category: models]]

70.41.222.228 at 06:10, 1 February 2009

2009-02-01T06:10:31Z

← Older revision		Revision as of 08:10, 1 February 2009
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	{{stub-general}}		{{stub-general}}
	'''Parallel hard cubes'''		'''Parallel hard cubes'''

			==Usefulness of the Model==
			Parallel hard cubes provide a simple statistical mechanical model, amenable to analytic and theoretical work as well as computer simulation.
			Beginning with Geilikman's work in Russia in 1950 a succession of studies have clarified [1] the low-density virial expansion (pressure as
			a power series in the density) {2} the ordering phase transition at moderate density, [3] the properties of mixtures of cubes of different
			sizes, and [4] the conceptual role of the model as an ideal-gas thermometer capable of measuring separately the tensor components of
			temperature, <math>\{ T_{xx}, T_{yy}, T_{zz} \}</math>.
	==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)]		#[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)]

70.41.222.228: /* References */

2009-02-01T05:52:49Z

References

← Older revision		Revision as of 07:52, 1 February 2009
Line 7:		Line 7:
	==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)]		#[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)]
	#[http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCPSA6000036000012003141000001&idtype=cvips&gifs=Yes		#[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).]
	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://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).]
	#[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).]
	#[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)]
	[[category: models]]		[[category: models]]

70.41.222.228: /* References */

2009-02-01T05:50:47Z

References

← Older revision		Revision as of 07:50, 1 February 2009
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	==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)]		#[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)]
	#[http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCPSA6000036000012003141000001&idtype=cvips&gifs=Yes]		#[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).		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://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCPSA6000040000004000937000001&idtype=cvips&gifs=Yes]		#[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).]		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)]
	[[category: models]]		[[category: models]]

70.41.222.228: /* References */

2009-02-01T05:49:05Z

References

← Older revision		Revision as of 07:49, 1 February 2009
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	==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)]		#[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)]
	#[http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCPSA6000036000012003141000001&idtype=cvips&gifs=Yes		#[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).]		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://~~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://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).]
	#[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)]
	[[category: models]]		[[category: models]]

70.41.222.228: /* References */

2009-02-01T05:45:11Z

References

← Older revision		Revision as of 07:45, 1 February 2009
Line 7:		Line 7:
	==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)]		#[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)]
	#[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).]		#[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).]
	#[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://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)]
	[[category: models]]		[[category: models]]