Editing Quantum hard spheres

The great usefulness of the [[hard sphere model]] for representing particles in classical [[statistical mechanics]] is very well known and its study has provided guidance in the understanding of classical fluids and solids. This model assumes that [[Intermolecular pair potential |pairwise interactions]] between particles are singular in that they become an infinite repulsion for distances smaller than the diameter of the spheres, being identically zero otherwise. Perhaps, the most remarkable feature of classical hard spheres is that they show a [[Solid-liquid phase transitions |fluid-solid transition]], which was first predicted with [[Computer simulation techniques |computer simulation]], and confirmed later with experiments on [[colloids |colloidal particles]] (see [http://dx.doi.org/10.1038/320340a0 Pusey & van Megen]). From the [[Classical thermodynamics |thermodynamic]] point of view the states of this model only need one parameter to be characterized: the (number) density. This classical state of affairs implies that the quantum [[De Broglie thermal wavelength |thermal de Broglie wavelength]] of the particles is zero. With the use of reduced units (unit length= hard-sphere diameter) the results arising from this singular interaction potential (hard core) can be transferred between systems differing in the size of their spheres. Among the many interesting features displayed by this model one should mention that there is “contact” between particles at distances equal to the  diameter (<math>d^+</math>) (they are like hard billiards), which reflects in the fact that the main peak of the pair [[Radial distribution function |radial correlation function]] is located just at that “contact” point.

	Nevertheless, the switching on of the quantum conditions upon this system (i.e. non-zero de Broglie wavelengths) changes dramatically the classical properties. To illustrate this three examples will suffice. First, the characterization of the state points requires an additional parameter, the thermal wavelength, which contains the [[temperature]], the mass of the particles, and [[Planck constant |Planck’s constant]] (once again, using reduced units allows one to transfer results between situations at the same values of the density and of the de Broglie wavelength). Secondly, the above classical “contact” is forbidden, as quantum hard spheres repel each other before getting into (classical) contact. And thirdly, the fluid-solid phase transition is driven by energy in the quantum limit of zero temperature, this being different from the classical case which is driven by [[entropy]]. Furthermore, quantum hard spheres seem appropriate to understand the low temperature properties of ultra-hard materials and colloids. In this endeavour Feynman’s [[Path integral formulation |path-integrals]] combined with computer simulations provide a very powerful tool to undertake the pertinent calculations. Thus, apart from being an appealing mathematical problem, quantum hard spheres can be very useful from a practical standpoint for the design of new materials.

Attention has been focused upon the equilibrium properties, thermodynamic and structural. It is worth realizing that, while there is only one pair radial correlation function in the classical case, the quantum de-localisation brings about three different pair radial correlation functions in the quantum case, each of them possessing a definite physical meaning. Great emphasis has been placed on the study of the different structural functions, in both the [[r-correlation]] and the k-Fourier spaces that can be determined in a quantum many-body system, as they open an alternative way to carry out computations leading to the fixing of the [[Equations of state |equation of state]]. This effort has helped to clarify the role of the path-integral centroids in (equilibrium) quantum statistical mechanics, and also the possibilities of utilizing [[Ornstein-Zernike relation |Ornstein-Zernike]] classical frameworks in dealing with quantum fluids.

==References==
#[http://dx.doi.org/10.1103/PhysRev.106.412 M. Fierz "Connection between Pair Density and Pressure for a Bose Gas Consisting of Rigid Spherical Atoms", Physical Review '''106''' 412 - 413 (1957)]
#[http://dx.doi.org/10.1063/1.1705099 Elliott H. Lieb "Calculation of Exchange Second Virial Coefficient of a Hard-Sphere Gas by Path Integrals", Journal of Mathematical Physics '''8''' pp. 43-52 (1967)]
#[http://dx.doi.org/10.1080/00268977500101711 W. G. Gibson "Quantum corrections to the properties of a dense fluid with non-analytic intermolecular potential function I. The general case" Molecular Physics '''30''' pp. 1-11 (1975)]
#[http://dx.doi.org/10.1080/00268977500101721 W. G. Gibson "Quantum corrections to the properties of a dense fluid with non-analytic intermolecular potential function II. Hard spheres", Molecular Physics '''30''' pp. 13-30 (1975)]
#[http://dx.doi.org/10.1063/1.437829 J. A. Barker "A quantum-statistical Monte Carlo method; path integrals with boundary conditions", Journal of Chemical Physics '''70''' pp. 2914-2918 (1979)]
#[http://dx.doi.org/10.1063/1.446134 G. Jacucci and E. Omerti "Monte Carlo calculation of the radial distribution function of quantum hard spheres at finite temperatures using path integrals with boundary conditions", Journal of Chemical Physics '''79''' pp. 3051-3054 (1983)]
#[http://dx.doi.org/10.1103/PhysRevB.38.135 Karl J. Runge and Geoffrey V. Chester "Solid-fluid phase transition of quantum hard spheres at finite temperatures", Physical Review B '''38''' 135 - 162 (1988)]
#[http://dx.doi.org/10.1063/1.463076      J. Cao and B. J. Berne "A new quantum propagator for hard sphere and cavity systems", Journal of Chemical Physics  '''97''' pp. 2382-2385 (1992)]
#[http://dx.doi.org/10.1103/PhysRevLett.79.3549      Peter Grüter, David Ceperley and Frank Laloë "Critical Temperature of Bose-Einstein Condensation of Hard-Sphere Gases", Physical Review Letters '''79''' 3549 - 3552 (1997)]
#[http://dx.doi.org/10.1063/1.2718523  Luis M. Sesé "Computational study of the melting-freezing transition in the quantum hard-sphere system for intermediate densities. I. Thermodynamic results", Journal of Chemical Physics '''126''' 164508 (2007)]
#[http://dx.doi.org/10.1063/1.2718525     Luis M. Sesé and Lorna E. Bailey "Computational study of the melting-freezing transition in the quantum hard-sphere system for intermediate densities. II. Structural features",  Journal of Chemical Physics '''126''' 164509 (2007)]
#[http://dx.doi.org/10.1063/1.2753837  Luis M. Sesé and Lorna E. Bailey "Erratum: “Computational study of the melting-freezing transition in the quantum hard-sphere system for intermediate densities. II. Structural features” <nowiki>[</nowiki> Journal of Chemical Physics '''126''' 164509 (2007)<nowiki>]</nowiki>, Journal of Chemical Physics '''127''' 049901 (2007)]
#[http://dx.doi.org/10.1063/1.2009733 Luis M. Sesé "Triplet correlations in the quantum hard-sphere fluid", Journal of Chemical Physics '''123''' 104507 (2005)]
#[http://dx.doi.org/10.1063/1.1776114 Luis M. Sesé "Computation of the equation of state of the quantum hard-sphere fluid utilizing several path-integral strategies", Journal of Chemical Physics  '''121''' 3702 (2004)]
#[http://dx.doi.org/10.1063/1.1808115 L. E. Bailey and L. M. Sesé "The decay of pair correlations in quantum hard-sphere fluids", Journal of Chemical Physics '''121''' 10076 (2004)]
#[http://dx.doi.org/10.1063/1.1618731 L. M. Sesé and L. E. Bailey "A simulation study of the quantum hard-sphere Yukawa fluid",  Journal of Chemical Physics '''119''' 10256 (2003)]
#[http://dx.doi.org/10.1080/0026897031000094470 L. M. Sesé "The compressibility theorem for quantum simple fluids at equilibrium", Molecular Physics '''101''' 1455 (2003)]
#[http://dx.doi.org/10.1063/1.1468223 L. M. Sesé "Properties of the path-integral quantum hard-sphere fluid in k space",  Journal of Chemical Physics '''116''' 8492 (2002)]
#[http://dx.doi.org/10.1063/1.1401818 L. E. Bailey and L. M. Sesé "The asymptotic decay of pair correlations in the path-integral quantum hard-sphere fluid", Journal of Chemical Physics '''115''' 6557 (2001)]
#[http://dx.doi.org/10.1063/1.1328751 L. M. Sesé "Path-integral Monte Carlo study of the structural and mechanical properties of quantum fcc and bcc hard-sphere solids", Journal of Chemical Physics '''114''' 1732 (2001)]
#[http://dx.doi.org/10.1063/ L. M. Sesé "Thermodynamic ans structural properties of the path-integral quantum hard-sphere fluid", Journal of Chemical Physics '''108''' 9086 (1998)]
#[http://dx.doi.org/10.1063/ L. M. Sesé and R. Ledesma "Computation of the static structure factor of the path-integral quantum hard-sphere fluid", Journal of Chemical Physics '''106''' 1134 (1997)]
#[http://dx.doi.org/10.1063/ L. M. Sesé and R. Ledesma "Path-integral energy and structure of the quantum hard-sphere system using efficient propagators", Journal of Chemical Physics  '''102''' 3776 (1995)]
[[category: hard sphere]]
@@ Line 4: / Line 4: @@
 Attention has been focused upon the equilibrium properties, thermodynamic and structural. It is worth realizing that, while there is only one pair radial correlation function in the classical case, the quantum de-localisation brings about three different pair radial correlation functions in the quantum case, each of them possessing a definite physical meaning. Great emphasis has been placed on the study of the different structural functions, in both the [[r-correlation]] and the k-Fourier spaces that can be determined in a quantum many-body system, as they open an alternative way to carry out computations leading to the fixing of the [[Equations of state |equation of state]]. This effort has helped to clarify the role of the path-integral centroids in (equilibrium) quantum statistical mechanics, and also the possibilities of utilizing [[Ornstein-Zernike relation |Ornstein-Zernike]] classical frameworks in dealing with quantum fluids.
-==Isothermal compressibility==
-[[Compressibility#Isothermal compressibility|Isothermal compressibility]] <ref>[http://dx.doi.org/10.1063/1.4729254 Luis M. Sesé "On the accurate direct computation of the isothermal compressibility for normal quantum simple fluids: Application to quantum hard spheres", Journal of Chemical Physics '''136''' 244504 (2012)]</ref>.
-==Crystallization line==
-The structural regularities along the crystallization line has been studied by way of [[Path integral formulation | path integral Monte Carlo simulations]] and the [[Ornstein-Zernike relation | Ornstein-Zernike pair equation]] <ref>[http://dx.doi.org/10.1063/1.4943005  Luis M. Sesé "Path-integral and Ornstein-Zernike study of quantum fluid structures on the crystallization line", Journal of Chemical Physics '''144''' 094505 (2016)]</ref>.
 ==References==
-<references/>
+#[http://dx.doi.org/10.1103/PhysRev.106.412 M. Fierz "Connection between Pair Density and Pressure for a Bose Gas Consisting of Rigid Spherical Atoms", Physical Review '''106''' 412 - 413 (1957)]
-;Related reading
+#[http://dx.doi.org/10.1063/1.1705099 Elliott H. Lieb "Calculation of Exchange Second Virial Coefficient of a Hard-Sphere Gas by Path Integrals", Journal of Mathematical Physics '''8''' pp. 43-52 (1967)]
-*[http://dx.doi.org/10.1103/PhysRev.106.412 M. Fierz "Connection between Pair Density and Pressure for a Bose Gas Consisting of Rigid Spherical Atoms", Physical Review '''106''' 412 - 413 (1957)]
+#[http://dx.doi.org/10.1080/00268977500101711 W. G. Gibson "Quantum corrections to the properties of a dense fluid with non-analytic intermolecular potential function I. The general case" Molecular Physics '''30''' pp. 1-11 (1975)]
-*[http://dx.doi.org/10.1063/1.1705099 Elliott H. Lieb "Calculation of Exchange Second Virial Coefficient of a Hard-Sphere Gas by Path Integrals", Journal of Mathematical Physics '''8''' pp. 43-52 (1967)]
+#[http://dx.doi.org/10.1080/00268977500101721 W. G. Gibson "Quantum corrections to the properties of a dense fluid with non-analytic intermolecular potential function II. Hard spheres", Molecular Physics '''30''' pp. 13-30 (1975)]
-*[http://dx.doi.org/10.1080/00268977500101711 W. G. Gibson "Quantum corrections to the properties of a dense fluid with non-analytic intermolecular potential function I. The general case" Molecular Physics '''30''' pp. 1-11 (1975)]
+#[http://dx.doi.org/10.1063/1.437829 J. A. Barker "A quantum-statistical Monte Carlo method; path integrals with boundary conditions", Journal of Chemical Physics '''70''' pp. 2914-2918 (1979)]
-*[http://dx.doi.org/10.1080/00268977500101721 W. G. Gibson "Quantum corrections to the properties of a dense fluid with non-analytic intermolecular potential function II. Hard spheres", Molecular Physics '''30''' pp. 13-30 (1975)]
+#[http://dx.doi.org/10.1063/1.446134 G. Jacucci and E. Omerti "Monte Carlo calculation of the radial distribution function of quantum hard spheres at finite temperatures using path integrals with boundary conditions", Journal of Chemical Physics '''79''' pp. 3051-3054 (1983)]
-*[http://dx.doi.org/10.1063/1.437829 J. A. Barker "A quantum-statistical Monte Carlo method; path integrals with boundary conditions", Journal of Chemical Physics '''70''' pp. 2914-2918 (1979)]
+#[http://dx.doi.org/10.1103/PhysRevB.38.135 Karl J. Runge and Geoffrey V. Chester "Solid-fluid phase transition of quantum hard spheres at finite temperatures", Physical Review B '''38''' 135 - 162 (1988)]
-*[http://dx.doi.org/10.1063/1.446134 G. Jacucci and E. Omerti "Monte Carlo calculation of the radial distribution function of quantum hard spheres at finite temperatures using path integrals with boundary conditions", Journal of Chemical Physics '''79''' pp. 3051-3054 (1983)]
+#[http://dx.doi.org/10.1063/1.463076      J. Cao and B. J. Berne "A new quantum propagator for hard sphere and cavity systems", Journal of Chemical Physics  '''97''' pp. 2382-2385 (1992)]
-*[http://dx.doi.org/10.1103/PhysRevB.38.135 Karl J. Runge and Geoffrey V. Chester "Solid-fluid phase transition of quantum hard spheres at finite temperatures", Physical Review B '''38''' 135 - 162 (1988)]
+#[http://dx.doi.org/10.1103/PhysRevLett.79.3549      Peter Grüter, David Ceperley and Frank Laloë "Critical Temperature of Bose-Einstein Condensation of Hard-Sphere Gases", Physical Review Letters '''79''' 3549 - 3552 (1997)]
-*[http://dx.doi.org/10.1063/1.463076      J. Cao and B. J. Berne "A new quantum propagator for hard sphere and cavity systems", Journal of Chemical Physics  '''97''' pp. 2382-2385 (1992)]
+#[http://dx.doi.org/10.1063/1.2718523  Luis M. Sesé "Computational study of the melting-freezing transition in the quantum hard-sphere system for intermediate densities. I. Thermodynamic results", Journal of Chemical Physics '''126''' 164508 (2007)]
-*[http://dx.doi.org/10.1103/PhysRevLett.79.3549      Peter Grüter, David Ceperley and Frank Laloë "Critical Temperature of Bose-Einstein Condensation of Hard-Sphere Gases", Physical Review Letters '''79''' 3549 - 3552 (1997)]
+#[http://dx.doi.org/10.1063/1.2718525     Luis M. Sesé and Lorna E. Bailey "Computational study of the melting-freezing transition in the quantum hard-sphere system for intermediate densities. II. Structural features",  Journal of Chemical Physics '''126''' 164509 (2007)]
-*[http://dx.doi.org/10.1063/1.2718523  Luis M. Sesé "Computational study of the melting-freezing transition in the quantum hard-sphere system for intermediate densities. I. Thermodynamic results", Journal of Chemical Physics '''126''' 164508 (2007)]
+#[http://dx.doi.org/10.1063/1.2753837  Luis M. Sesé and Lorna E. Bailey "Erratum: “Computational study of the melting-freezing transition in the quantum hard-sphere system for intermediate densities. II. Structural features” <nowiki>[</nowiki> Journal of Chemical Physics '''126''' 164509 (2007)<nowiki>]</nowiki>, Journal of Chemical Physics '''127''' 049901 (2007)]
-*[http://dx.doi.org/10.1063/1.2718525     Luis M. Sesé and Lorna E. Bailey "Computational study of the melting-freezing transition in the quantum hard-sphere system for intermediate densities. II. Structural features",  Journal of Chemical Physics '''126''' 164509 (2007)]
+#[http://dx.doi.org/10.1063/1.2009733 Luis M. Sesé "Triplet correlations in the quantum hard-sphere fluid", Journal of Chemical Physics '''123''' 104507 (2005)]
-*[http://dx.doi.org/10.1063/1.2753837  Luis M. Sesé and Lorna E. Bailey "Erratum: “Computational study of the melting-freezing transition in the quantum hard-sphere system for intermediate densities. II. Structural features” <nowiki>[</nowiki> Journal of Chemical Physics '''126''' 164509 (2007)<nowiki>]</nowiki>, Journal of Chemical Physics '''127''' 049901 (2007)]
+#[http://dx.doi.org/10.1063/1.1776114 Luis M. Sesé "Computation of the equation of state of the quantum hard-sphere fluid utilizing several path-integral strategies", Journal of Chemical Physics  '''121''' 3702 (2004)]
-*[http://dx.doi.org/10.1063/1.2009733 Luis M. Sesé "Triplet correlations in the quantum hard-sphere fluid", Journal of Chemical Physics '''123''' 104507 (2005)]
+#[http://dx.doi.org/10.1063/1.1808115 L. E. Bailey and L. M. Sesé "The decay of pair correlations in quantum hard-sphere fluids", Journal of Chemical Physics '''121''' 10076 (2004)]
-*[http://dx.doi.org/10.1063/1.1776114 Luis M. Sesé "Computation of the equation of state of the quantum hard-sphere fluid utilizing several path-integral strategies", Journal of Chemical Physics  '''121''' 3702 (2004)]
+#[http://dx.doi.org/10.1063/1.1618731 L. M. Sesé and L. E. Bailey "A simulation study of the quantum hard-sphere Yukawa fluid",  Journal of Chemical Physics '''119''' 10256 (2003)]
-*[http://dx.doi.org/10.1063/1.1808115 L. E. Bailey and L. M. Sesé "The decay of pair correlations in quantum hard-sphere fluids", Journal of Chemical Physics '''121''' 10076 (2004)]
+#[http://dx.doi.org/10.1080/0026897031000094470 L. M. Sesé "The compressibility theorem for quantum simple fluids at equilibrium", Molecular Physics '''101''' 1455 (2003)]
-*[http://dx.doi.org/10.1063/1.1618731 L. M. Sesé and L. E. Bailey "A simulation study of the quantum hard-sphere Yukawa fluid",  Journal of Chemical Physics '''119''' 10256 (2003)]
+#[http://dx.doi.org/10.1063/1.1468223 L. M. Sesé "Properties of the path-integral quantum hard-sphere fluid in k space",  Journal of Chemical Physics '''116''' 8492 (2002)]
-*[http://dx.doi.org/10.1080/0026897031000094470 L. M. Sesé "The compressibility theorem for quantum simple fluids at equilibrium", Molecular Physics '''101''' 1455 (2003)]
+#[http://dx.doi.org/10.1063/1.1401818 L. E. Bailey and L. M. Sesé "The asymptotic decay of pair correlations in the path-integral quantum hard-sphere fluid", Journal of Chemical Physics '''115''' 6557 (2001)]
-*[http://dx.doi.org/10.1063/1.1468223 L. M. Sesé "Properties of the path-integral quantum hard-sphere fluid in k space",  Journal of Chemical Physics '''116''' 8492 (2002)]
+#[http://dx.doi.org/10.1063/1.1328751 L. M. Sesé "Path-integral Monte Carlo study of the structural and mechanical properties of quantum fcc and bcc hard-sphere solids", Journal of Chemical Physics '''114''' 1732 (2001)]
-*[http://dx.doi.org/10.1063/1.1401818 L. E. Bailey and L. M. Sesé "The asymptotic decay of pair correlations in the path-integral quantum hard-sphere fluid", Journal of Chemical Physics '''115''' 6557 (2001)]
+#[http://dx.doi.org/10.1063/ L. M. Sesé "Thermodynamic ans structural properties of the path-integral quantum hard-sphere fluid", Journal of Chemical Physics '''108''' 9086 (1998)]
-*[http://dx.doi.org/10.1063/1.1328751 L. M. Sesé "Path-integral Monte Carlo study of the structural and mechanical properties of quantum fcc and bcc hard-sphere solids", Journal of Chemical Physics '''114''' 1732 (2001)]
+#[http://dx.doi.org/10.1063/ L. M. Sesé and R. Ledesma "Computation of the static structure factor of the path-integral quantum hard-sphere fluid", Journal of Chemical Physics '''106''' 1134 (1997)]
-*[http://dx.doi.org/10.1063/ L. M. Sesé "Thermodynamic and structural properties of the path-integral quantum hard-sphere fluid", Journal of Chemical Physics '''108''' 9086 (1998)]
+#[http://dx.doi.org/10.1063/ L. M. Sesé and R. Ledesma "Path-integral energy and structure of the quantum hard-sphere system using efficient propagators", Journal of Chemical Physics  '''102''' 3776 (1995)]
-*[http://dx.doi.org/10.1063/ L. M. Sesé and R. Ledesma "Computation of the static structure factor of the path-integral quantum hard-sphere fluid", Journal of Chemical Physics '''106''' 1134 (1997)]
-*[http://dx.doi.org/10.1063/ L. M. Sesé and R. Ledesma "Path-integral energy and structure of the quantum hard-sphere system using efficient propagators", Journal of Chemical Physics  '''102''' 3776 (1995)]
-*[http://dx.doi.org/10.1063/1.4813635 Luis M. Sesé "Path integral Monte Carlo study of quantum-hard sphere solids", Journal of Chemical Physics '''139''' 044502 (2013)]
 [[category: hard sphere]]