http://www.sklogwiki.org/SklogWiki/index.php?title=Quantum_hard_spheres&feed=atom&action=historyQuantum hard spheres - Revision history2024-03-28T23:45:53ZRevision history for this page on the wikiMediaWiki 1.41.0http://www.sklogwiki.org/SklogWiki/index.php?title=Quantum_hard_spheres&diff=19207&oldid=prevCarl McBride: Added a section regarding the Crystallization line2016-03-09T14:47:29Z<p>Added a section regarding the Crystallization line</p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Isothermal compressibility==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Isothermal compressibility==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[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>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[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>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Crystallization line==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">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>.</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td></tr>
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</table>Carl McBridehttp://www.sklogwiki.org/SklogWiki/index.php?title=Quantum_hard_spheres&diff=13896&oldid=prevCarl McBride: /* References */ Added a recent publication2013-11-06T13:18:12Z<p><span dir="auto"><span class="autocomment">References: </span> Added a recent publication</span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 14:18, 6 November 2013</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*[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)]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*[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)]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*[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)]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*[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)]</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*[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)]</ins></div></td></tr>
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</table>Carl McBridehttp://www.sklogwiki.org/SklogWiki/index.php?title=Quantum_hard_spheres&diff=12946&oldid=prevCarl McBride: Added a recent publication2012-06-26T13:42:54Z<p>Added a recent publication</p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 14:42, 26 June 2012</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Isothermal compressibility==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[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>.</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><references/></ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">;Related reading</ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*</ins>[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)]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*</ins>[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)]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*</ins>[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)]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*</ins>[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)]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*</ins>[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)]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*</ins>[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)]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*</ins>[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)]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*</ins>[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)]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*</ins>[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)]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*</ins>[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)]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*</ins>[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)]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*</ins>[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)]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*</ins>[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)]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*</ins>[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)]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#</del>[http://dx.doi.org/10.1080/0026897031000094470 L. M. Sesé "The compressibility theorem for quantum simple fluids at equilibrium", Molecular Physics '''101''' 1455 (2003)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*</ins>[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)]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*</ins>[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)]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*</ins>[http://dx.doi.org/10.1080/0026897031000094470 L. M. Sesé "The compressibility theorem for quantum simple fluids at equilibrium", Molecular Physics '''101''' 1455 (2003)]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*</ins>[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)]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#</del>[http://dx.doi.org/10.1063/ L. M. Sesé "Thermodynamic <del style="font-weight: bold; text-decoration: none;">ans </del>structural properties of the path-integral quantum hard-sphere fluid", Journal of Chemical Physics '''108''' 9086 (1998)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*</ins>[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)]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*</ins>[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)]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#</del>[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)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*</ins>[http://dx.doi.org/10.1063/ L. M. Sesé "Thermodynamic <ins style="font-weight: bold; text-decoration: none;">and </ins>structural properties of the path-integral quantum hard-sphere fluid", Journal of Chemical Physics '''108''' 9086 (1998)]</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*</ins>[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)]</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*</ins>[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)]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[category: hard sphere]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[category: hard sphere]]</div></td></tr>
</table>Carl McBridehttp://www.sklogwiki.org/SklogWiki/index.php?title=Quantum_hard_spheres&diff=4828&oldid=prevCarl McBride: Slight tidy up.2007-11-06T17:39:55Z<p>Slight tidy up.</p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 18:39, 6 November 2007</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Line 1:</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>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]<del style="font-weight: bold; text-decoration: none;">] [Wood & Jacobson, J. CHEM. PHYS. 27, 1207 (1957); Alder & Wainwright, J. CHEM. PHYS. 27, 1208 (1957)</del>], and confirmed later with experiments on [[colloids |colloidal particles]] (Pusey & van Megen<del style="font-weight: bold; text-decoration: none;">, NATURE 320, 340 (1986)</del>). 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.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>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]] (<ins style="font-weight: bold; text-decoration: none;">see [http://dx.doi.org/10.1038/320340a0 </ins>Pusey & van Megen<ins style="font-weight: bold; text-decoration: none;">]</ins>). 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.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> 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.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> 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.</div></td></tr>
</table>Carl McBridehttp://www.sklogwiki.org/SklogWiki/index.php?title=Quantum_hard_spheres&diff=4602&oldid=prevCarl McBride: /* References */2007-10-17T13:29:05Z<p><span dir="auto"><span class="autocomment">References</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 14:29, 17 October 2007</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>#M. Fierz, <del style="font-weight: bold; text-decoration: none;">Phys. Rev. </del>106<del style="font-weight: bold; text-decoration: none;">, </del>412 (1957)</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>#<ins style="font-weight: bold; text-decoration: none;">[http://dx.doi.org/10.1103/PhysRev.106.412 </ins>M. Fierz <ins style="font-weight: bold; text-decoration: none;">"Connection between Pair Density and Pressure for a Bose Gas Consisting of Rigid Spherical Atoms"</ins>, <ins style="font-weight: bold; text-decoration: none;">Physical Review '''</ins>106<ins style="font-weight: bold; text-decoration: none;">''' </ins>412 <ins style="font-weight: bold; text-decoration: none;">- 413 </ins>(1957)<ins style="font-weight: bold; text-decoration: none;">]</ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>#<del style="font-weight: bold; text-decoration: none;">E</del>. H. Lieb, <del style="font-weight: bold; text-decoration: none;">J</del>. <del style="font-weight: bold; text-decoration: none;">Math. Phys. 8, </del>43 (1967)</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>#<ins style="font-weight: bold; text-decoration: none;">[http://dx.doi.org/10.1063/1</ins>.<ins style="font-weight: bold; text-decoration: none;">1705099 Elliott </ins>H. Lieb <ins style="font-weight: bold; text-decoration: none;">"Calculation of Exchange Second Virial Coefficient of a Hard-Sphere Gas by Path Integrals"</ins>, <ins style="font-weight: bold; text-decoration: none;">Journal of Mathematical Physics '''8''' pp</ins>. 43<ins style="font-weight: bold; text-decoration: none;">-52 </ins>(1967)<ins style="font-weight: bold; text-decoration: none;">]</ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>#W. G. Gibson<del style="font-weight: bold; text-decoration: none;">, Molec</del>. <del style="font-weight: bold; text-decoration: none;">Phys</del>. 30<del style="font-weight: bold; text-decoration: none;">, </del>13 (1975)</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>#<ins style="font-weight: bold; text-decoration: none;">[http://dx.doi.org/10.1080/00268977500101711 </ins>W. G. Gibson <ins style="font-weight: bold; text-decoration: none;">"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)]</ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>#J. A. Barker, <del style="font-weight: bold; text-decoration: none;">J</del>. <del style="font-weight: bold; text-decoration: none;">Chem. Phys. 70, </del>2914 (1979)</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">#[http://dx.doi.org/10.1080/00268977500101721 W</ins>. <ins style="font-weight: bold; text-decoration: none;">G</ins>. <ins style="font-weight: bold; text-decoration: none;">Gibson "Quantum corrections to the properties of a dense fluid with non-analytic intermolecular potential function II. Hard spheres", Molecular Physics '''</ins>30<ins style="font-weight: bold; text-decoration: none;">''' pp. </ins>13<ins style="font-weight: bold; text-decoration: none;">-30 </ins>(1975)<ins style="font-weight: bold; text-decoration: none;">]</ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>#G. Jacucci and E. Omerti, <del style="font-weight: bold; text-decoration: none;">J. Chem</del>. <del style="font-weight: bold; text-decoration: none;">Phys. 79, </del>3051 (1983)<del style="font-weight: bold; text-decoration: none;">.</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>#<ins style="font-weight: bold; text-decoration: none;">[http://dx.doi.org/10.1063/1.437829 </ins>J. A. Barker <ins style="font-weight: bold; text-decoration: none;">"A quantum-statistical Monte Carlo method; path integrals with boundary conditions"</ins>, <ins style="font-weight: bold; text-decoration: none;">Journal of Chemical Physics '''70''' pp</ins>. 2914<ins style="font-weight: bold; text-decoration: none;">-2918 </ins>(1979)<ins style="font-weight: bold; text-decoration: none;">]</ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>#<del style="font-weight: bold; text-decoration: none;">K</del>. J. Runge and <del style="font-weight: bold; text-decoration: none;">G. </del>V. Chester, <del style="font-weight: bold; text-decoration: none;">Phys. Rev. </del>B 38<del style="font-weight: bold; text-decoration: none;">, </del>135 (1988)</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>#<ins style="font-weight: bold; text-decoration: none;">[http://dx.doi.org/10.1063/1.446134 </ins>G. Jacucci and E. Omerti <ins style="font-weight: bold; text-decoration: none;">"Monte Carlo calculation of the radial distribution function of quantum hard spheres at finite temperatures using path integrals with boundary conditions"</ins>, <ins style="font-weight: bold; text-decoration: none;">Journal of Chemical Physics '''79''' pp</ins>. 3051<ins style="font-weight: bold; text-decoration: none;">-3054 </ins>(1983)<ins style="font-weight: bold; text-decoration: none;">]</ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>#J. Cao and B. J. Berne, <del style="font-weight: bold; text-decoration: none;">J. Chem. Phys</del>. <del style="font-weight: bold; text-decoration: none;">97, </del>2382 (1992)</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>#<ins style="font-weight: bold; text-decoration: none;">[http://dx.doi.org/10.1103/PhysRevB.38</ins>.<ins style="font-weight: bold; text-decoration: none;">135 Karl </ins>J. Runge and <ins style="font-weight: bold; text-decoration: none;">Geoffrey </ins>V. Chester <ins style="font-weight: bold; text-decoration: none;">"Solid-fluid phase transition of quantum hard spheres at finite temperatures"</ins>, <ins style="font-weight: bold; text-decoration: none;">Physical Review </ins>B <ins style="font-weight: bold; text-decoration: none;">'''</ins>38<ins style="font-weight: bold; text-decoration: none;">''' </ins>135 <ins style="font-weight: bold; text-decoration: none;">- 162 </ins>(1988)<ins style="font-weight: bold; text-decoration: none;">]</ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>#<del style="font-weight: bold; text-decoration: none;">P</del>. Grüter, <del style="font-weight: bold; text-decoration: none;">D. </del>Ceperley and <del style="font-weight: bold; text-decoration: none;">F. Lalöe</del>, <del style="font-weight: bold; text-decoration: none;">Phys. Rev. Lett. </del>79<del style="font-weight: bold; text-decoration: none;">, </del>3549 (1997)</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>#<ins style="font-weight: bold; text-decoration: none;">[http://dx.doi.org/10.1063/1.463076 </ins>J. Cao and B. J. Berne <ins style="font-weight: bold; text-decoration: none;">"A new quantum propagator for hard sphere and cavity systems"</ins>, <ins style="font-weight: bold; text-decoration: none;">Journal of Chemical Physics '''97''' pp</ins>. 2382<ins style="font-weight: bold; text-decoration: none;">-2385 </ins>(1992)<ins style="font-weight: bold; text-decoration: none;">]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>#<ins style="font-weight: bold; text-decoration: none;">[http://dx</ins>.<ins style="font-weight: bold; text-decoration: none;">doi.org/10.1103/PhysRevLett.79.3549 Peter </ins>Grüter, <ins style="font-weight: bold; text-decoration: none;">David </ins>Ceperley and <ins style="font-weight: bold; text-decoration: none;">Frank Laloë "Critical Temperature of Bose-Einstein Condensation of Hard-Sphere Gases"</ins>, <ins style="font-weight: bold; text-decoration: none;">Physical Review Letters '''</ins>79<ins style="font-weight: bold; text-decoration: none;">''' </ins>3549 <ins style="font-weight: bold; text-decoration: none;">- 3552 </ins>(1997)<ins style="font-weight: bold; text-decoration: none;">]</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#[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)]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#[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)]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#[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)]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#[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)]</div></td></tr>
</table>Carl McBridehttp://www.sklogwiki.org/SklogWiki/index.php?title=Quantum_hard_spheres&diff=4594&oldid=prevCarl McBride: /* References */2007-10-16T17:47:31Z<p><span dir="auto"><span class="autocomment">References</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 18:47, 16 October 2007</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#[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)]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#[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)]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#[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)]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#[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)]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>#[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><del style="font-weight: bold; text-decoration: none;">J. Chem. Phys. </del>126<del style="font-weight: bold; text-decoration: none;">, </del>164509 (2007)<nowiki>]</nowiki>, Journal of Chemical Physics '''127''' 049901 (2007)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>#[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> <ins style="font-weight: bold; text-decoration: none;">Journal of Chemical Physics '''</ins>126<ins style="font-weight: bold; text-decoration: none;">''' </ins>164509 (2007)<nowiki>]</nowiki>, Journal of Chemical Physics '''127''' 049901 (2007)]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>#[http://dx.doi.org/10.1063/1.2009733 Luis M. Sesé "Triplet correlations in the quantum hard-sphere fluid", <del style="font-weight: bold; text-decoration: none;">J. Chem. Phys. </del>'''123''' 104507 (2005)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>#[http://dx.doi.org/10.1063/1.2009733 Luis M. Sesé "Triplet correlations in the quantum hard-sphere fluid", <ins style="font-weight: bold; text-decoration: none;">Journal of Chemical Physics </ins>'''123''' 104507 (2005)]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>#[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", <del style="font-weight: bold; text-decoration: none;">J. J.Chem. Phys. </del>'''121''' 3702 (2004)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>#[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", <ins style="font-weight: bold; text-decoration: none;">Journal of Chemical Physics </ins> '''121''' 3702 (2004)]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>#[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", <del style="font-weight: bold; text-decoration: none;">J. Chem. Phys. </del>'''121''' 10076 (2004)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>#[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", <ins style="font-weight: bold; text-decoration: none;">Journal of Chemical Physics </ins>'''121''' 10076 (2004)]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>#[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", <del style="font-weight: bold; text-decoration: none;">J. Chem. Phys. </del>'''119''' 10256 (2003)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>#[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", <ins style="font-weight: bold; text-decoration: none;"> Journal of Chemical Physics </ins>'''119''' 10256 (2003)]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>#[http://dx.doi.org/10.1080/0026897031000094470 L. M. Sesé "The compressibility theorem for quantum simple fluids at equilibrium", <del style="font-weight: bold; text-decoration: none;">Molec. Phys </del>'''101''' 1455 (2003)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>#[http://dx.doi.org/10.1080/0026897031000094470 L. M. Sesé "The compressibility theorem for quantum simple fluids at equilibrium", <ins style="font-weight: bold; text-decoration: none;">Molecular Physics </ins>'''101''' 1455 (2003)]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>#[http://dx.doi.org/10.1063/1.1468223 L. M. Sesé "Properties of the path-integral quantum hard-sphere fluid in k space", <del style="font-weight: bold; text-decoration: none;">J. Chem. Phys. </del>'''116''' 8492 (2002)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>#[http://dx.doi.org/10.1063/1.1468223 L. M. Sesé "Properties of the path-integral quantum hard-sphere fluid in k space", <ins style="font-weight: bold; text-decoration: none;"> Journal of Chemical Physics </ins>'''116''' 8492 (2002)]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>#[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", <del style="font-weight: bold; text-decoration: none;">J. Chem. Phys. </del>'''115''' 6557 (2001)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>#[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", <ins style="font-weight: bold; text-decoration: none;">Journal of Chemical Physics </ins>'''115''' 6557 (2001)]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>#[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", <del style="font-weight: bold; text-decoration: none;">J. Chem. Phys. </del>'''114''' 1732 (2001)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>#[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", <ins style="font-weight: bold; text-decoration: none;">Journal of Chemical Physics </ins>'''114''' 1732 (2001)]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>#[http://dx.doi.org/10.1063/ L. M. Sesé "Thermodynamic ans structural properties of the path-integral quantum hard-sphere fluid", <del style="font-weight: bold; text-decoration: none;">J. Chem. Phys. </del>'''108''' 9086 (1998)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>#[http://dx.doi.org/10.1063/ L. M. Sesé "Thermodynamic ans structural properties of the path-integral quantum hard-sphere fluid", <ins style="font-weight: bold; text-decoration: none;">Journal of Chemical Physics </ins>'''108''' 9086 (1998)]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>#[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", <del style="font-weight: bold; text-decoration: none;">J. Chem. Phys. </del>'''106''' 1134 (1997)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>#[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", <ins style="font-weight: bold; text-decoration: none;">Journal of Chemical Physics </ins>'''106''' 1134 (1997)]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>#[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", <del style="font-weight: bold; text-decoration: none;"> J.Chem. Phys. </del>'''102''' 3776 (1995)]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>#[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", <ins style="font-weight: bold; text-decoration: none;">Journal of Chemical Physics </ins>'''102''' 3776 (1995)]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[category: hard sphere]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[category: hard sphere]]</div></td></tr>
</table>Carl McBridehttp://www.sklogwiki.org/SklogWiki/index.php?title=Quantum_hard_spheres&diff=4591&oldid=prevCarl McBride at 16:51, 16 October 20072007-10-16T16:51:16Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 17:51, 16 October 2007</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>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 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 [Wood & Jacobson, J. CHEM. PHYS. 27, 1207 (1957); Alder & Wainwright, J. CHEM. PHYS. 27, 1208 (1957)], and confirmed later with experiments on [[colloids |colloidal particles]] (Pusey & van Megen, NATURE 320, 340 (1986)). From the 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 <del style="font-weight: bold; text-decoration: none;">= </del>diameter (+) (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.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>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 <ins style="font-weight: bold; text-decoration: none;">[[Intermolecular pair potential |</ins>pairwise interactions<ins style="font-weight: bold; text-decoration: none;">]] </ins>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 <ins style="font-weight: bold; text-decoration: none;">[[Computer simulation techniques |</ins>computer simulation<ins style="font-weight: bold; text-decoration: none;">]] </ins>[Wood & Jacobson, J. CHEM. PHYS. 27, 1207 (1957); Alder & Wainwright, J. CHEM. PHYS. 27, 1208 (1957)], and confirmed later with experiments on [[colloids |colloidal particles]] (Pusey & van Megen, NATURE 320, 340 (1986)). From the <ins style="font-weight: bold; text-decoration: none;">[[Classical thermodynamics |</ins>thermodynamic<ins style="font-weight: bold; text-decoration: none;">]] </ins>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 <ins style="font-weight: bold; text-decoration: none;">equal to the </ins>diameter (<ins style="font-weight: bold; text-decoration: none;"><math>d^</ins>+<ins style="font-weight: bold; text-decoration: none;"></math></ins>) (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.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> Nevertheless, the switching on of the quantum conditions upon this system (i.e. <del style="font-weight: bold; text-decoration: none;">nonzero </del>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 <del style="font-weight: bold; text-decoration: none;">ultrahard </del>materials and colloids. In this endeavour Feynman’s [[Path integral formulation |path-integrals]] combined with <del style="font-weight: bold; text-decoration: none;">[[Computer simulation techniques |</del>computer simulations<del style="font-weight: bold; text-decoration: none;">]] </del>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.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> Nevertheless, the switching on of the quantum conditions upon this system (i.e. <ins style="font-weight: bold; text-decoration: none;">non-zero </ins>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 <ins style="font-weight: bold; text-decoration: none;">ultra-hard </ins>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.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>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 <del style="font-weight: bold; text-decoration: none;">delocalization </del>brings about three different pair radial correlation functions in the quantum case, each of them possessing a definite physical meaning. <del style="font-weight: bold; text-decoration: none;">A great </del>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 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.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>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 <ins style="font-weight: bold; text-decoration: none;">de-localisation </ins>brings about three different pair radial correlation functions in the quantum case, each of them possessing a definite physical meaning. <ins style="font-weight: bold; text-decoration: none;">Great </ins>emphasis has been placed on the study of the different structural functions, in both the <ins style="font-weight: bold; text-decoration: none;">[[</ins>r-correlation<ins style="font-weight: bold; text-decoration: none;">]] </ins>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 <ins style="font-weight: bold; text-decoration: none;">[[Equations of state |</ins>equation of state<ins style="font-weight: bold; text-decoration: none;">]]</ins>. 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.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td></tr>
</table>Carl McBridehttp://www.sklogwiki.org/SklogWiki/index.php?title=Quantum_hard_spheres&diff=4588&oldid=prevNice and Tidy at 16:26, 16 October 20072007-10-16T16:26:26Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 17:26, 16 October 2007</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">UNED GROUP PAPERS ON QUANTUM HARD SPHERES</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">#M. Fierz, Phys. Rev. 106, 412 (1957)</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">#E. H. Lieb, J. Math. Phys. 8, 43 (1967)</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">#W. G. Gibson, Molec. Phys. 30, 13 (1975)</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">#J. A. Barker, J. Chem. Phys. 70, 2914 (1979)</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">#G. Jacucci and E. Omerti, J. Chem. Phys. 79, 3051 (1983).</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">#K. J. Runge and G. V. Chester, Phys. Rev. B 38, 135 (1988)</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">#J. Cao and B. J. Berne, J. Chem. Phys. 97, 2382 (1992)</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">#P. Grüter, D. Ceperley and F. Lalöe, Phys. Rev. Lett. 79, 3549 (1997)</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#[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)]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#[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)]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#[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)]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#[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)]</div></td></tr>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#[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", J. Chem. Phys. '''106''' 1134 (1997)]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#[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", J. Chem. Phys. '''106''' 1134 (1997)]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#[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", J.Chem. Phys. '''102''' 3776 (1995)]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>#[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", J.Chem. Phys. '''102''' 3776 (1995)]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">SOME BASIC QUANTUM HARD-SPHERE REFERENCES</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#M. Fierz, Phys. Rev. 106, 412 (1957)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#E. H. Lieb, J. Math. Phys. 8, 43 (1967)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#W. G. Gibson, Molec. Phys. 30, 13 (1975)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#J. A. Barker, J. Chem. Phys. 70, 2914 (1979)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#G. Jacucci and E. Omerti, J. Chem. Phys. 79, 3051 (1983).</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#K. J. Runge and G. V. Chester, Phys. Rev. B 38, 135 (1988)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#J. Cao and B. J. Berne, J. Chem. Phys. 97, 2382 (1992)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#P. Grüter, D. Ceperley and F. Lalöe, Phys. Rev. Lett. 79, 3549 (1997)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">GENERAL REFERENCES ON PATH-INTEGRALS & APPLICATIONS</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">a) Reviews</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#R. P. Feynman and A. R. Hibbs, Path-integrals and Quantum Mechanics (McGraw-Hill, New York, 1965)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#R. P. Feynman, Statistical Mechanics (Benjamin, Reading, Mass., 1972)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#D. Chandler and P. G. Wolynes, J. Chem. Phys. 74, 4078 (1981)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#B. J. Berne and D. Thirumalai, Annu. Rev. Phys. Chem. 37, 401 (1986)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#D. M. Ceperley, Rev. Mod. Phys. 67, 229 (1995)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">b) Applications (Phase transitions, Quantum Dynamics, Centroids)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#J. R. Melrose and K. Singer, Molec. Phys. 66, 1203 (1989)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#J. Cao and G. A. Voth, J. Chem. Phys. 100, 5093 (1994); 104, 273 (1996)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#R. Ramírez and T. López-Ciudad, J. Chem. Phys. 111, 3339 (1999)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#C. Chakravarty and R. M. Lynden-Bell, J. Chem. Phys. 113, 9239 (2000)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
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</table>Nice and Tidyhttp://www.sklogwiki.org/SklogWiki/index.php?title=Quantum_hard_spheres&diff=4586&oldid=prevNice and Tidy at 16:21, 16 October 20072007-10-16T16:21:36Z<p></p>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The great usefulness of the hard<del style="font-weight: bold; text-decoration: none;">-</del>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 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 fluid-solid transition, which was first predicted with computer simulation [Wood & Jacobson, J. CHEM. PHYS. 27, 1207 (1957); Alder & Wainwright, J. CHEM. PHYS. 27, 1208 (1957)], and confirmed later with experiments on colloidal particles (Pusey & van Megen, NATURE 320, 340 (1986). From the 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 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 = diameter (+) (they are like hard billiards), which reflects in the fact that the main peak of the pair radial correlation function is located just at that “contact” point.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The great usefulness of the <ins style="font-weight: bold; text-decoration: none;">[[</ins>hard sphere model<ins style="font-weight: bold; text-decoration: none;">]] </ins>for representing particles in classical <ins style="font-weight: bold; text-decoration: none;">[[</ins>statistical mechanics<ins style="font-weight: bold; text-decoration: none;">]] </ins>is very well known and its study has provided guidance in the understanding of classical fluids and solids. This model assumes that 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 <ins style="font-weight: bold; text-decoration: none;">[[Solid-liquid phase transitions |</ins>fluid-solid transition<ins style="font-weight: bold; text-decoration: none;">]]</ins>, which was first predicted with computer simulation [Wood & Jacobson, J. CHEM. PHYS. 27, 1207 (1957); Alder & Wainwright, J. CHEM. PHYS. 27, 1208 (1957)], and confirmed later with experiments on <ins style="font-weight: bold; text-decoration: none;">[[colloids |</ins>colloidal particles<ins style="font-weight: bold; text-decoration: none;">]] </ins>(Pusey & van Megen, NATURE 320, 340 (1986<ins style="font-weight: bold; text-decoration: none;">)</ins>). From the 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 <ins style="font-weight: bold; text-decoration: none;">[[De Broglie thermal wavelength |</ins>thermal de Broglie wavelength<ins style="font-weight: bold; text-decoration: none;">]] </ins>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 = diameter (+) (they are like hard billiards), which reflects in the fact that the main peak of the pair <ins style="font-weight: bold; text-decoration: none;">[[Radial distribution function |</ins>radial correlation function<ins style="font-weight: bold; text-decoration: none;">]] </ins>is located just at that “contact” point.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> Nevertheless, the switching on of the quantum conditions upon this system (i.e. nonzero 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’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 ultrahard materials and colloids. In this endeavour Feynman’s 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.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> Nevertheless, the switching on of the quantum conditions upon this system (i.e. nonzero 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 <ins style="font-weight: bold; text-decoration: none;">[[</ins>temperature<ins style="font-weight: bold; text-decoration: none;">]]</ins>, the mass of the particles, and <ins style="font-weight: bold; text-decoration: none;">[[Planck constant |</ins>Planck’s constant<ins style="font-weight: bold; text-decoration: none;">]] </ins>(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 <ins style="font-weight: bold; text-decoration: none;">[[</ins>entropy<ins style="font-weight: bold; text-decoration: none;">]]</ins>. Furthermore, quantum hard spheres seem appropriate to understand the low temperature properties of ultrahard materials and colloids. In this endeavour Feynman’s <ins style="font-weight: bold; text-decoration: none;">[[Path integral formulation |</ins>path-integrals<ins style="font-weight: bold; text-decoration: none;">]] </ins>combined with <ins style="font-weight: bold; text-decoration: none;">[[Computer simulation techniques |</ins>computer simulations<ins style="font-weight: bold; text-decoration: none;">]] </ins>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.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"> The UNED group is engaged in the study of this system and over the years has published a number of research papers dealing with the properties of the quantum hard-sphere system under conditions covering from the fluid to the solid states. The titles of these papers in the list given in "References" are a good summary of the findings and the results obtained. </del>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 delocalization brings about three different pair radial correlation functions in the quantum case, each of them possessing a definite physical meaning. A 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 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 classical frameworks in dealing with quantum fluids.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>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 delocalization brings about three different pair radial correlation functions in the quantum case, each of them possessing a definite physical meaning. A 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 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 <ins style="font-weight: bold; text-decoration: none;">[[Ornstein-Zernike relation |</ins>Ornstein-Zernike<ins style="font-weight: bold; text-decoration: none;">]] </ins>classical frameworks in dealing with quantum fluids.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td></tr>
</table>Nice and Tidyhttp://www.sklogwiki.org/SklogWiki/index.php?title=Quantum_hard_spheres&diff=4584&oldid=prevNice and Tidy at 16:11, 16 October 20072007-10-16T16:11:02Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 17:11, 16 October 2007</td>
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<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">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 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 fluid-solid transition, which was first predicted with computer simulation [Wood & Jacobson, J. CHEM. PHYS. 27, 1207 (1957); Alder & Wainwright, J. CHEM. PHYS. 27, 1208 (1957)], and confirmed later with experiments on colloidal particles (Pusey & van Megen, NATURE 320, 340 (1986). From the 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 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 = diameter (+) (they are like hard billiards), which reflects in the fact that the main peak of the pair radial correlation function is located just at that “contact” point.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> Nevertheless, the switching on of the quantum conditions upon this system (i.e. nonzero 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’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 ultrahard materials and colloids. In this endeavour Feynman’s 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.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> The UNED group is engaged in the study of this system and over the years has published a number of research papers dealing with the properties of the quantum hard-sphere system under conditions covering from the fluid to the solid states. The titles of these papers in the list given in "References" are a good summary of the findings and the results obtained. 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 delocalization brings about three different pair radial correlation functions in the quantum case, each of them possessing a definite physical meaning. A 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 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 classical frameworks in dealing with quantum fluids.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>UNED GROUP PAPERS ON QUANTUM HARD SPHERES</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>UNED GROUP PAPERS ON QUANTUM HARD SPHERES</div></td></tr>
</table>Nice and Tidy