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where <math> \Phi_{12}\left(r \right) </math> is the [[intermolecular pair potential]] between two spheres at a distance <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math>, and <math> \sigma </math> is the diameter of the sphere. | where <math> \Phi_{12}\left(r \right) </math> is the [[intermolecular pair potential]] between two spheres at a distance <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math>, and <math> \sigma </math> is the diameter of the sphere. | ||
The hard sphere model can be considered to be a special case of the [[hard ellipsoid model]], where each of the semi-axes has the same length, <math>a=b=c</math>. | The hard sphere model can be considered to be a special case of the [[hard ellipsoid model]], where each of the semi-axes has the same length, <math>a=b=c</math>. | ||
==First simulations of hard spheres | ==First simulations of hard spheres== | ||
The hard sphere model, along with its two-dimensional manifestation [[hard disks]], was one of the first ever systems studied using [[computer simulation techniques]] with a view | The hard sphere model, along with its two-dimensional manifestation [[hard disks]], was one of the first ever systems studied using [[computer simulation techniques]] with a view | ||
to understanding the thermodynamics of the liquid and solid phases and their corresponding [[Phase transitions | phase transition]] | to understanding the thermodynamics of the liquid and solid phases and their corresponding [[Phase transitions | phase transition]] | ||
<ref>[http://dx.doi.org/10.1063/1.1740207 Marshall N. Rosenbluth and Arianna W. Rosenbluth "Further Results on Monte Carlo Equations of State", Journal of Chemical Physics '''22''' pp. 881-884 (1954)]</ref> | <ref>[http://dx.doi.org/10.1063/1.1740207 Marshall N. Rosenbluth and Arianna W. Rosenbluth "Further Results on Monte Carlo Equations of State", Journal of Chemical Physics '''22''' pp. 881-884 (1954)]</ref> | ||
<ref>[http://dx.doi.org/10.1063/1.1743956 W. W. Wood and J. D. Jacobson "Preliminary Results from a Recalculation of the Monte Carlo Equation of State of Hard Spheres", Journal of Chemical Physics '''27''' pp. 1207-1208 (1957)]</ref> | <ref>[http://dx.doi.org/10.1063/1.1743956 W. W. Wood and J. D. Jacobson "Preliminary Results from a Recalculation of the Monte Carlo Equation of State of Hard Spheres", Journal of Chemical Physics '''27''' pp. 1207-1208 (1957)]</ref> | ||
<ref>[http://dx.doi.org/10.1063/1.1743957 B. J. Alder and T. E. Wainwright "Phase Transition for a Hard Sphere System", Journal of Chemical Physics '''27''' pp. 1208-1209 (1957)]</ref> | <ref>[http://dx.doi.org/10.1063/1.1743957 B. J. Alder and T. E. Wainwright "Phase Transition for a Hard Sphere System", Journal of Chemical Physics '''27''' pp. 1208-1209 (1957)]</ref> | ||
==Liquid phase radial distribution function== | ==Liquid phase radial distribution function== | ||
The following are a series of plots of the hard sphere [[radial distribution function]] <ref>The [[total correlation function]] data was produced using the [https://old.vscht.cz/fch/software/hsmd/hspline-8-2004.zip computer code] written by [https://web.vscht.cz/~kolafaj/ Jiří Kolafa]</ref> shown for different values of the number density <math>\rho</math>. The horizontal axis is in units of <math>\sigma</math> where <math>\sigma</math> is set to be 1. Click on image of interest to see a larger view. | The following are a series of plots of the hard sphere [[radial distribution function]] <ref>The [[total correlation function]] data was produced using the [https://old.vscht.cz/fch/software/hsmd/hspline-8-2004.zip computer code] written by [https://web.vscht.cz/~kolafaj/ Jiří Kolafa]</ref> shown for different values of the number density <math>\rho</math>. The horizontal axis is in units of <math>\sigma</math> where <math>\sigma</math> is set to be 1. Click on image of interest to see a larger view. |