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Hard spheres are widely used as model particles in the statistical mechanical theory of fluids and solids. They are defined simply as impenetrable spheres that cannot overlap in space. They mimic the extremely strong ("infinitely elastic bouncing") repulsion that atoms and spherical molecules experience at very close distances. Hard spheres systems are studied by analytical means, by
molecular dynamics Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of t ...
simulations, and by the experimental study of certain colloidal model systems. The hard-sphere system provides a generic model that explains the quasiuniversal structure and dynamics of simple liquids.


Formal definition

Hard spheres of diameter \sigma are particles with the following pairwise interaction potential: :V(\mathbf_1,\mathbf_2)=\left\{ \begin{matrix}0 & \mbox{if}\quad , \mathbf{r}_1-\mathbf{r}_2, \geq \sigma \\ \infty & \mbox{if}\quad, \mathbf{r}_1-\mathbf{r}_2, < \sigma \end{matrix} \right. where \mathbf{r}_1 and \mathbf{r}_2 are the positions of the two particles.


Hard-spheres gas

The first three
virial coefficient Virial coefficients B_i appear as coefficients in the virial expansion of the pressure of a many-particle system in powers of the density, providing systematic corrections to the ideal gas law. They are characteristic of the interaction potenti ...
s for hard spheres can be determined analytically :{, , \frac{B_2}{v_0}, , =, , 4{\frac{}{ , - , \frac{B_3} , - , \frac{B_4}{35 \pi}+\frac{4131}{35 \pi} \arccos{\frac{1}{\sqrt{3}\approx 18.365 Higher-order ones can be determined numerically using
Monte Carlo integration In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algorithms usually evaluate the integrand a ...
. We list :{, , \frac{B_5}{{v_0}^4}, , =, , 28.24 \pm 0.08 , - , \frac{B_6}{{v_0}^5}, , =, , 39.5 \pm 0.4 , - , \frac{B_7}{{v_0}^6}, , =, , 56.5 \pm 1.6 A table of virial coefficients for up to eight dimensions can be found on the pag
Hard sphere: virial coefficients
{{cite journal , last1=Clisby , first1=Nathan , last2=McCoy , first2=Barry M. , title=Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions , journal=Journal of Statistical Physics , date=January 2006 , volume=122 , issue=1 , pages=15–57 , doi=10.1007/s10955-005-8080-0, arxiv=cond-mat/0503525 , bibcode=2006JSP...122...15C , s2cid=16278678 The hard sphere system exhibits a fluid-solid phase transition between the volume fractions of freezing \eta_\mathrm{f}\approx 0.494 and melting \eta_\mathrm{m}\approx 0.545. The pressure diverges at
random close pack Random close packing (RCP) of spheres is an empirical parameter used to characterize the maximum volume fraction of solid objects obtained when they are packed randomly. For example, when a solid container is filled with grain, shaking the containe ...
ing \eta_\mathrm{rcp}\approx 0.644 for the metastable liquid branch and at
close packing In geometry, close-packing of equal spheres is a dense arrangement of congruent spheres in an infinite, regular arrangement (or lattice). Carl Friedrich Gauss proved that the highest average density – that is, the greatest fraction of space occu ...
\eta_\mathrm{cp}=\sqrt{2}\pi/6 \approx 0.74048 for the stable solid branch.


Hard-spheres liquid

The
static structure factor In condensed matter physics and crystallography, the static structure factor (or structure factor for short) is a mathematical description of how a material scatters incident radiation. The structure factor is a critical tool in the interpretation ...
of the hard-spheres liquid can be calculated using the Percus–Yevick approximation.


See also

*
Classical fluid Classical fluidsR. Balescu, ''Equilibrium and Nonequilibrium Statistical Mechanics'', (John Wiley, 1975) are systems of particles which retain a definite volume, and are at sufficiently high temperatures (compared to their Fermi energy) that quantum ...


Literature

*J. P. Hansen and I. R. McDonald ''Theory of Simple Liquids'' Academic Press, London (1986)
Hard sphere model
page on SklogWiki.


References

{{Reflist Statistical mechanics Conceptual models