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 simulations, and by the experimental study of certain
colloid
A colloid is a mixture in which one substance consisting of microscopically dispersed insoluble particles is suspended throughout another substance. Some definitions specify that the particles must be dispersed in a liquid, while others extend ...
al 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
are particles with the following pairwise interaction potential:
:
where
and
are the positions of the two particles.
Hard-spheres gas
The first three
virial coefficients for hard spheres can be determined analytically
:{,
,
, , =, ,
, -
,
, -
,
Higher-order ones can be determined numerically using
Monte Carlo integration. We list
:{,
,
, , =, ,
, -
,
, , =, ,
, -
,
, , =, ,
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
and melting
. The pressure diverges at
random close packing
for the metastable liquid branch and at
close packing for the stable solid branch.
Hard-spheres liquid
The
static structure factor of the hard-spheres liquid can be calculated using the
Percus–Yevick approximation In statistical mechanics the Percus–Yevick approximation is a closure relation to solve the Ornstein–Zernike equation. It is also referred to as the Percus–Yevick equation. It is commonly used in fluid theory to obtain e.g. expressions for t ...
.
See also
*
Classical fluid
Literature
*J. P. Hansen and I. R. McDonald ''Theory of Simple Liquids'' Academic Press, London (1986)
Hard sphere modelpage on SklogWiki.
References
{{Reflist
Statistical mechanics
Conceptual models