Hard spheres

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 colloidal model systems. 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 colloidal model systems. Hard spheres of diameter σ {displaystyle sigma } are particles with the following pairwise interaction potential: where r 1 {displaystyle mathbf {r} _{1}} and r 2 {displaystyle mathbf {r} _{2}} are the positions of the two particles. 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 page Hard sphere: virial coefficients. The hard sphere system exhibits a fluid-solid phase transition between the volume fractions of freezing η f ≈ 0.494 {displaystyle eta _{mathrm {f} }approx 0.494} and melting η m ≈ 0.545 {displaystyle eta _{mathrm {m} }approx 0.545} . The pressure diverges at random close packing η r c p ≈ 0.644 {displaystyle eta _{mathrm {rcp} }approx 0.644} for the metastable liquid branch and at close packing η c p = 2 π / 6 ≈ 0.74048 {displaystyle eta _{mathrm {cp} }={sqrt {2}}pi /6approx 0.74048} for the stable solid branch. The static structure factor of the hard-spheres liquid can be calculated using the Percus–Yevick approximation.