# Hard bullet model

Hard spheres are a common particle model for fluids and solids in statistical mechanics . They are defined as non-penetrable spheres in space that cannot overlap and model the strong repulsion that atoms and spherical molecules experience at very small distances from one another. Hard spheres are examined using analytical methods, by simulating molecular dynamics and the experimental investigation of certain colloid model systems.

## Formal definition

Hard spheres with a diameter are particles with the following pairwise interaction potential: ${\ displaystyle \ sigma}$

${\ displaystyle V \ left (\ mathbf {r} _ {1}, \ mathbf {r} _ {2} \ right) = {\ begin {cases} 0 & \ left | \ mathbf {r} _ {1} - \ mathbf {r} _ {2} \ right | \ geq \ sigma \\\ infty & \ left | \ mathbf {r} _ {1} - \ mathbf {r} _ {2} \ right | <\ sigma \ end {cases}}}$

where and describe the positions of the two particles. ${\ displaystyle \ mathbf {r} _ {1}}$${\ displaystyle \ mathbf {r} _ {2}}$

## Hard balls model for a gas

The first three virial coefficients for hard spheres can be determined analytically:

 ${\ displaystyle {\ frac {B_ {2}} {v_ {0}}}}$ = ${\ displaystyle 4 {\ frac {} {}}}$ ${\ displaystyle {\ frac {B_ {3}} {{v_ {0}} ^ {2}}}}$ = ${\ displaystyle 10 {\ frac {} {}}}$ ${\ displaystyle {\ frac {B_ {4}} {{v_ {0}} ^ {3}}}}$ = ${\ displaystyle - {\ frac {712} {35}} + {\ frac {219 {\ sqrt {2}}} {35 \ pi}} + {\ frac {4131} {35 \ pi}} \ arccos { \ frac {1} {\ sqrt {3}}} \ approx 18 {,} 365}$

Higher order coefficients can be found numerically by Monte Carlo integration. The following are listed as examples:

 ${\ displaystyle {\ frac {B_ {5}} {{v_ {0}} ^ {4}}}}$ = ${\ displaystyle 28 {,} 24 \ pm 0 {,} 08}$ ${\ displaystyle {\ frac {B_ {6}} {{v_ {0}} ^ {5}}}}$ = ${\ displaystyle 39 {,} 5 \ pm 0 {,} 4}$ ${\ displaystyle {\ frac {B_ {7}} {{v_ {0}} ^ {6}}}}$ = ${\ displaystyle 56 {,} 5 \ pm 1 {,} 6}$

A table of virial coefficients for up to eight dimensions can be found in the SklogWiki.

The hard sphere system forms a liquid-solid phase transition between the packing densities for freezing and melting . The pressure randomly diverges for the tightest packing for the metastable liquid branch and for the tightest packing for the stable solid branch. ${\ displaystyle \ eta _ {\ mathrm {f}} \ approx 0 {,} 494}$${\ displaystyle \ eta _ {\ mathrm {m}} \ approx 0 {,} 545}$${\ displaystyle \ eta _ {\ mathrm {rcp}} \ approx 0 {,} 644}$${\ displaystyle \ eta _ {\ mathrm {cp}} = {\ sqrt {2}} \ pi / 6 \ approx 0 {,} 74048}$

## Hard spheres model for a liquid

The structure factor for a liquid consisting of hard spheres can be calculated using the Percus- Yevick approximation.

Phase diagram of a system of hard spheres (solid line - stable branch, dashed line - metastable branch): pressure as a function of the
packing density (crystallography)${\ displaystyle P}$ ${\ displaystyle \ eta}$

## Generalizations

Not only spheres can be equipped with a hard interaction potential, but also bodies of any geometry.

## literature

• JP Hansen, IR McDonald: Theory of Simple Liquids. 4th edition, Academic Press, London 2013, ISBN 978-0-12-387032-2 .