Hard-sphere dynamics

by Nikolai V. Shokhirev

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Introduction

The dynamics of hard spheres can not be easily treated in terms of the Lagrangian or Hamiltonian mechanics because of discontinuities in inter-particle interaction:

(1)

See also the graph below.

Hard-sphere potential

Dynamics

The dynamics of particles between collisions is trivial:

(2)

A collision takes place when the distance between two particles is equal to d₂₁:

(3)

If

(4)

then the Eq. (3) has the following solution for the collision time:

(5)

Collision

The collision between hard spheres is considered to be instantaneous and elastic.

The component of the relative velocity, which is parallel to  ,  instantaneously changes its sign. The perpendicular component remains unchanged (elasticity). 

This process is governed by the momentum conservation law

(6)

and the energy conservation law

(7)

The above equations can be rewritten as

(8)

and

(9)

Rearranging the terms in Eq. (9) we get the equation for p

(10)

(11)

Here n₂₁ is the unit vector along d₂₁.
 

References

1. Molecular dynamics of pure hard spheres http://www.vscht.cz/fch/software/hsmd/
2. Molecular Dynamics Simulation of Hard Spheres http://www.cs.princeton.edu/introcs/assignments/collisions.html
3. Molecular Dynamics Simulation of A Simple Hard Sphere  System (pdf)
4. Molecular dynamics simulations of crystallization of hard spheres http://www.ifpan.edu.pl/~cieplak/papers/volkov.pdf
5. MD simulations http://condor.wesleyan.edu/courses/2006s/chem388/01/mdcourse/LectureSlides/Lecture%25204-1.ppt
6.  

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