by Nikolai Shokhirev
Up: Physics
 Mechanics
 Hardsphere dynamics
 MD Simulation
 Kinetics

This is the simplest molecular dynamics problem. It is of great interest because of its fundamental nature [17]. The problem can be reduced to the elementary act of the collision of two particles. I was not satisfied with the formulas I found [17]. Some of them are limited, the others are overcomplicated. Therefore I decided to derive the formulas myself.
The dynamics of hard spheres can not be easily treated in terms of the Lagrangian or Hamiltonian mechanics because of discontinuities in interparticle interaction:
(1) 
See also the graph below.
Hardsphere potential
The dynamics of particles between collisions is trivial:
(2) 
A collision takes place when the distance between two particles is equal to d_{21}:
(3) 
If
(4) 
then the Eq. (3) has the following solution for the collision time:
(5) 
The collision between hard spheres is considered to be instantaneous and elastic.
Approaching and collision.
The component of the relative velocity, which is parallel to , instantaneously changes its sign. The perpendicular component remains unchanged (elasticity).
Collision and separation.
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) 
The solution vector p should be along d_{21}.
(11) 
Here n_{21} is the unit vector along d_{21}.
Up: Physics
 Mechanics
 Hardsphere dynamics
 MD Simulation
 Kinetics

Please email me at nikolai@shokhirev.com 
Nikolai Shokhirev, 20012016