Hard-sphere dynamics

by Nikolai Shokhirev

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Introduction

This is the simplest molecular dynamics problem. It is of great interest because of its fundamental nature [1-7]. The problem can be reduced to the elementary act of the collision of two particles. I was not satisfied with the formulas I found [1-7]. Some of them are limited, the others are overcomplicated. Therefore I decided to derive the formulas myself.

Problem statement

 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.


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)

 

Solution

The solution vector p should be along d21.

     (11)

Here n21 is the unit vector along d21.
 

References

  1. What are the general solutions to a hard sphere collision?
  2. Jared F. Bennatt Hard Sphere Collisions
  3. Fabrizio Ferri, Elastic collisions of balls, January 9, 2013
  4. Molecular Dynamics Simulation of Hard Spheres Algorithms (COS 226 Programming Assignment)
  5. Hard-Sphere Molecular Dynamics
  6. MD simulation of Hard Spheres
  7. Molecular Dynamics Simulation (Hard Spheres / Hard disks, Continuous Potentials, Thermostats / Nosť-Hoover method)

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Up: Physics

- Mechanics
- Hard-sphere dynamics
- MD Simulation
- Kinetics
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