Energy Distribution in 3D Including Rotors and Atoms

Introduction

We've already animated the energy and motion of two dimensional (2D) rotors (i.e. rotors with discs at the ends) and 2D atoms and shown that the energy distribution of the 2D rotors agrees with that of just 3D atoms alone. In the present animation, we allow the rotor to have 2 more degrees of freedom- linear motion in the z direction and rotation about both the two axes (call them yaw and roll) perpendicular to the line between the end spheres. The rotation about this line is not important since

a. For central forces, collisions do not induce rotation about the line.

b. The moment of inertia of the rotor about the line is much smaller than the yaw and roll moments of inertia.

For the 3D rotors, this results in a total of 5 degrees of freedom and an energy distribution that is quite different from that of simple atoms.

Energy Distributions

We include large numbers of both free atoms and rotors in the same box and fit the energy distributions of each to the following curves:

Free Atoms:

$\frac{N (E)}{d E} = N_{0} \sqrt{E} \exp (- b E)$

(1.1)

3D Rotors:

$\frac{N (E)}{d E} = N_{0} E^{\frac{3}{2}} \exp (- b E)$

(1.2)

The two relations above stem from the densities of speeds:

Free Atoms:

$\frac{d N (v)}{d v} = N_{0} v^{2} \exp (- b v^{2})$

(1.3)

3D Rotors:

$\frac{d N (v)}{d v} = N_{0} v^{4} \exp (- b v^{2})$

(1.4)

since

$d E = m v d v$

(1.5)

where m is the mass.