## Introduction

This program uses molecular dynamics to simulate the motion of particles in two dimensions interacting
via the Lennard-Jones potential. The output includes the velocity distribution and the kinetic temperature,
which is proportional to the mean kinetic energy per particle. The program also uses the demon algorithm to make
an independent measure of the temperature.

See: [**Introduction to STP**] and [**Introduction to Ideal Thermometer**]

## Algorithm

According to the equipartition
theorem, the mean kinetic energy of a particle per degree of freedom is
kT/2, where
k is Boltzmann's constant and T is the temperature. We can generalize
this relation to define the temperature at time t by

kT(t) = (2/d) K(t)/N, (1)

where K is the kinetic energy of the system at time t and d is the spatial dimension. In the following we will
consider d = 2.

The temperature that is usually measured in a molecular dynamics simulation is
the time average of T(t) over many configurations of the particles. We will refer to this temperature as the
kinetic temperature. For
two dimensions we write the kinetic temperature as

(2)

(The bar denotes the time average.)

Equation (2) holds only if the momentum of the center of mass of the system is fixed. In a simulation,
we impose the constraint that the center of mass
momentum (in each of the d directions) be zero. Consequently, the
system has
dN - d independent velocity components rather than dN
components, so that the corrected expression for the kinetic temperature is

(3)

The presence of the factor (N - 2) rather than N in
is an example of a finite size
correction that becomes unimportant for large N.

We can use the demon algorithm to demonstrate that the demon gives an independent measure of the temperature. In
this case the demon chooses a particle at random and changes its velocity by a random amount. Because the finite
size effects are different for the two measurements, you will find that the results of the two measurements are
slightly different.

## Problems

- Run the program with the default parameters. After you have run for a sufficient time for equilibrium to be
established, click on
`Zero Averages`. How does the kinetic temperature compare to the mean demon
energy? Why is the latter proportional to the temperature?

Updated 20 March 2020.