This Exercise Set has been submitted for peer review, but it has not yet been accepted for publication in the PICUP collection.

+
Microcanonical Ensemble

Developed by Shafat Mubin

In a microcanonical ensemble, a fixed amount of energy is distributed among different energy levels of a system and the number of possible configurations are counted. The occupation probability of each energy level is determined from the total number of different configurations, and subsequently used to generate the energy and velocity distributions of the system. The same energy and velocity distributions apply to any system consisting of a large number of interacting particles that evolve over a sufficiently long time interval through Newtonian mechanics. If the trajectories of these particles are calculated using only Newton's laws and the energies and velocity of every particle are listed at every timestep of the trajectory, the energy and velocity distributions can be obtained. In this exercise, the energy and velocity distributions of a microcanonical generated from Newtonian mechanics are compared with those calculated by statistical mechanics to demonstrate consistency of results.
Subject Area Thermal & Statistical Physics Beyond the First Year and Advanced Python Students will be able to: - generate the trajectory of a single particle using Newton’s laws in a 1D harmonic potential - plot histograms of velocity, kinetic energy and potential energy of the above particle - repeat analysis for a system consisting of multiple interacting particles - infer the expected shapes of the velocity and energy distributions of large systems, and compare with theoretical results
$\bf{Exercise \ 1: Single \ Particle \ Harmonic \ Oscillator}$ Numerically calculate the trajectory of a single particle located in a harmonic potential well. Assign the following values in arbitrary units: $\quad \quad \quad$ Mass of particle: m = 1.0 $\quad \quad \quad$ Timestep: $\Delta t$ = 0.1 $\quad \quad \quad$ Spring constant: k = 1.0 Also, assign arbitrary initial position and velocity. Run simulation for at least 100,000 iterations and generate histograms of velocity, potential energy and kinetic energy. --- $\bf{Exercise \ 2: System \ of \ Multiple \ Interacting \ Particles}$ Repeat above analysis for a 1D chain of particles, in which each particle only interacts with its connected neighbors via a Lennard-Jones (LJ) pair potential energy function: $\quad \quad U_{LJ} (r) = 4\varepsilon \left [ \left(\frac{\sigma}{r} \right)^{12} - \left(\frac{\sigma}{r} \right)^6 \right]$ where $r$ is the separation between the pair of particles, $\varepsilon$ is the minimum value of the energy profile between the pair and $\sigma$ is the distance at which energy is zero. Use the following parameters (in arbitrary units): $\quad \quad \quad$ m = 2.0 $\quad \quad \quad$ $\Delta t$ = 0.001 $\quad \quad \quad$ $\varepsilon$ = 2.4 $\quad \quad \quad$ $\sigma$ = 8.9 Assign different initial velocities and positions to each particle (use a random number generator if convenient) and observe the evolution of the chain. Note: If initial velocities are too large or if initial separations between particles are too small, system may disintegrate owing to enormous forces. Analyse the shape of the Lennard-Jones potential to understand the directions and relative magnitudes of forces at different separations. Create a system of two particles to validate your setup. Once your code is functioning satisfactorily, run simulations with more particles. For your final results, run simulation for at least 100,000 iterations with at least 100 particles, and generate histograms of velocity, potential energy and kinetic energy as before. --- $\bf{Exercise \ 3: Analysis}$ Repeat Exercise 2 with a different potential function (e.g. harmonic, double well) and compare the velocity and energy histograms with those of Exercise 2. - Is there a common pattern in the histograms? - How do the histograms compare with theoretical statistical mechanics predictions of the microcanonical ensemble? - How would the histograms change if the simulations were performed: (i) in 2D or 3D? (ii) with an even larger number of particles?