Approach to equilibrium: Molecular dynamics

Introduction

The program implements a molecular dynamics simulation of a Lennard-Jones fluid in two dimensions. The particles are initially in the middle third of the simulation cell. The number of particles in each of the three cells is plotted.

See: [Introduction to STP] and [Approach to Equilibrium]

Algorithm

The algorithm for simulating the evolution of the model can be summarized by the following steps:

  1. The particles are placed at random in the center of the box with the constraint that no two particles can be closer than the length σ = 1 in the Lennard-Jones potential. This constraint prevents the initial force between any two particles from being too big, which would lead to the breakdown of the numerical method used to solve the differential equations. The velocity of each particle is assigned at random and then the velocity of the center of mass is set to zero.
  2. The total force on each particle is computed and the positions and velocities are updated according to the Verlet algorithm.
  3. The time is increased by the time step Δt.

Problems

  1. Choose N = 27, n1 = 0, n2 = N, and n3 = 0. What is the qualitative behavior of n1, n2, and n3, the number of particles in each third of the box, as a function of the time t? Does the system appear to show a direction of time? Choose various values of N that are multiples of three up to N = 270. Is the direction of time better defined for larger N?
  2. Suppose that we made a video of the motion of the particles considered in Problem 1. Would you be able to tell if the video were played forward or backward for the various values of N? Would you be willing to make an even bet about the direction of time? Does your conclusion about the direction of time become more certain as N increases?
  3. How long does it take before the system appears to be in equilibrium? What is your criterion for equilibrium?
  4. From watching the motion of the particles, describe the nature of the boundary conditions that are used in the simulation.
  5. *Run the system with N = 270 until t = 5. Then press the Reverse button, which reverses all the velocities, and continue the simulation. Does the system return to its initial state? Repeat for other times. Is there a time at which the system does not return to its initial state? Repeat for N = 27. What can you conclude about the N dependence of reversibility?

Updated 20 March 2020.