Einstein solid in equilibrium with a heat bath

Introduction

Consider a simple system, commonly known as an Einstein or harmonic solid. The energy of each particle in an Einstein solid is restricted to the positive integers. That is, each particle may have energy 0, 1, 2, … The particles do not interact. These particles are equivalent to the quanta of the harmonic oscillator, which have energy εn = (n + 1/2)hν. If we measure the energies from the lowest energy state, hν/2, and choose units such that hν = 1, we have εn = n.

    In the following we will explore the properties of a Einstein solid in equilibrium with a heat bath at temperature T. We will also compare our results with analytical calculations of the thermodynamic properties of the Einstein solid.

Algorithm

The program implements the Metropolis algorithm by choosing a particle at random and randomly increasing or decreasing its energy by one unit. If the energy is decreased, the change is accepted. If the energy is increased, the program generates a number r at random in the unit interval and accepts the change if r ≤ e-β1;, where β = 1/T. (As usual, we choose units such that Boltzmann's constant k = 1.) If a trial change is not accepted, the existing microstate is counted in all averages.

Problems

  1. Start from the formal expression for the partition function and find an analytical expression for the mean energy and specific heat of the Einstein solid.
  2. Use the default values for the number of particle N in the solid and the temperature T. Run for a sufficient number of Monte Carlo steps per particle so that the mean energy becomes well defined. When you are satisfied that it is, press the Accept E and CV button. Then consider a range of values of T and determine the T-dependence of the mean energy and the specific heat.
  3. What is the limiting value of the mean energy and the specific heat for high and low temperatures?

References

Java Classes

Updated 19 March 2007.