### The Einstein solid

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.

We consider the Einstein solid in the following contexts:

- Two Einstein solids that can exchange energy. The program counts the number of ways that the energy can be distributed between the two systems and helps to develop an intuitive understanding of temperature.
- Two Einstein solids that can exchange particles. The program counts the number of ways that the particles can be distributed between the two systems and and helps to develop an intuitive understanding of the chemical potential.

model are available for one dimension, square lattice (J > 0), square lattice (J < 0), and hexagonal lattice (J < 0). Also available are numerical solutions of mean-field theory, the direct simulation of the density of states, the direct simulation of the partition function, and a simulation of hysteresis.

References

Each year many hundreds of papers are published that apply the Ising model to problems in such diverse fields as neural networks, protein folding, biological membranes, and social behavior.

1. A biographical note about Ising's life is at <www.bradley.edu/las/phy/personnel/ising.html>. Also see S. Kobe, "Ernst Ising 1900–1998," Braz. J. Phys. 30 (4), 649–654 (2000). Available from various Web sites.
2. See <en.wikipedia.org/wiki/Lars_Onsager> for a summary of Onsager's life and contributions.
3. Stephen G. Brush, "History of the Lenz-Ising model," Rev. Mod. Phys. 39, 883–893 (1967).
4. Many textbooks on statistical mechanics discuss the Ising model. See for example, M. E. J. Newman and G. T. Barkema, Monte Carlo Methods in Statistical Physics, Oxford University Press (1999).
5. H. Gould, J. Tobochnik, and W. Christian, An Introduction to Computer Simulation Methods: Applications to Physical Systems, Addison-Wesley (2006), Chapter 15.

Updated 5 March 2009.