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
ε_{n} = (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 ε_{n} = n.

The advantage of the Einstein solid is that it is easy to calculate the number of microstates Ω(E,N) for a given number of particles N and energy E.

.

Consider two Einstein solids A and B that can exchange energy with one another, but are isolated from their surroundings. That is, the two systems are surrounded by insulating, rigid, and
impermeable outer walls and are separated from each other by a conducting,
rigid, and impermeable wall. The program counts the number
of ways that the energy can be distributed between the two systems. N_{A} represents the number of
oscillators in system A and N_{B} the number in system B.

The total number of microstates Ω(E_{A},E_{A}) accessible to the composite system with subsystems A and N with energy E_{A} and E_{B} (and fixed number of particles N_{A} and N_{B}) is

Ω(E_{A}, E_{B}) = Ω_{A}(E_{A})Ω_{B}(E_{B}).

The total energy E = E_{A} + E_{B} is fixed. Because the composite system is isolated, its accessible
microstates are equally probable. Hence, the probability P_{A}(E_{A}) that subsystem A has energy E_{A} is

P_{A}(E_{A}) = Ω_{A}(E_{A})Ω_{B}(E - E_{A})/Ω.

The output of the program is the mean energy of system A and the probability P(E_{A}) that system A
has energy E_{A}.

Problems

- Suppose that N
_{A}= 2, N_{B}= 2 and initially E_{A}= 5 and E_{B}= 1. What is the initial number of microstates for the composite system? The internal constraint is then removed so that the two subsystems can exchange energy. Determine the probability P(E_{A}) that system A has energy E_{A}, and the most probable energy of system A. What is the total number of microstates after the internal constraint has been removed? Discuss the qualitative dependence of P(E_{A}) on the energy E_{A}. The corresponding data can be obtained by choosing`DataTable`from the`Views`menu. Use this data to calculate the mean and variance of the energy of each subsystem. - Answer the same questions as in Problem 1 with N
_{A}= 20, N_{B}= 20, E_{A}= 100, and E_{B}= 20. - Answer the same questions as in Problem 1 with N
_{A}= 20, N_{B}= 40, E_{A}= 100, and E_{B}= 20. - If the two subsystems have equal numbers of particles, it is reasonable to conclude that the "hotter" system has higher energy. What is the probability that energy goes from the hotter to the colder system after the internal constraint has been removed?
- Consider successively larger systems until you have satisfied yourself that you understand the qualitative behavior of the various quantities. Discuss your general conclusions.
- Consider a special subsystem with only one particle, N
_{A}= 1. Suppose that N_{B}= 5, E_{A}= 0, and E_{B}= 12. If we assume that the subsystem A can exchange with the much larger system B, what is the probability that system A has energy E_{A}? What is the probability that system A is in a particular microstate with energy n where n is an integer? The probability in this case is called the*Boltzmann probability*. Why is the form of this probability different than the probability that you found in the other problems.

References

- Harvey Gould and Jan Tobochnik,
*Statistical and Thermal Physics,*Chapter 4, online notes. - Thomas A. Moore and Daniel V. Schroeder, "A different
approach to introducing statistical mechanics," Am. J. Phys.
**65**, 26–36 (1997). - Daniel V. Schroeder,
*An Introduction to Thermal Physics,*Addison-Wesley (1999).

Java Classes

- EinsteinSolid
- EinsteinSolidApp

Updated 9 March 2009.