Reference: Daniel V. Schroeder, *An Introduction to Thermal Physics*, (Addison-Wesley, 2000) – Problems 2.9 – 2.10.

Here are a few more examples of the probabilities of various macrostates in two interacting Einstein solids. As before, we have two solids, and , containing and oscillators and and quanta of energy, with . For any particular partition of the quanta, that is, for particular values of and , the total number of microstates available to the compound system is

The overall number of microstates is

Example 1Consider a simple system with and as shown in Fig 2.4 in Schroeder. Using Maple to calculate the binomial coefficients (Maple has a ‘binomial’ function that does this automatically) and produce the plot, we have

Plugging in the numbers, we get

0 1 28 28 0.061 1 3 21 63 0.136 2 6 15 90 0.195 3 10 10 100 0.216 4 15 6 90 0.195 5 21 3 63 0.136 6 28 1 28 0.061 A bar chart of the probabilities looks like this:

As before, the most likely state is when the energy is equally distributed between the two solids.

Example 2Now we’ll see what happens if one solid has more oscillators to store energy than the other one. We’ll take , and . We now have

Plugging in the numbers, we get

0 1 84 84 0.017 1 6 56 336 0.067 2 21 35 735 0.147 3 56 20 1120 0.224 4 126 10 1260 0.252 5 252 4 1008 0.201 6 462 1 462 0.092 A bar chart of the probabilities looks like this:

Since solid has more oscillators, the probabilities are skewed towards more of the quanta being stored in solid than in solid .

Example 3Now we’ll ramp things up a bit and consider two solids with , and . As there are 101 macrostates, we won’t list all the various states. The formulas in this case are

The plot is

The maximum probability of 0.073 occurs at and the minimum of at . As solid contains of the oscillators, the maximum probability is when of the energy is stored in solid . Notice how vanishing small is the chance of finding the system in a macrostate with anything less than about . Thus even though all microstates are equally probable, it is overwhelmingly likely that the energy will be more or less evenly distributed over all the oscillators.

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