Interacting Einstein solids: a few examples

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, {A} and {B}, containing {N_{A}} and {N_{B}} oscillators and {q_{A}} and {q_{B}} quanta of energy, with {q_{A}+q_{B}=q=\mbox{constant}}. For any particular partition of the quanta, that is, for particular values of {q_{A}} and {q_{B}}, the total number of microstates available to the compound system is

\displaystyle \Omega_{total}=\Omega_{A}\Omega_{B}=\binom{q_{A}+N_{A}-1}{q_{A}}\binom{q_{B}+N_{B}-1}{q_{B}} \ \ \ \ \ (1)

The overall number of microstates is

\displaystyle \Omega_{overall}=\binom{q+N_{A}+N_{B}-1}{q} \ \ \ \ \ (2)

Example 1 Consider a simple system with {N_{A}=N_{B}=3} and {q=6} 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

\displaystyle \Omega_{overall} \displaystyle = \displaystyle \binom{6+3+3-1}{6}=462\ \ \ \ \ (3)
\displaystyle \Omega_{A} \displaystyle = \displaystyle \binom{q_{A}+2}{q_{A}}\ \ \ \ \ (4)
\displaystyle \Omega_{B} \displaystyle = \displaystyle \binom{q-q_{A}+2}{q-q_{A}}=\binom{8-q_{A}}{6-q_{A}}\ \ \ \ \ (5)
\displaystyle Prob\left(q_{A}\right) \displaystyle = \displaystyle \frac{\Omega_{A}\Omega_{B}}{\Omega_{overall}} \ \ \ \ \ (6)

Plugging in the numbers, we get

{q_{A}} {\Omega_{A}} {\Omega_{B}} {\Omega_{total}} {Prob\left(q_{A}\right)}
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 2 Now we’ll see what happens if one solid has more oscillators to store energy than the other one. We’ll take {N_{A}=6}, {N_{B}=4} and {q=6}. We now have

\displaystyle \Omega_{overall} \displaystyle = \displaystyle \binom{6+6+4-1}{6}=5005\ \ \ \ \ (7)
\displaystyle \Omega_{A} \displaystyle = \displaystyle \binom{q_{A}+5}{q_{A}}\ \ \ \ \ (8)
\displaystyle \Omega_{B} \displaystyle = \displaystyle \binom{q-q_{A}+3}{q-q_{A}}=\binom{9-q_{A}}{6-q_{A}}\ \ \ \ \ (9)
\displaystyle Prob\left(q_{A}\right) \displaystyle = \displaystyle \frac{\Omega_{A}\Omega_{B}}{\Omega_{overall}} \ \ \ \ \ (10)

Plugging in the numbers, we get

{q_{A}} {\Omega_{A}} {\Omega_{B}} {\Omega_{total}} {Prob\left(q_{A}\right)}
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 {A} has more oscillators, the probabilities are skewed towards more of the quanta being stored in solid {A} than in solid {B}.

Example 3 Now we’ll ramp things up a bit and consider two solids with {N_{A}=200}, {N_{B}=100} and {q=100}. As there are 101 macrostates, we won’t list all the various states. The formulas in this case are

\displaystyle \Omega_{overall} \displaystyle = \displaystyle \binom{200+100+100-1}{100}=1.681\times10^{96}\ \ \ \ \ (11)
\displaystyle \Omega_{A} \displaystyle = \displaystyle \binom{q_{A}+199}{q_{A}}\ \ \ \ \ (12)
\displaystyle \Omega_{B} \displaystyle = \displaystyle \binom{q-q_{A}+99}{q-q_{A}}=\binom{199-q_{A}}{100-q_{A}}\ \ \ \ \ (13)
\displaystyle Prob\left(q_{A}\right) \displaystyle = \displaystyle \frac{\Omega_{A}\Omega_{B}}{\Omega_{overall}} \ \ \ \ \ (14)

The plot is

The maximum probability of 0.073 occurs at {q_{A}=67} and the minimum of {2.69\times10^{-38}} at {q_{A}=0}. As solid {A} contains {\frac{2}{3}} of the oscillators, the maximum probability is when {\frac{2}{3}} of the energy is stored in solid {A}. Notice how vanishing small is the chance of finding the system in a macrostate with anything less than about {q_{A}=45}. 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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