Einstein solid

Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problems 2.5 – 2.7.

A simple model of a solid proposed by Einstein in 1907 is that it consists of a collection of {N} oscillators with quantized energy units. We can think of each oscillator as a quantum harmonic oscillator, and each energy unit as a quantum of size {\hbar\omega}, but the concept applies to any system with energy units that are all the same size. In general, a solid with {N} oscillators can have {q} energy units to distribute amongst them, so the number of possible microstates of such a system is the number of ways of distributing {q} balls into {N} bins. This is a standard problem in combinatorics, and the solution goes as follows.

We can represent the {q} balls by Xs and the {N} bins by {N-1} vertical bars, where each bar serves to separate the contents of one bin from its neighbour. Thus if we have {N=3} and {q=4}, the possible microstates are

\displaystyle   ||XXXX
\displaystyle  |XXXX|
\displaystyle  XXXX||
\displaystyle  |X|XXX
\displaystyle  |XX|XX
\displaystyle  |XXX|X
\displaystyle  X||XXX
\displaystyle  XX||XX
\displaystyle  XXX||X
\displaystyle  X|XXX|
\displaystyle  XX|XX|
\displaystyle  XXX|X|
\displaystyle  X|X|XX
\displaystyle  X|XX|X
\displaystyle  XX|X|X

In general, the number of microstates is the number of ways of choosing {q} (or {N-1}) objects from a total of {q+N-1} objects, without regard to order, which is just the binomial coefficient {\binom{q+N-1}{q}}. For the example just given,

\displaystyle  \binom{q+N-1}{q}=\binom{6}{4}=15 \ \ \ \ \ (1)

which corresponds to the 15 cases listed above.

In his problem 2.5, Schroeder asks us to list the microstates for several other values of {N} and {q}, but this gets pretty tedious and the general idea should be obvious from the above. We’ll just list the number of microstates for each case.

{N} {q} {\binom{q+N-1}{q}}
3 5 21
3 6 28
4 2 10
4 3 20
1 anything 1
anything 1 {N}
30 30 59132290782430712

Admittedly, Schroeder does tell us not to attempt to list all the microstates for the last line(!)

Well OK, just one more example, with {N=4} and {q=2}.

\displaystyle   |||XX
\displaystyle  ||XX|
\displaystyle  |XX||
\displaystyle  XX|||
\displaystyle  ||X|X
\displaystyle  |X|X|
\displaystyle  X|X||
\displaystyle  |X||X
\displaystyle  X|||X
\displaystyle  X||X|