# 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 ${q=2}$.

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

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