Einstein Model of a Solid
The conceptual Einstein solid is useful for examining the idea of multiplicity in the distribution of energy among the available energy states of the system. All of the energy levels are considered to be equally probable within the constraint of having q units of energy and N oscillators.
As an example, consider q=3 units of energy distributed in an Einstein solid with N=4 oscillators.
The entropy of the Einstein solid can be expressed in terms of the multiplicity.
To make the multiplicity expression manageable, consider the following steps:
The numbers q and N are presumed large and the 1 is dropped. Now making use of Stirling's approximation to evaluate the factorials
Now making the physical assumption that the number of energy units is much larger than the number of oscillators, q>>N, the expression can be further simplified. With rearrangement
Besides making use of log combination rules, the above steps make use of the series expansion of ln(1+N/q) which can be approximated by the first term when q>>N: (approximately, ln(1+x)=x if x<<1). Substitution gives an expression for the entropy of the Einstein solid:
Finally we can make a connection to something akin to a real-world solid. The internal energy U can be represented by q times the oscillator energy unit hf = ε. The last term above will be negligible under the assumption q>>N, so the entropy expression becomes
Using the definition of temperature as a function of entropy gives
This is what is expected from equipartition of energy. Each oscillator has two degrees of freedom, and each should represent kT/2 of energy, giving U=NkT.