Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problems 2.20 – 2.21.
The number of microstates in an Einstein solid with oscillators and energy quanta is
For large systems at high temperatures, so that we have the approximate formula
If we now have two such solids and allow them to interact, the number of microstates for the combined system for any given macrostate (that is, a given division of the total energy between the two solids) is just the product of the numbers for the two separate solids:
We’ve already seen that the most probable state for a pair of interacting solids is the state in which the energy quanta are distributed evenly between the two systems, so that . Our goal in this post is to investigate how likely it is that the distribution of energy will deviate significantly from this most probable state.
To make things simpler, we’ll take so that both solids are the same size. Then
With held constant, the shape of this curve is determined by the factor. If we pull out a factor of , we get
We can get a feel for how the curve’s shape changes as we increase by plotting for several values of . itself ranges from 0 to 1 and has a maximum value of 0.25 at , so we can scale the graph to a vertical range of 0 to 1 by inserting a factor of 4 inside the parentheses. That is, we plot , as shown:
The curves are for (red), (blue), (yellow), (green) and (violet).
The actual width of the curve can be approximated by the calculation given by Schroeder in section 2.4. The peak of the curve occurs at so we can investigate the behaviour for and where . Schroeder shows that becomes, near the peak:
which is a Gaussian curve. The width can be defined as the distance in between the points where the curve is of its maximum value, which occurs when
so the width of the curve is twice this, or . Since we’re assuming , this width is still a large number, but since the total width of the graph is , the width of the peak as a fraction of the total width of the graph is around , which for any macroscopic value of , is vanishingly small. For example, if is on the order of Avogadro’s number, , and we drew the graph so that the total width of the graph fits on a page 10 cm wide, the width of the central peak is around , which is about the size of a hydrogen atom.