Problem 2.22 (Schroeder’s Intro to Thermal Physics)

Problem 2.22

This problem gives an alternative approach to estimating the width of the peak of the multiplicity function for a system of two large Einstein solids.

(a) Consider two identical Einstein solids, each with N oscillators, in thermal contact with each other. Suppose that the total number of energy units in the combined system is exactly 2N. How many different macrostates (that is, possible values for the total energy in the first solid) are there for this combined system?

(b) Use the result of Problem 2.18 to find an approximate expression for the
total number of microstates for the combined system. (Hint: Treat the
combined system as a single Einstein solid. Do not throw away factors
of “large” numbers, since you will eventually be dividing two “very large”
numbers that are nearly equal. Answer: .)

(c) The most likely macrostate for this system is (of course) the one in which
the energy is shared equally between the two solids. Use the result of
Problem 2.18 to find an approximate expression for the multiplicity of this
macrostate. (Answer: .)

(d) You can get a rough idea of the “sharpness” of the multiplicity function
by comparing your answers to parts (b) and (c). Part (c) tells you the
height of the peak, while part (b) tells you the total area under the entire
graph. As a very crude approximation, pretend that the peak’s shape is
rectangular. In this case, how wide would it be? Out of all the macrostates,
what fraction have reasonably large probabilities? Evaluate this fraction
numerically for the case

Solution:

Problem 2.22 Solution (Download)

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