Problem 2.30 (Schroeder’s Intro to Thermal Physics)

Problem 2.30

Consider again the system of two large, identical Einstein solids
treated in Problem 2.22.

(a) For the case , compute the entropy of this system (in terms of Boltzmann’s constant), assuming that all of the microstates are allowed. (This is the system’s entropy over long time scales.)
(b) Compute the entropy again, assuming that the system is in its most likely
macrostate. (This is the system’s entropy over short time scales, except when there is a large and unlikely fluctuation away from the most likely macrostate.)
(c) Is the issue of time scales really relevant to the entropy of this system?
(d) Suppose that, at a moment when the system is near its most likely macrostate, you suddenly insert a partition between the solids so that they can no longer exchange energy. Now, even over long time scales, the entropy is given by your answer to part (b). Since this number is less than your answer to part (a), you have, in a sense, caused a violation of the second law of thermodynamics. Is this violation significant? Should we lose any sleep over it?

Solution:

Problem 2.30 Solution (Download)

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