Consider again the system of two large, identical Einstein solids treated in below Problem 1.
a) For the case N = 1023, 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?
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 below Problem 2 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.
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 below Problem 2 to find an approximate expression for the multiplicity of this macrostate.
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 yon 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 N = 1023.
Use Stirling’s approximation to show that the multiplicity of an Einstein solid, for any large values of N and q, is approximately
The square root in the denominator is merely large, and can often be neglected. However, it is needed in above Problem 1. (Hint: First show that Do not neglect the in Stirling’s approximation.)
Step 1 :
In this question , we need to find the entropy of a system, assuming all microstates are allowed that is for large time scale
In the second part, we need to find the entropy of a system, assuming macrostate are allowed, here we are considering small time scale
In the third part, we need to explain if the time scale relevant in entropy of a system
And finally, if a partition inserted in between solid , entropy of macrostate is found to be lesser than the entropy of a macrostate, here second law of thermodynamics is violated, we need to explain if this violation significant