Question

In this problem, you will estimate the with 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 macro states (that is, possible values for the total energy in the first solid) are there for this combined system?

b) Find an approximate expression for the total number of microstates for the combined system (Hint: treat the entire system as a single Einstein solid. Start from the multiplicity function and apply Stirling’s approximation; Do not throw away factors of “large” numbers, since you will eventually be dividing by large numbers.)

c) The most likely macro state for this system is (of course) the one in which the total energy is shared equally between the two solids. Find an approximate expression for the multiplicity of this macro state.

d) You can get a rough idea of the “sharpness” of the multiplicity function by comparing your answers of 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 of the macro states, what fraction have reasonably large probabilities? Evaluate this fraction numerically for the case N = 1023. What happens to this width in the thermodynamic limit (N → ∞; remember, q=2N for this problem.)

e) The classical statement of the 2nd law of thermodynamics is: energy will not spontaneously flow from a cold object to a hot object.; Re-interpret this in terms of probabilities and the results of parts (a-d)

Answer 1