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Chapter 2: Q. 2.6 (page 55)
Calculate the multiplicity of an Einstein solid with oscillators and units of energy. (Do not attempt to list all the microstates.)
An Einstein solid with oscillators and energy units has a multiplicity of
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Use a computer to reproduce the table and graph in Figure: two Einstein solids, each containing three harmonic oscillators, with a total of six units of energy. Then modify the table and graph to show the case where one Einstein solid contains six harmonic oscillators and the other contains four harmonic oscillators (with the total number of energy units still equal to six). Assuming that all microstates are equally likely, what is the most probable macrostate, and what is its probability? What is the least probable macrostate, and what is its probability?
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 oscillators, in thermal contact with each other. Suppose that the total number of energy units in the combined system is exactly . 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 Problemto 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 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 .
Consider a two-state paramagnet with elementary dipoles, with the total energy fixed at zero so that exactly half the dipoles point up and half point down.
(a) How many microstates are "accessible" to this system?
(b) Suppose that the microstate of this system changes a billion times per second. How many microstates will it explore in ten billion years (the age of the universe)?
(c) Is it correct to say that, if you wait long enough, a system will eventually be found in every "accessible" microstate? Explain your answer, and discuss the meaning of the word "accessible."
Calculate the number of possible five-card poker hands, dealt from a deck of 52 cards. (The order of cards in a hand does not matter.) A royal flush consists of the five highest-ranking cards (ace, king, queen, jack, 10) of any one of the four suits. What is the probability of being dealt a royal flush (on the first deal)?
Suppose you flip fair coins.
(a) How many possible outcomes (microstates) are there?
(b) What is the probability of getting the sequence HTHHTTTHTHHHTHHHHTHT (in exactly that order)?
(c) What is the probability of getting heads and tails (in any order)?
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