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Use a computer to produce a table and graph, like those in this section, for the case where one Einstein solid contains oscillators, the other contains oscillators, and there are units of energy in total. What is the most probable macrostate, and what is its probability? What is the least probable macrostate, and what is its probability?
Use a computer to produce a table and graph, like those in this section, for two interacting two-state paramagnets, each containing elementary magnetic dipoles. Take a "unit" of energy to be the amount needed to flip a single dipole from the "up" state (parallel to the external field) to the "down" state (antiparallel). Suppose that the total number of units of energy, relative to the state with all dipoles pointing up, is; this energy can be shared in any way between the two paramagnets. What is the most probable macrostate, and what is its probability? What is the least probable macrostate, and what is its probability?
The natural logarithm function, , is defined so that for any positive number.Sketch a graph of the natural logarithm function. Prove the identitiesand(c) Prove that.(d)
Fun with logarithms. Simplify the expression. (That is, write it in a way that doesn't involve logarithms.) Assuming that , prove that . (Hint: Factor out the from the argument of the logarithm, so that you can apply the approximation of part of the previous proble
Write in the form , for some
Use a pocket calculator to check the accuracy of Stirling's approximation for . Also check the accuracy of equation for
Suppose you flip coins. What is the probability of getting exactly heads and tails? (Hint: First write down a formula for the total number of possible outcomes. Then, to determine the "multiplicity" of the "macrostate," use Stirling's approximation. If you have a fancy calculator that makes Stirling's approximation unnecessary, multiply all the numbers in this problem by , or , or, until Stirling's approximation becomes necessary.) What is the probability of getting exactly heads and
Use the methods of this section to derive a formula, similar to equation, for the multiplicity of an Einstein solid in the "low-temperature" limit,
Use Stirling's approximation to show that the multiplicity of an Einstein solid, for any large values ofand,is approximately
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