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 problem.)
The expressions is
The equation is proved.
We know that,
The logarithm rule is
Consider and factor as
Using Taylor formula as
Hence,the equation is proved.
Use Stirling's approximation to show that the multiplicity of an Einstein solid, for any large values ofandlocalid="1650383388983" ,is approximately
The square root in the denominator is merely large, and can often be neglected. However, it is needed in Problem . (Hint: First show that . Do not neglect the in Stirling's approximation.)
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."
Suppose you flip fair coins.
(a) How many possible outcomes (microstates) are there?
(b) How many ways are there of getting exactly heads and tails?
(c) What is the probability of getting exactly heads and tails?
(d) What is the probability of getting exactly heads and tails?
(e) What is the probability of getting exactly heads and 10 tails?
(f) What is the probability of getting heads and no tails?
(g) Plot a graph of the probability of getting n heads, as a function of n.
For a single large two-state paramagnet, the multiplicity function is very sharply peaked about .
(a) Use Stirling's approximation to estimate the height of the peak in the multiplicity function.
(b) Use the methods of this section to derive a formula for the multiplicity function in the vicinity of the peak, in terms of . Check that your formula agrees with your answer to part (a) when .
(c) How wide is the peak in the multiplicity function?
(d) Suppose you flip coins. Would you be surprised to obtain heads and 499,000 tails? Would you be surprised to obtain 510,000 heads and 490,000 tails? Explain.
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