END; include ("common_short.php"); physics_header("",""); print <<Physics H133: Problem Set #\$ps_num Here are some hints, suggestions, and comments on the problem set.

## Two-Minute Problems

Remember to give a good explanation, no longer than two sentences.

1. T5T.5: How is probability related to multiplicity? What happens when you take the ratio of probabilities? How is multiplicity related to entropy?
2. T6T.1: What happens to the temperature of the object as its thermal energy goes to zero? How is this related to the shape of the graph U vs. S?

## Chapter T Problems

• T5B.6: Review the definition of entropy and its relation to multiplicity. Is entropy an extensive (additive) quantity?
• T5B.7: Is it possible to move from a macropartition with larger number of microstates to one with smaller number? What does this imply for the entropy? Why are small and large systems different in this respect?
• T5S.8: (a) Convert the difference of entropies into a ratio of probabilities. (b) When physicists say "never" it means something different than if a mathematician says "never". How so?
• T5R.2: (a) Continue the argument started in the problem statement, following the hint. (b) How many characters will all monkeys together have typed until today? Each of these characters (except for the last 99,999) could be the first of a string that represents Hamlet. Estimate the probability that, within the long string of characters typed by all monkeys over all times and lined up into a single sequence, you find a string of 100,000 that equals the text of Hamlet.
• T6S.1: (a) Analyze the units. (b) Divide the system in subsystems A and B with different temperatures T_new,A and T_new,B. Consider a process where thermal energy of A increases. See whether you can get the second law to work for you which says that the entropy cannot decrease in this process.
• T5A.1: (a) Rewrite U_A and U_B in terms of U and x. Then find Omega_{AB} = Omega_A * Omega_B in terms of these variables. Note: There is a typo here: B depends on U, but not on x. (b) The Taylor expansion of (1-z)^b starts 1 - b*z + ... Is it a good approximation to use just these terms? (c) A typical spontaneous fluctuation should be of the order of x_{1/2}, because much further will have too small a multiplicity.