PHY3355 Assignment 1 due Jan. 19, 2024 A. Use the first law of thermodynamics for a reversible process to show that S = V ∂P ∂T and µ N = V ∂P ∂µ Hint: G = µN T B. Consider tossing a coin, with each possible state (head or tail) given equal probability. a) If three coins are tossed, how much less likely are 3 heads (or 3 tails) than 2 of one and 1 of the other? b) If six coins are tossed, how much less likely are 6 heads (or 6 tails) than 3 of each? C. If N distinguishable coins are tossed at once, the number of distinct outcomes giving N1 heads up is w = N !/N1 !(N − N1 )!, where N1 is the number of heads and N − N1 is the number of tails. a) Assuming N is large enough that Sterling’s approximation i.e. lnN ! ≈ NlnN − N is valid, show that lnw is a maximum for N1 = N/2. b) Show that wmax ≈ eN ln2. c) Let P (N1 ) be the probability of finding N1 heads up. Using the approximation in (a) for large N1 , N and N − N1 , find σ satisfying ln P (N/2 ± σ)/Pmax = −1/2. d) Improve on (b) by using Eq. (A.2) of the text for the factorials, to evaluate wmax and Pmax for large N1 etc. Then check whether Pmax agrees with what you would expect based on (c) and the limiting form of P (N1 ) given by the central limit theorem.
