The Boltzmann's equation S(t) = kp log W(t) is a fundamental equation in Statistical Mechanics that relates the entropy S(t) to W(t), which is the number of microstates that constitute a state/configuration t. Here k > 0 is a universal constant called the Boltzmann constant. In this problem, you will use counting (and the Boltzmann equation) to identify the most stable configuration for a collection of n particles.
Suppose n is even, and we have n different particles, each of which can have one of two spins: up spin or down spin. The state of the system is the number of particles t that have spin up. The number of microstates W(t) of a state t corresponds to the number of ways in which exactly t out of the n particles have spin up.
(i) Give an expression for the number of microstates W(t) corresponding to the state t. What is the number of microstates for the state n - t? (1 point)
(ii) Prove that the state t that has the maximum number of microstates is t = n/2, i.e. the value of t that maximizes the number of ways in which exactly t out of n particles have state up is t = n/2. (2 points)
Hint: How does the number of microstates in state i compare to the number of microstates in state (i - 1)?
(iii) Prove that for the most stable state (the state with the maximum number of microstates): the number of microstates W(t) > 2/(n + 1). Then, use the Boltzmann equation to prove that the entropy of the most stable state is at least k(n - log(n + 1)). (2 points)
Hint: What is the value of n + n + ... + n?
