Question

2. Microcanonical ensemble: One-dimensional chain. (24 pts.) Consider a one-dimensional chain consisting of N segments as illus- trated in Figure 1. Let the length of each segment be a when the long dimension of the segment is parallel to the chain and 0 when the long dimension is normal to the chain direction. Each segment has just two non-degenerate states: long dimension parallel to the chain or perpen- dicular to the chain. Now consider a macrostate of the chain in which there are n segments with their long dimension parallel to the chain and the rest segments with their long dimension perpendicular to the chain. We call this macrostate n-state. Figure 1: A one-dimensional chain consisting of segments. (a) What is the length L of the chain when it is in the n-state? (b) One particular partitioning of N segments into two groups, with a given set of n segments with their long dimension parallel to the chain while the rest N - n segments with their long dimension perpendicular to the chain, is called one microstate (configuration) Note that for two microstates corresponding to the same n-macrostate, if the identities (or indices) of the n segments with their long dimension parallel to the chain are different, then these two microstates are different. What is the number of microstates Ω(N,n), for the chain when it is in the n-state? (c) What is the entropy S as a function of N and n of the chain when it is in the n-state? Derive an asymptotic formula valid when N > n 3-1 (Note: Sterling's formula In r! ~エInr-r for n> 1). (d) Identify the value of n that maximizes S (e) Express S you obtain in (c) as a function of N and L. Find out the = ( )N effective temperature T of the system defined via (f) Express L as a function of T and N. Determine and interpret the asymptotic values of L(T-+0+), L(T → +oo), L(T →-oo), and L(T0) (Hint: How are different states populated by the segments in each limit ?)

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