1) Assuming that the total number of microstates accessible to a given statistic
ID: 2180064 • Letter: 1
Question
1) Assuming that the total number of microstates accessible to a given statistical system is omega. show that the entropy of the system, as given by equation S=-k<lnPr>=-ksum(PrlnPr), is maximum when all omega states are equally likely to occur.
2) If , on the other hand, we have an ensemble of system sharing energy(with mean value E), then show that the entropy, as given by the same formal expression, is maximum when Pr proportional to exp(-?Er),? being a constant to be determined by the given value of E.
Explanation / Answer
On using the method of lagrange multipliers to solve this problem: follow along with page 43 of Pathria, only we are not trying to find the set nr that makes our function maximal, but Pr, so replace lnW with S and nr with Pr. part a) constraint: sum over Pr =1. Only one lagrange mulitplier, get that Pr is contant part b)two constraints: sum Pr=1 an sum ErPr=Ebar, two lagrange multipliers, get that Pr is equal to same constant as in a) but now multiplied by exp(-betaEr)
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