that the variables must obey. The(n,, variables are also constrained to give the total energy E of the M-container system when summed as E We have two problems: 1. how to maximize Q2 and ii. how to impose these constraints Because Q2 takes on values greater than unity for any choice of the in,), Q2 will experience its maximum where InQ2 has its maximum, so we can maximize In Q2 if doing so helps. Because the n, variables are assumed to take on large numbers(when M is large), we can use Sterlings approximation In X! =X In x-X to approximate In Q2 as follows n9=lnM!-∑{ nIn n-n1) This expression will prove useful because we can take its derivative with respect to the n, variables which we need to do to search for the maximum of in Q To impose the constraints 2,n,=Mand >, n, s=e we use the technique of Lagrange multipliers. That is, we seek to find values of (,) that maximize the following function PAGE 10PAGE 10 SJ nJ = M that the variables must obey. The {nj} variables are also constrained to give the total energy E of the M-container system when summed as SJ nJeJ = E. We have two problems: i. how to maximize W and ii. how to impose these constraints. Because W takes on values greater than unity for any choice of the {nj}, W will experience its maximum where lnW has its maximum, so we can maximize ln W if doing so helps. Because the nJ variables are assumed to take on large numbers (when M is large), we can use Sterling’s approximation ln X! = X ln X – X to approximate ln W as follows: ln W = ln M! - SJ {nJ ln nJ – nJ ). This expression will prove useful because we can take its derivative with respect to the nJ variables, which we need to do to search for the maximum of ln W. To impose the constraints SJ nJ = M and SJ nJ eJ = E we use the technique of Lagrange multipliers. That is, we seek to find values of {nJ} that maximize the following function: