which state can be permuted without altering the total energy E. The formula for how many ways the M container states can be permuted such that there are n, containers appearing in single-container state J, with i.a total ofM containers,is 92(n)=M!/{I1n1!} Here n=n,, n,, n3,..,,... denote the number of containers existing in single- container states 1, 2, 3,..J,.. This combinatorial formula reflects the permutational degeneracy arising from placing n, containers into state 1, n, containers into state 2, etc If we imagine an extremely large number of containers and we view M as well as the (n, as being large numbers(n b, we will soon see that this is the case), we can ask for what choices of the variables(n,,n2, n3,.n,,...) is this degeneracy function Q2(n)a maximum. Moreover, we can examine @2(n)at its maximum and compare its value at values of the (n) parameters changed only slightly from the values that maximized Q(n) As we will see, Q2 is very strongly peaked at its maximum and decreases extremely rapidly for values of in that differ only slightly from the "optimal" values. It is this property that gives rise to the very narrow energy distribution discussed earlier in this Section. So, lets take a closer look at how this energy distribution formula arises We want to know what values of the variables (n,,n2, n3,.n,...) make Q2= M!/(IIn !i a maximum. However, all of the n, n,, n3,.,,... variables are not ndependent; they must add up to m, the total number of containers, so we have a constraint Page 9PAGE 9 which state can be permuted without altering the total energy E. The formula for how many ways the M container states can be permuted such that: i. there are nJ containers appearing in single-container state J, with ii. a total of M containers, is W(n) = M!/{PJnJ !}. Here n = {n1 , n2 , n3 , …nJ , …} denote the number of containers existing in single￾container states 1, 2, 3, … J, …. This combinatorial formula reflects the permutational degeneracy arising from placing n1 containers into state 1, n2 containers into state 2, etc. If we imagine an extremely large number of containers and we view M as well as the {nJ} as being large numbers (n.b., we will soon see that this is the case), we can ask for what choices of the variables {n1 , n2 , n3 , …nJ , …} is this degeneracy function W(n) a maximum. Moreover, we can examine W(n) at its maximum and compare its value at values of the {n} parameters changed only slightly from the values that maximized W(n). As we will see, W is very strongly peaked at its maximum and decreases extremely rapidly for values of {n} that differ only slightly from the “optimal” values. It is this property that gives rise to the very narrow energy distribution discussed earlier in this Section. So, let’s take a closer look at how this energy distribution formula arises. We want to know what values of the variables {n1 , n2 , n3 , …nJ , …} make W = M!/{PJnJ !} a maximum. However, all of the {n1 , n2 , n3 , …nJ , …} variables are not independent; they must add up to M, the total number of containers, so we have a constraint