Question

Given the distribution of `N` particles in `n` systems, `t = prod_N prod_i (n!)/(n_(N_i)!)`, and the following constraints, derive the grand canonical partition function and the distribution law for the grand ensemble?

Answer 1

Alright, so the first thing is to define the Lagrangian using our constraints:

`ℒ = lnt + alphasum_Nsum_i n_(N_i) - betasum_Nsum_i n_(N_i)E_(N_i) + ln lambdasum_Nsum_i n_(N_i)N`

For brevity and minimization of eyesores, I'll denote `sum_N sum_i` as `sum_(N,i)`. Hence, we have:

`ℒ = lnt + alphasum_(N,i) n_(N_i) - betasum_(N,i) n_(N_i)E_(N_i) + ln lambdasum_(N,i) n_(N_i)N`

The idea is that are finding the optimal distribution of systems, `n_(N_i)^"*"`, such that the Lagrangian is maximized. That means we have to perform the following derivative and set it equal to 0:

`((delℒ)/(deln_(N_i)))_(n_(N_(j ne i))) = 0`

Next, we would then need to convert the distribution expression into its `ln` form. The reason for this is that it is ridiculous to take the derivative of a factorial.

`ln(prod_N prod_i (n!)/(n_(N_i)!))`

`= sum_(N,i) ln((n!)/(n_(N_i)!))`

Since the sum is not over `n`, but the `n_(N_i)` systems and `N_i` particles, respectively, we can float the `n!` out.

`=> ln(n!) - sum_(N,i) ln(n_(N_i)!)`

Stirling's approximation, `ln(n!) ~~ nlnn - n`, is useful here. For statistical mechanics, the number of systems can get very large, so `n` can get very large. When that is the case, to a good approximation:

`lnt = nlnn - n - sum_(N,i) n_(N_i)ln(n_(N_i)) - n_(N_i)`

Now, we can take the derivative. Since the derivative is of a specific `n_(N_i)` (at a constant `n_(N_(j ne i))`), the sums all go away, and we focus on only the `n_(N_j)` where `j = i`.

`((delℒ)/(deln_(N_i)))_(n_(N_(j ne i))) = -(n_(N_i)*1/(n_(N_i)) + ln(n_(N_i)) - 1) + alpha (deln_(N_i))/(deln_(N_i)) - beta(deln_(N_i)E_(N_i))/(deln_(N_i)) + ln lambda (deln_(N_i)N)/(deln_(N_i))`

`= -ln(n_(N_i)) + alpha - betaE_(N_i) + Nln lambda = 0`

That was the hard part! Solving for `n_(N_i)^"*"`, we get:

`n_(N_i)^"*" = lambda^N e^(alpha)e^(-betaE_(N_i))`

Therefore, going back to the first constraint, we have:

`sum_(N,i) n_(N_i) = n = e^(alpha)sum_(N,i) lambda^N e^(-betaE_(N_i))`

We then define the grand canonical partition function as:

`color(blue)(Xi(lambda,beta,V) = sum_(N,i) lambda^N e^(-betaE_(N_i)))`

As a result, `e^(alpha) = n/Xi`, and if we want to find the distribution law, we multiply by `(n_(N_i))/(n_(N_i))` to get:

`e^alpha = n/Xi * (n_(N_i))/(n_(N_i))`

Therefore, the distribution law `p_(N_i) = n_(N_i)/n` is:

`color(blue)(p_(N_i)) = n_(N_i)/n = (n_(N_i))/(e^(alpha)Xi)`

`= (lambda^N cancel(e^(alpha))e^(-betaE_(N_i)))/(cancel(e^(alpha))sum_(N,i)lambda^N e^(-betaE_(N_i)))`

`= color(blue)((lambda^Ne^(-betaE_(N_i)))/(sum_(N,i)lambda^N e^(-betaE_(N_i))))`