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I'm trying to review some statistical mechanics from the following link

In doing so on the topic of variational theory and maximization with Lagrange multipliers, the author states that given a function:

$$ h(x_i) = f(x_i) - \lambda g(x_i), $$

then to extremize we require the condition:

$$ \frac{\partial h}{\partial x_i} = 0. $$

Though this is phrased a little differently than I am used to, the conclusion is the same and so I am okay with it.

However, later on the author solves this example with the case of a Grand Canonical Ensemble (GCE). In my own understanding, the GCE is described by the number of particles, and the energy. I arrived at the same constrain equations as the author however, based on my understanding the the parameteres of the same are described by $(E,N)$, I would have expected that there should have been two equations set to equal zero:

$$ \frac{\partial}{\partial E_n} \left[S - \lambda_E \sum P(E,N) E - \lambda_N \sum P(E,N) N - \lambda_1 \sum P(E,N) \right] = 0, $$ and

$$ \frac{\partial}{\partial N_n} \left[S - \lambda_E \sum P(E,N) E - \lambda_N \sum P(E,N) N - \lambda_1 \sum P(E,N) \right] = 0. $$

I was able to find another text that arrives at the same solutions for the probability function, with different variable substitutions (in terms of $\mu$ and $\beta$) so I presume that this procedure is the correct one. I suspect my confusion is related to idea of: what is a function of what. Which has been a bane of my existence through all of my courses on thermodynamics and statistical mechanics.

Any explanation would be greatly appreciated! :)

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    $\begingroup$ should be pi and E and N should be $E_i$ and $N_i$ ... In the Link(www2.ph.ed.ac.uk/~mevans/sp/sp2.pdf) they have for Grand Canonical Ensemble (GCE) not $p_i$, but $p_{i,N}$ and so not $N_i$ but $N$ (see (12) at p.8) $\endgroup$
    – Aleksey Druggist
    Commented Feb 13, 2022 at 1:13
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    $\begingroup$ @AlekseyDruggist You're right, I did not look at the author's notation, so I removed my comment to not add another layer of confusion ;) . (I prefer to sum over microstates $i$ with no constraints on the particle number and consider an observable $i\mapsto N_i$, which is of course equivalent. The author seems to prefer to sum over $N$ and over microstates with $N$ particles. This amount to the same thing, of course.) $\endgroup$
    – Yvan Velenik
    Commented Feb 13, 2022 at 8:21
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    $\begingroup$ Yes, the variables are $\mu$ and $\beta$ (although the identification of these constants with the Lagrange mutipliers should be done separately). Derivatives w.r.t. $\lambda_E$ and $\lambda_N$ (and $\lambda_1$) will just yield the three constraints (normalization, fixed expected energy, fixed expected number of particles). $\endgroup$
    – Yvan Velenik
    Commented Feb 13, 2022 at 15:00
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    $\begingroup$ The derivatives w.r.t. $p_i$ (or rather $p_{Ni}$ in the notation of your reference) will lead to an equation determining the $p_i$: $\frac{\partial}{\partial p_i} \bigl(\sum_i p_i\log p_i - \lambda_E \sum_i p_i E_i - \lambda_N \sum_i p_i N_i - \lambda_1 \sum_i p_i \bigr) = 0 $ yields $\log p_i + 1 - \lambda_E E_i -\lambda_N N_i - \lambda_1 = 0$, that is, $p_i \propto \exp(\lambda_E E_i + \lambda_N N_i)$ as you want. You still have to argue that $\lambda_E = -\beta$ and $\lambda_N = \beta\mu$ if you wish to obtain the usual form, of course. $\endgroup$
    – Yvan Velenik
    Commented Feb 13, 2022 at 15:00
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    $\begingroup$ We discuss this in detail in Section 1.2 of our book. (See also Section 1.3 for the link between the Lagrange multipliers and the relevant physical quantities.) $\endgroup$
    – Yvan Velenik
    Commented Feb 13, 2022 at 15:03

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There are three constraints in the grand canonical ensemble: $$ \sum_{n,j} a(n,j) = \mathcal{N} $$ $$ \sum_{n,j} a(n,j) E_j= \mathcal{N} \bar{E} $$ $$ \sum_{n,j} a(n,j) N_n= \mathcal{N} \bar{N} $$ $a(n,j)$ represents the number of samples with $N_n$ particles and energy $E_j$ in the giant canonical ensemble. $\mathcal{N}$ is the total number of samples. $w=\dfrac{\mathcal{N}!}{\Pi_{n,j}a(n,j)!}$ is the total number of microstates.
When we use the Lagrangian method, the function that needs to be constructed is: $$f({a(n,j),\alpha,\beta, \gamma} = \ln w -\alpha ( \sum_{n,j} a(n,j) - \mathcal{N} )-\beta( \sum_{n,j} a(n,j) E_j - \mathcal{N} \bar{E} ) - \gamma (\sum_{n,j} a(n,j) N_n-\mathcal{N} \bar{N}) $$ The independent variables of the function f are only $ a(n,j),\alpha,\beta, \gamma $.
$ \bar{E}, \bar{N}$ is the expectation of the number of particles and energy given in advance, $E_j, N_n$ is our artificial standard, such as $N_n$ must be $0,1,2,3...$ so they are constants rather than self-energy, and there is no need to take partial derivatives for them.

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