Question regarding Blundell: Concepts in Thermal Physics

[WWCY]

Hi all, I have an issue with understanding the following passage in the aforementioned book. I have uploaded the relevant passage as an image below. Any assistance is greatly appreciated!

The paragraph under equation (22.57) says that the total Helmholtz function $F$ is a sum of Helmholtz functions of both positrons and electrons. This means $F = F_+ (T,V,N_+) + F_- (T,V,N_-)$, which I guess means
$$\frac{\partial F_- (T,V,N_+ - N)}{\partial N_-}|_{T,V,N} + \frac{\partial F_+ (T,V,N + N_-)}{\partial N_+}|_{T,V,N} = 0$$
Question 1: How then, do I go from this expression to (22.58), where the first term is a partial derivative of $F$, and instead of $N$ being kept constant, we keep $N_+$ constant, and where the second term is a partial derivative over $F$ as well?

Question 2: The book then goes on to define the chemical potential for positrons and electrons in (22.59) and (22.60). Why are they not defined in terms of their respective Helmholtz functions instead of the total Helmholtz function? So for example:
$$\frac{\partial F_- (T,V,N_-)}{\partial N_-}|_{T,V} = \mu _-$$
Since the general definition for chemical potential under constant $T$ and $V$ is
$$\frac{\partial F (T,V,N)}{\partial N} |_{T,V} = \mu $$

Edit: Upon more careful reading, I have found out that the answer to question 2 lies in the fact that $dF =-pdV - SdT + \sum_i \mu _i dN_i$ for the multi-species case, which I forgot about. I still can't see how to resolve question 1 though.

 

[Lord Jestocost]

N = N₊ - N_- is always fixed owing to charge conservation.
To see how to formulate the Helmholtz function F in this case, have a look at:
Equilibrium and Non-Equilibrium Statistical Thermodynamics

 

[Lord Jestocost]

Regarding such problems, one can also use the method of undetermined multipliers.

Write for the Helmholtz Function of the electron-positron “gas”:

$F=F(T,V;N_-,N_+)$

The constraint due to charge conservation:

$N_0=N_--N_+=const.$

Define formally a "new" function $\tilde{F}$ where $λ$ is the undetermined multiplier:

$\tilde{F}=F(T,V;N_-,N_+)+λ(N_0-(N_--N_+))$

Search for the minimum of $\tilde{F}$ with respect to variations in $N_-$ and $N_+$:

$\frac{\partial \tilde{F}}{\partial N_-}|_{N_+}=0$ ⇒ $\frac{\partial F(T,V; N_-,N_+)}{\partial N_-}|_{T,V,N_+}-λ=\mu _--λ=0$

$\frac{\partial \tilde{F}}{\partial N_+}|_{N_-}=0$ ⇒ $\frac{\partial F(T,V; N_-,N_+)}{\partial N_+}|_{T,V,N_-}+λ=\mu_++λ=0$

Eliminating $λ$ gives:

$\mu_-+\mu_+=0$