Editing Semi-grand ensembles

== General features ==
Semi-grand ensembles are used in Monte Carlo simulation of mixtures.

In these ensembles the total number of molecules is fixed, but the composition can change.

== Canonical Ensemble: fixed volume, temperature and number(s) of molecules ==

We shall consider a system consisting of ''c'' components;. 
In the [[Canonical ensemble|canonical ensemble]], the differential
equation energy for the [[Helmholtz energy function]] can be written as:

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \sum_{i=1}^c (\beta \mu_i) d N_i </math>,
where:

*<math> A </math> is the [[Helmholtz energy function]]
*<math> \beta \equiv 1/k_B T </math>
*<math> k_B</math> is the [[Boltzmann constant]]
*<math> T </math> is the [[absolute temperature]]
*<math> E </math> is the [[internal energy]]
*<math>  p </math> is the [[pressure]]
*<math> \mu_i </math> is the [[Chemical potential|chemical potential]] of the species <math>i</math>
*<math> N_i </math> is the number of molecules of the species <math>i</math>

== Semi-grand ensemble at fixed volume and temperature ==

Consider now that we want to consider a system with fixed total number of particles, <math> N </math>

: <math> \left. N = \sum_{i=1}^c N_i  \right. </math>; 

but the composition can change, from the thermodynamics we can apply a Legendre's transform [HAVE TO CHECK ACCURACY]
to the differential equation written above in terms of <math> A (T,V,N_1,N_2) </math>. 

* Consider the variable change <math> N_1 \rightarrow N </math> i.e.: <math> \left. N_1 = N-  \sum_{i=2}^c N_i  \right. </math>


: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V +  \beta \mu_1 d N - \beta \mu_1 \sum_{i=2}^c d N_i + \sum_{i=2}^c \beta \mu_2 d N_2; </math>


: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V +  \beta \mu_1 d N + \sum_{i=2}^c \beta (\mu_2-\mu_i) d N_i; </math>

Or:

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V +  \beta \mu_1 d N + \sum_{i=2}^c \beta \mu_{i1} d N_i; </math>

where <math> \left. \mu_{i1} \equiv  \mu_i - \mu_1 \right. </math>.
* Now considering the thermodynamical potential: <math> \beta A - \sum_{i=2}^c \left( N_i \beta \mu_{i1} \right) </math>

:<math> d \left[ \beta A - \sum_{i=2}^c ( \beta \mu_{i1} N_i ) \right] = E d \beta - \left( \beta p \right) d V + \beta \mu_{1} d N - N_2 d \left( \beta \mu_{21} \right).
</math>

== Fixed pressure and temperature == 

In the [[Isothermal-Isobaric ensemble]]: <math> (N_1,N_2, \cdots, N_c, p, T) </math> ensemble we can write:

<math> d (\beta G) = E d \beta + V d (\beta p) + \sum_{i=1}^c \left( \beta \mu_i \right) d N_i </math>

where:

* <math> G </math> is the [[Gibbs energy function]]

==  Fixed pressure and temperature: Semi-grand ensemble ==

Following the procedure described above we can write:

<math> \beta G (\beta,\beta p, N_1, N_2,  \cdots N_c ) \rightarrow \beta \Phi (\beta, \beta p, N, \beta \mu_{21}, \cdots, \beta \mu_{c1} ) </math>, 
where the ''new'' thermodynamical Potential <math> \beta \Phi </math> is given by:

<math> d (\beta \Phi)  = d \left[ \beta G - \sum_{i=2}^c (\beta \mu_{i1} N_i ) \right] = E d \beta + V d (\beta p) + \beta \mu_1 d N
- \sum_{i=2}^c N_i d (\beta \mu_{i1} ).
</math>


==  Fixed pressure and temperature: Semi-grand ensemble: Partition function ==


TO BE CONTINUED SOON