Semi-grand ensembles

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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.

[edit] Canonical ensemble: fixed volume, temperature and number(s) of molecules

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

$d \left( \beta A \right) = E d \beta - (\beta p) d V + \sum_{i=1}^c (\beta \mu_i) d N_i$ ,

where:

$A$ is the Helmholtz energy function
$β: = 1 / k B T$
$k B$ is the Boltzmann constant
$T$ is the absolute temperature
$E$ is the internal energy
$p$ is the pressure
$μ i$ is the chemical potential of the species $i$
$N i$ is the number of molecules of the species $i$

[edit] Semi-grand ensemble at fixed volume and temperature

Consider now that we wish to consider a system with fixed total number of particles, $N$

$\left. N = \sum_{i=1}^c N_i \right.$ ;

but the composition can change, from thermodynamic considerations one can apply a Legendre transform [HAVE TO CHECK ACCURACY] to the differential equation written above in terms of $A (T, V, N 1, N 2)$ .

Consider the variable change $N_1 \rightarrow N$ i.e.: $\left. N_1 = N- \sum_{i=2}^c N_i \right.$

$d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N - \beta \mu_1 \sum_{i=2}^c d N_i + \sum_{i=2}^c \beta \mu_i d N_i;$

$d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N + \sum_{i=2}^c \beta (\mu_i-\mu_1) d N_i;$

or,

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

where $\left. \mu_{i1} \equiv \mu_i - \mu_1 \right.$ .

Now considering the thermodynamic potential: $\beta A - \sum_{i=2}^c \left( N_i \beta \mu_{i1} \right)$

$d \left[ \beta A - \sum_{i=2}^c ( \beta \mu_{i1} N_i ) \right] = E d \beta - \left( \beta p \right) d V + \beta \mu_{1} d N - \sum_{i=2}^c N_i d \left( \beta \mu_{i1} \right).$

[edit] Fixed pressure and temperature

In the isothermal-isobaric ensemble: $(N_1,N_2, \cdots, N_c, p, T)$ one can write:

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

where:

$G$ is the Gibbs energy function

[edit] Fixed pressure and temperature: Semi-grand ensemble

Following the procedure described above one can write:

$\beta G (\beta,\beta p, N_1, N_2, \cdots N_c ) \rightarrow \beta \Phi (\beta, \beta p, N, \beta \mu_{21}, \cdots, \beta \mu_{c1} )$ ,

where the new thermodynamic potential $βΦ$ is given by:

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

[edit] Fixed pressure and temperature: Semi-grand ensemble: partition function

In the fixed composition ensemble one has:

$Q_{N_i,p,T} = \frac{ \beta p }{\prod_{i=1}^c \left( \Lambda_i^{3N_i} N_i! \right) } \int_{0}^{\infty} dV e^{-\beta p V } V^N \int \left( \prod_{i=1}^c d (R_i^*)^{3N_i} \right) \exp \left[ - \beta U \left( V, (R_1^*)^{3N_1} , \cdots \right) \right].$