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.

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

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 ,


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

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;


 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).

Fixed pressure and temperature[edit]

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


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

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  \beta \Phi 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} ).

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

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].


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