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:
,
where:
Semi-grand ensemble at fixed volume and temperature[edit]
Consider now that we wish to consider a system with fixed total number of particles,
;
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
.
- Consider the variable change
i.e.: 


or,

where
.
- Now considering the thermodynamic potential:

![{\displaystyle d\left[\beta A-\sum _{i=2}^{c}(\beta \mu _{i1}N_{i})\right]=Ed\beta -\left(\beta p\right)dV+\beta \mu _{1}dN-\sum _{i=2}^{c}N_{i}d\left(\beta \mu _{i1}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/872cabdfec304b53920fd81de6fd247d65fbea96)
Fixed pressure and temperature[edit]
In the isothermal-isobaric ensemble:
one can write:

where:
Fixed pressure and temperature: Semi-grand ensemble[edit]
Following the procedure described above one can write:
,
where the new thermodynamic potential
is given by:
![{\displaystyle d(\beta \Phi )=d\left[\beta G-\sum _{i=2}^{c}(\beta \mu _{i1}N_{i})\right]=Ed\beta +Vd(\beta p)+\beta \mu _{1}dN-\sum _{i=2}^{c}N_{i}d(\beta \mu _{i1}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9ecb6a938d9b13ec7be5a28a8ecdf34dd065651)
Fixed pressure and temperature: Semi-grand ensemble: partition function[edit]
In the fixed composition ensemble one has:
![{\displaystyle Q_{N_{i},p,T}={\frac {\beta p}{\prod _{i=1}^{c}\left(\Lambda _{i}^{3N_{i}}N_{i}!\right)}}\int _{0}^{\infty }dVe^{-\beta pV}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].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04e0111c9152df95a0f85133726f8a9fb4e1c809)
References[edit]
- Related reading