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, the differential equation energy for the Helmholtz energy function can be written as:

$d(\beta A)=Ed\beta -(\beta p)dV+\sum _{i=1}^c(\beta {\mu }_i)d{N}_i$,

where:

• $A$ is the Helmholtz energy function
• $\beta \equiv 1/k_{B}T$
• $k_B$ is the Boltzmann constant
• $T$ is the absolute temperature
• $E$ is the internal energy
• $p$ is the pressure
• ${\mu }_i$ is the chemical potential of the species $i$
• $N_i$ is the number of molecules of the species $i$

Semi-grand ensemble at fixed volume and temperature

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

$N=\sum _{i=1}^c{N}_i$;

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 $N_1\to N$ i.e.: $N_1=N-\sum _{i=2}^c{N}_i$

$d(\beta A)=Ed\beta -(\beta p)dV+\beta {\mu }_{1}dN-\beta {\mu }_1\sum _{i=2}^{c}d{N}_i+\sum _{i=2}^c\beta {\mu }_{i}d{N}_i;$

$d(\beta A)=Ed\beta -(\beta p)dV+\beta {\mu }_{1}dN+\sum _{i=2}^c\beta ({\mu }_i-{\mu }_1)d{N}_i;$

or,

$d(\beta A)=Ed\beta -(\beta p)dV+\beta {\mu }_{1}dN+\sum _{i=2}^c\beta {\mu }_{i1}d{N}_i;$

where ${\mu }_{i1}\equiv {\mu }_i-{\mu }_1$.

• Now considering the thermodynamical potential: $\beta A-\sum _{i=2}^c(N_i\beta {\mu }_{i1})$

$d[\beta A-\sum _{i=2}^c(\beta {\mu }_{i1}{N}_i)]=Ed\beta -(\beta p)dV+\beta {\mu }_{1}dN-\sum _{i=2}^c{N}_{i}d(\beta {\mu }_{i1}).$

Fixed pressure and temperature

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

$d(\beta G)=Ed\beta +Vd(\beta p)+\sum _{i=1}^c(\beta {\mu }_i)d{N}_i$

where:

• $G$ is the Gibbs energy function

Fixed pressure and temperature: Semi-grand ensemble

Following the procedure described above one can write:

$\beta G(\beta ,\beta p,N_1,N_2,\cdots N_c)\to \beta \mathrm{\Phi }(\beta ,\beta p,N,\beta {\mu }_{21},\cdots ,\beta {\mu }_{c1})$,

where the new thermodynamical Potential $\beta \mathrm{\Phi }$ is given by:

$d(\beta \mathrm{\Phi })=d[\beta G-\sum _{i=2}^c(\beta {\mu }_{i1}{N}_i)]=Ed\beta +Vd(\beta p)+\beta {\mu }_{1}dN-\sum _{i=2}^c{N}_{i}d(\beta {\mu }_{i1}).$

Fixed pressure and temperature: Semi-grand ensemble: partition function

In the fixed composition ensemble one has:

$Q_{N_i,p,T}=\frac{\beta p}{\prod _{i=1}^c(\underset{i}{\overset{3N_i}{\mathrm{\Lambda }}}{N}_i!)}\underset{0}{\overset{\mathrm{\infty }}{\int }}dV{e}^{-\beta pV}{V}^N\int (\prod _{i=1}^{c}d(R_i^{*}{)}^{3N_i})\exp [-\beta U(V,(R_1^{*}{)}^{3N_1},\cdots )].$