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:

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

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, $N$

$\left.N=\sum _{i=1}^{c}N_{i}\right.$;

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.: $\left.N_{1}=N-\sum _{i=2}^{c}N_{i}\right.$

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

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

or,

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

where $\left.\mu _{i1}\equiv \mu _{i}-\mu _{1}\right.$.

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

$d\left[\beta A-\sum _{i=2}^{c}(\beta \mu _{i1}N_{i})\right]=Ed\beta -\left(\beta p\right)dV+\beta \mu _{1}dN-N_{2}d\left(\beta \mu _{21}\right).$

Fixed pressure and temperature

In the isothermal-isobaric ensemble: $(N_{1},N_{2},\cdots ,N_{c},p,T)$ one can write:

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

where:

• $G$ is the Gibbs energy function

Fixed pressure and temperature: Semi-grand ensemble

Following the procedure described above one can write:

$\beta G(\beta ,\beta p,N_{1},N_{2},\cdots N_{c})\rightarrow \beta \Phi (\beta ,\beta p,N,\beta \mu _{21},\cdots ,\beta \mu _{c1})$,

where the new thermodynamical Potential Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta \Phi } is given by:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

In the fixed composition ensemble one has:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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]. }