Semi-grand ensembles
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
- \( \beta := 1/k_B T \)
- \( k_B\) 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 \(i\)
[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 \( \beta \Phi \) is given by:
\[ d (\beta \Phi) = d \left[ \beta G - \sum_{i=2}^c (\beta \mu_{i1} N_i ) \right] = E d \beta + V d (\beta p) + \beta \mu_1 d N - \sum_{i=2}^c N_i d (\beta \mu_{i1} ). \]
[edit] Fixed pressure and temperature: Semi-grand ensemble: partition function
In the fixed composition ensemble one has:
\[ Q_{N_i,p,T} = \frac{ \beta p }{\prod_{i=1}^c \left( \Lambda_i^{3N_i} N_i! \right) } \int_{0}^{\infty} dV e^{-\beta p V } V^N \int \left( \prod_{i=1}^c d (R_i^*)^{3N_i} \right) \exp \left[ - \beta U \left( V, (R_1^*)^{3N_1} , \cdots \right) \right]. \]
[edit] References
- Related reading
