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| \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]. | | \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]. |
| </math> | | </math> |
|
| |
| ... to be continued ...
| |
|
| |
|
| ==References== | | ==References== |
| [[category: Statistical mechanics]] | | [[category: Statistical mechanics]] |
Revision as of 10:53, 7 September 2007
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:
- ,
where:
Semi-grand ensemble at fixed volume and temperature
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 thermodynamical potential:
Fixed pressure and temperature
In the isothermal-isobaric ensemble: one can write:
where:
Fixed pressure and temperature: Semi-grand ensemble
Following the procedure described above one can write:
- ,
where the new thermodynamical Potential is given by:
Fixed pressure and temperature: Semi-grand ensemble: partition function
In the fixed composition ensemble one has:
References