Semi-grand ensembles: Difference between revisions

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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].  
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==References==
[[category: Statistical mechanics]]

Revision as of 15:10, 28 May 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