Semi-grand ensembles

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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[edit]

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:



Semi-grand ensemble at fixed volume and temperature[edit]

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.:


where .

  • Now considering the thermodynamic potential:

Fixed pressure and temperature[edit]

In the isothermal-isobaric ensemble: one can write:


Fixed pressure and temperature: Semi-grand ensemble[edit]

Following the procedure described above one can write:


where the new thermodynamic potential is given by:

Fixed pressure and temperature: Semi-grand ensemble: partition function[edit]

In the fixed composition ensemble one has:


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