Semi-grand ensembles: Difference between revisions

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== 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.
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 ==
== Canonical ensemble: fixed volume, temperature and number(s) of molecules ==
We shall consider a system consisting of ''c'' components;.  
We shall consider a system consisting of ''c'' components;.  
In the [[Canonical ensemble|canonical ensemble]], the differential
In the [[Canonical ensemble|canonical ensemble]], the differential
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*<math> A </math> is the [[Helmholtz energy function]]
*<math> A </math> is the [[Helmholtz energy function]]
*<math> \beta \equiv 1/k_B T </math>
*<math> \beta := 1/k_B T </math>
*<math> k_B</math> is the [[Boltzmann constant]]
*<math> k_B</math> is the [[Boltzmann constant]]
*<math> T </math> is the [[absolute temperature]]
*<math> T </math> is the absolute [[temperature]]
*<math> E </math> is the [[internal energy]]
*<math> E </math> is the [[internal energy]]
*<math>  p </math> is the [[pressure]]
*<math>  p </math> is the [[pressure]]
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where <math> \left. \mu_{i1} \equiv  \mu_i - \mu_1 \right. </math>.
where <math> \left. \mu_{i1} \equiv  \mu_i - \mu_1 \right. </math>.
* Now considering the thermodynamical potential: <math> \beta A - \sum_{i=2}^c \left( N_i \beta \mu_{i1} \right) </math>
* Now considering the thermodynamic potential: <math> \beta A - \sum_{i=2}^c \left( N_i \beta \mu_{i1} \right) </math>


:<math> 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 - N_2 d \left( \beta \mu_{21} \right).
:<math> 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).
</math>
</math>


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:<math> \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} ) </math>,  
:<math> \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} ) </math>,  


where the ''new'' thermodynamical Potential <math> \beta \Phi </math> is given by:
where the ''new'' thermodynamic potential <math> \beta \Phi </math> is given by:


:<math> 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
:<math> 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
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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>
==References==
==References==
<references/>
;Related reading
*[http://dx.doi.org/10.1063/1.3677193 Yiping Tang "A new method of semigrand canonical ensemble to calculate first-order phase transitions for binary mixtures", Journal of Chemical Physics '''136''' 034505 (2012)]
[[category: Statistical mechanics]]
[[category: Statistical mechanics]]

Latest revision as of 13:05, 20 January 2012

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:

,

where:

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.:



or,

where .

  • Now considering the thermodynamic potential:

Fixed pressure and temperature[edit]

In the isothermal-isobaric ensemble: one can write:

where:

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:

References[edit]

Related reading