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.
== Canonical ensemble: fixed volume, temperature and number(s) of molecules ==
 
We shall consider a system consisting of ''c'' components;.  
In these ensembles the total number of molecules is fixed, but the composition can change.
In the [[Canonical ensemble|canonical ensemble]], the differential
 
== Canonical Ensemble: fixed volume, temperature and number(s) of molecules ==
 
We will consider a system with "c" components;.  
In the Canonical Ensemble, the differential
equation energy for the [[Helmholtz energy function]] can be written as:
equation energy for the [[Helmholtz energy function]] can be written as:


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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]]
*<math> \mu_i </math> is the chemical potential of the species "i"
*<math> \mu_i </math> is the [[Chemical potential|chemical potential]] of the species <math>i</math>
*<math> N_i </math> is the number of molecules of the species "i"
*<math> N_i </math> is the number of molecules of the species <math>i</math>


== Semi-grand ensemble at fixed volume and temperature ==
== Semi-grand ensemble at fixed volume and temperature ==


Consider now that we want to consider a system with fixed total number of particles, <math> N </math>
Consider now that we wish to consider a system with fixed total number of particles, <math> N </math>


: <math> \left. N = \sum_{i=1}^c N_i  \right. </math>;  
: <math> \left. N = \sum_{i=1}^c N_i  \right. </math>;  


but the composition can change, from the thermodynamics we can apply a Legendre's transform [HAVE TO CHECK ACCURACY]
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 <math> A (T,V,N_1,N_2) </math>.  
to the differential equation written above in terms of <math> A (T,V,N_1,N_2) </math>.  


# Consider the variable change <math> N_1 \rightarrow N </math> i.e.: <math> \left. N_1 = N-  \sum_{i=2}^c N_i  \right. </math>
* Consider the variable change <math> N_1 \rightarrow N </math> i.e.: <math> \left. N_1 = N-  \sum_{i=2}^c N_i  \right. </math>




: <math> 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_2 d N_2; </math>
: <math> 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; </math>




: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V +  \beta \mu_1 d N + \sum_{i=2}^c \beta (\mu_2-\mu_i) d N_i; </math>
: <math> 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; </math>


Or:
or,


: <math> 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; </math>
: <math> 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; </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>
where <math> \left. \mu_{i1} \equiv  \mu_i - \mu_1 \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>


== Fixed pressure and temperature ==  
== Fixed pressure and temperature ==  


In the [[Isothermal-Isobaric ensemble]]: <math> (N_1,N_2, \cdots, N_c, p, T) </math> ensemble we can write:
In the [[isothermal-isobaric ensemble]]: <math> (N_1,N_2, \cdots, N_c, p, T) </math> one can write:


<math> d (\beta G) = E d \beta + V d (\beta p) + \sum_{i=1}^c \left( \beta \mu_i \right) d N_i </math>
:<math> d (\beta G) = E d \beta + V d (\beta p) + \sum_{i=1}^c \left( \beta \mu_i \right) d N_i </math>


where:
where:
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* <math> G </math> is the [[Gibbs energy function]]
* <math> G </math> is the [[Gibbs energy function]]


==  Fixed pressure and temperature: Semigrand ensemble ==
==  Fixed pressure and temperature: Semi-grand ensemble ==
 
Following the procedure described above one can write:


Following the procedure described above we can write:
:<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'' thermodynamic potential <math> \beta \Phi </math> is given by:
where the ''new'' thermodynamical 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
- \sum_{i=2}^c N_i d (\beta \mu_{i1} ).
- \sum_{i=2}^c N_i d (\beta \mu_{i1} ).
</math>
</math>


TO BE CONTINUED ... SOON
==  Fixed pressure and temperature: Semi-grand ensemble: partition function ==
 
In the fixed composition ensemble one has:
 
:<math> 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].  
</math>
 
==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]]

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