Grand canonical ensemble: Difference between revisions

Revision as of 10:08, 30 August 2011

The grand-canonical ensemble is particularly well suited to simulation studies of adsorption.

The classical grand canonical partition function for a one-component system in a three-dimensional space is given by:

Q_{μ V T} = \sum_{N = 0}^{\infty} \frac{\exp [β μ N]}{N! Λ^{3 N}} \int d (R^{*})^{3 N} \exp [- β U (V, (R^{*})^{3 N})]

where:

N is the number of particles
$Λ$ is the de Broglie thermal wavelength (which depends on the temperature)
$β = \frac{1}{k_{B} T}$ , with $k_{B}$ being the Boltzmann constant
U is the potential energy, which depends on the coordinates of the particles (and on the interaction model)
${(R^{*})}^{3 N}$ represent the $3 N$ position coordinates of the particles (reduced with the system size): i.e. $\int d (R^{*})^{3 N} = 1$

The corresponding thermodynamic potential, the grand potential, $Ω$ , for the aforementioned grand canonical partition function is:

Ω = A - μ N

,

where A is the Helmholtz energy function. Using the relation

U = T S - P V + μ N

one arrives at

Ω = - P V

i.e.:

p V = k_{B} T \log Q_{μ V T}

@@ Line 7: / Line 7: @@
 The classical grand canonical partition function for a one-component system in a three-dimensional space is given by:
-:<math> Q_{\mu VT} = \sum_{N=0}^{\infty} \frac{ \exp \left[ \beta \mu N \right] V^N}{N! \Lambda^{3N} } \int  d (R^*)^{3N} \exp \left[ - \beta U \left( V, (R^*)^{3N} \right) \right] </math>
+:<math> Q_{\mu VT} = \sum_{N=0}^{\infty} \frac{ \exp \left[ \beta \mu N \right]}{N! \Lambda^{3N} } \int  d (R^*)^{3N} \exp \left[ - \beta U \left( V, (R^*)^{3N} \right) \right] </math>
 where:
@@ Line 16: / Line 16: @@
 * ''U'' is the potential energy, which depends on the coordinates of the particles (and on the [[models | interaction model]])
 * <math> \left( R^*\right)^{3N} </math> represent the <math>3N</math> position coordinates of the particles (reduced with the system size): i.e. <math> \int d (R^*)^{3N} = 1 </math>
 == Helmholtz energy and partition function ==
 The  corresponding thermodynamic potential, the '''grand potential''', <math>\Omega</math>,