Grand canonical ensemble

2011-08-30T10:08:43Z

82.53.107.46: /* Grand canonical partition function */

The '''grand-canonical ensemble''' is particularly well suited to simulation studies of adsorption.
== Ensemble variables ==
* [[Chemical potential]], <math> \left. \mu \right. </math>
* Volume, <math> \left. V \right. </math>
* [[Temperature]], <math> \left. T \right. </math>
== Grand canonical partition function ==
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]}{N! \Lambda^{3N} } \int d (R^*)^{3N} \exp \left[ - \beta U \left( V, (R^*)^{3N} \right) \right] </math>

where:

* ''N'' is the number of particles
* <math> \left. \Lambda \right. </math> is the [[de Broglie thermal wavelength]] (which depends on the temperature)
* <math> \beta = \frac{1}{k_B T} </math>, with <math> k_B </math> being the [[Boltzmann constant]]
* ''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>,
for the aforementioned grand canonical partition function is:

: <math> \Omega = \left. A - \mu N \right. </math>,
where ''A'' is the [[Helmholtz energy function]].
Using the relation
:<math>\left.U\right.=TS -PV + \mu N</math>
one arrives at
: <math> \left. \Omega \right.= -PV</math>
i.e.:

:<math> \left. p V = k_B T \log Q_{\mu V T } \right. </math>
==See also==
*[[Monte Carlo in the grand-canonical ensemble]]
==References==
#[http://dx.doi.org/10.1103/PhysRev.57.1160 Richard C. Tolman "On the Establishment of Grand Canonical Distributions", Physical Review '''57''' pp. 1160-1168 (1940)]
[[Category:Statistical mechanics]]

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Grand canonical ensemble