Grand canonical ensemble: Difference between revisions

Revision as of 15:13, 28 February 2007

Classical Partition Function (one-component system) in a three-dimensional space: $Q_{\mu VT}$

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]

where:

$\left.\Lambda \right.$ is the de Broglie thermal wavelength (depends on the temperature)

$\left.U\right.$ is the potential energy, which depends on the coordinates of the particles (and on the interaction model)

$\left(R^{*}\right)^{3N}$ represent the 3N position coordinates of the particles (reduced with the system size): i.e. $\int d(R^{*})^{3N}=1$

The Helmholtz energy function is related to the canonical partition function via:

A\left(N,V,T\right)=-k_{B}T\log Q_{NVT}

@@ Line 16: / Line 16: @@
 where:
-* <math> \Lambda </math> is the [[de Broglie thermal wavelength]] (depends on the temperature)
+*<math> \left. N \right. </math> is the number of particles
+* <math> \left. \Lambda \right. </math> is the [[de Broglie thermal wavelength]] (depends on the temperature)
 * <math> \beta = \frac{1}{k_B T} </math>, with <math> k_B </math> being the [[Boltzmann constant]]
-* <math> U </math> is the potential energy, which depends on the coordinates of the particles (and on the interaction model)
+* <math> \left. U \right. </math> is the potential energy, which depends on the coordinates of the particles (and on the interaction model)
 * <math> \left( R^*\right)^{3N} </math> represent the 3N position coordinates of the particles (reduced with the system size): i.e. <math> \int d (R^*)^{3N} = 1 </math>