Canonical ensemble

Canonical Ensemble:

Variables:

• Number of Particles, $N$

• Volume, $V$

• Temperature, $T$

Partition Function

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

$Q_{NVT}={\frac {V^{N}}{N!\Lambda ^{3N}}}\int d(R^{*})^{3N}\exp \left[-\beta U\left(V,(R^{*})^{3N}\right)\right]$

where:

• $\Lambda$ is the de Broglie wavelength (depends on the temperature)

• Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \beta ={\frac {1}{k_{B}T}}} , with Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle k_{B}} being the Boltzmann constant

• $U$ 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. $\int d(R^{*})^{3N}=1$

Free energy and Partition Function

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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A\left(N,V,T\right)=-k_{B}T\log Q_{NVT}}