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: $Q_{NVT}$

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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

• $\beta ={\frac {1}{k_{B}T}}$, with $k_{B}$ being the Boltzmann constant,

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U } is the potential energy, which depends on the coordinates of the particles (and on the interaction model)

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( R^*\right)^{3N} } represent the 3N position coordinates of the particles (reduced with the system size): i.e. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int d (R^*)^{3N} = 1 }

Free energy

The Helmholtzz free energy is related to the canonical partition function as:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F\left(N,V,T \right) = - \log Q_{NVT} }