Grand canonical potential

The grand canonical potential (symbol in part also , or ; also Landau potential according to Lew Landau ) is a thermodynamic potential used in statistical mechanics , which is mainly used for irreversible processes in open systems . It is the adjusted thermodynamic potential for the μVT ensemble . ${\displaystyle \Omega }$ ${\displaystyle \Phi _{G}}$ ${\displaystyle J}$ ${\displaystyle K}$

definition

The grand canonical potential is defined by:

${\displaystyle \Omega :=F-\mu N=U-TS-\mu N}$

With

• of free energy ${\displaystyle F}$
• the chemical potential ${\displaystyle \mu }$
• the number of particles in the system ${\displaystyle N}$
• the inner energy ${\displaystyle U}$
• the absolute temperature of the system ${\displaystyle T}$
• of entropy . ${\displaystyle S}$

Alternatively, the grand-canonical potential can be defined using the grand-canonical partition function : ${\displaystyle {\mathcal {Z}}}$

${\displaystyle \Omega =-{\frac {1}{\beta }}\cdot \ln {\mathcal {Z}},}$

in which ${\displaystyle \beta ={\frac {1}{k_{\mathrm {B} }\cdot T}}}$

with the Boltzmann constant . ${\displaystyle k_{\mathrm {B} }}$

Because of the thermodynamic Euler equation , the grand-canonical potential is identical to

${\displaystyle \Omega =-pV}$

With

• the pressure ${\displaystyle p}$
• the volume of the system. ${\displaystyle V}$

An infinitesimal change in the grand-canonical potential is given by

${\displaystyle \mathrm {d} \Omega =-p\cdot \mathrm {d} V-S\cdot \mathrm {d} T-N\cdot \mathrm {d} \mu }$

At constant temperature ( ) and constant chemical potential ( ), the grand canonical potential of a thermodynamic system, which is left to its own devices without any work ( ), tends to a minimum ( ). ${\displaystyle \mathrm {d} T=0}$ ${\displaystyle \mathrm {d} \mu =0}$ ${\displaystyle -p\cdot \mathrm {d} V=0}$ ${\displaystyle \mathrm {d} \Omega =0}$

According to the above equation, the thermodynamic quantities entropy, pressure and number of particles can be obtained as follows:

${\displaystyle {\begin{pmatrix}S\\p\\N\end{pmatrix}}=-{\begin{pmatrix}\partial _{T}\\\partial _{V}\\\partial _{\mu }\end{pmatrix}}\Omega (T,V,\mu )}$

See also

• Gibbs energy