Grand potential

Definition and physical meaning

For a thermodynamic system with internal energy $U$, temperature $T$, entropy $S$, chemical potential $\mu$, and particle number $N$, the grand potential is the state function

Equivalently, $\Omega = F - \mu N$ in terms of the Helmholtz free energy $F$. It is the natural thermodynamic potential for an open system that can exchange both energy (as heat) and particles with large reservoirs that fix $T$ and $\mu$, while the volume $V$ is controlled.

Differential form and natural variables

For a simple compressible single-component system,

where $p$ is the pressure and $V$ the volume .

Thus $\Omega$ is naturally a function of $(T,V,\mu)$, and it generates:

• $S = -\left(\partial \Omega/\partial T\right)_{V,\mu}$,
• $p = -\left(\partial \Omega/\partial V\right)_{T,\mu}$,
• $N = -\left(\partial \Omega/\partial \mu\right)_{T,V}$.

At fixed $(T,V,\mu)$, equilibrium corresponds to minimizing $\Omega$ (the appropriate analog of minimizing $F$ at fixed $(T,V,N)$).

Pressure relation in the thermodynamic limit

For a uniform macroscopic system satisfying standard thermodynamic limit assumptions, $\Omega$ is extensive in $V$ and one has the important identity

which makes $\Omega$ a convenient generator of the equation of state in grand-canonical settings.

Relation to Legendre transforms and the grand-canonical ensemble

Starting from $F(T,V,N)$, the grand potential is obtained by a Legendre transform that trades the extensive $N$ for its conjugate intensive variable $\mu$.

In statistical mechanics (using the grand-canonical ensemble convention and natural logarithm convention ),

where $\Xi$ is the grand partition function, $k_B$ is the Boltzmann constant , and the temperature can also be packaged via the inverse temperature $\beta$.