The grand potential or Landau potential or Landau free energy is a quantity used in statistical mechanics, especially for irreversible processes in open systems.The grand potential is the characteristic state function for the grand canonical ensemble.
Grand potential is defined by
\Phi\rm \stackrel{def
The change in the grand potential is given by
\begin{align} d\Phi\rm&=dU-TdS-SdT-\mudN-Nd\mu\\ &=-PdV-SdT-Nd\mu\end{align}
dU=TdS-PdV+\mudN
When the system is in thermodynamic equilibrium, ΦG is a minimum. This can be seen by considering that dΦG is zero if the volume is fixed and the temperature and chemical potential have stopped evolving.
Some authors refer to the grand potential as the Landau free energy or Landau potential and write its definition as:[1] [2]
\Omega \stackrel{def
named after Russian physicist Lev Landau, which may be a synonym for the grand potential, depending on system stipulations. For homogeneous systems, one obtains
\Omega=-PV
In the case of a scale-invariant type of system (where a system of volume
λV
λ
V
\left( | \partial\langleP\rangle |
\partialV |
\right)\mu,T=0,
and all extensive quantities (particle number, energy, entropy, potentials, ...) must grow linearly with volume, e.g.
\left( | \partial\langleN\rangle |
\partialV |
\right)\mu,T=
N | |
V |
.
In this case we simply have
\Phi\rm=-\langleP\rangleV
G=\langleN\rangle\mu
\Phi\rm
\Phi\rm=-\langleP\rangleV
Such homogeneous scaling does not exist in many systems. For example, when analyzing the ensemble of electrons in a single molecule or even a piece of metal floating in space, doubling the volume of the space does double the number of electrons in the material.[4] The problem here is that, although electrons and energy are exchanged with a reservoir, the material host is not allowed to change.Generally in small systems, or systems with long range interactions (those outside the thermodynamic limit),
\PhiG ≠ -\langleP\rangleV