The grand potential or Landau potential or Landau free energy is a quantity used in statistical mechanics, especially for irreversible processes in open systems.
Grand potential is defined by where U is the internal energy, T is the temperature of the system, S is the entropy, μ is the chemical potential, and N is the number of particles in the system.
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] named after Russian physicist Lev Landau, which may be a synonym for the grand potential, depending on system stipulations.
The pressure, then, must be constant with respect to changes in volume: and all extensive quantities (particle number, energy, entropy, potentials, ...) must grow linearly with volume, e.g.
can be understood as the work that can be extracted from the system by shrinking it down to nothing (putting all the particles and energy back into the reservoir).
is negative implies that the extraction of particles from the system to the reservoir requires energy input.
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.