Grand potential

Statistical mechanics
Thermodynamics Kinetic theory
Particle statistics Spin–statistics theorem Indistinguishable particles Maxwell–Boltzmann Bose–Einstein Fermi–Dirac Parastatistics Anyonic statistics Braid statistics
Thermodynamic ensembles NVE Microcanonical NVT Canonical µVT Grand canonical NPH Isoenthalpic–isobaric NPT Isothermal–isobaric
Models Debye Einstein Ising Potts
Potentials Internal energy Enthalpy Helmholtz free energy Gibbs free energy Grand potential / Landau free energy
Scientists Maxwell Boltzmann Bose Gibbs Einstein Ehrenfest von Neumann Tolman Debye Fermi
v t e

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.

Definition

Grand potential is defined by

${\displaystyle \Phi _{\rm {G}}\ {\stackrel {\mathrm {def} }{=}}\ U-TS-\mu N}$

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.

The change in the grand potential is given by

${\displaystyle {\begin{aligned}d\Phi _{\rm {G}}&=dU-TdS-SdT-\mu dN-Nd\mu \\&=-PdV-SdT-Nd\mu \end{aligned}}}$

where P is pressure and V is volume, using the fundamental thermodynamic relation (combined first and second thermodynamic laws);

${\displaystyle dU=TdS-PdV+\mu dN}$

When the system is in thermodynamic equilibrium, Φ_G is a minimum. This can be seen by considering that dΦ_G is zero if the volume is fixed and the temperature and chemical potential have stopped evolving.

Landau free energy

Some authors refer to the grand potential as the Landau free energy or Landau potential and write its definition as:^[1][2]

${\displaystyle \Omega \ {\stackrel {\mathrm {def} }{=}}\ F-\mu N=U-TS-\mu N}$

named after Russian physicist Lev Landau, which may be a synonym for the grand potential, depending on system stipulations. For homogeneous systems, one obtains ${\displaystyle \Omega =-PV}$.^[3]

Homogeneous systems (vs. inhomogeneous systems)

In the case of a scale-invariant type of system (where a system of volume ${\displaystyle \lambda V}$ has exactly the same set of microstates as ${\displaystyle \lambda }$ systems of volume ${\displaystyle V}$), then when the system expands new particles and energy will flow in from the reservoir to fill the new volume with a homogeneous extension of the original system. The pressure, then, must be constant with respect to changes in volume:

${\displaystyle \left({\frac {\partial \langle P\rangle }{\partial V}}\right)_{\mu ,T}=0,}$

and all extensive quantities (particle number, energy, entropy, potentials, ...) must grow linearly with volume, e.g.

${\displaystyle \left({\frac {\partial \langle N\rangle }{\partial V}}\right)_{\mu ,T}={\frac {N}{V}}.}$

In this case we simply have ${\displaystyle \Phi _{\rm {G}}=-\langle P\rangle V}$, as well as the familiar relationship ${\displaystyle G=\langle N\rangle \mu }$ for the Gibbs free energy. The value of ${\displaystyle \Phi _{\rm {G}}}$ 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). The fact that ${\displaystyle \Phi _{\rm {G}}=-\langle P\rangle V}$ is negative implies that the extraction of particles from the system to the reservoir requires energy input.

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), ${\displaystyle \Phi _{G}\neq -\langle P\rangle V}$.^[5]

See also

• Gibbs energy
• Helmholtz energy

References

External links

• Grand Potential (Manchester University)

This page was last edited on 12 February 2023, at 11:46