Grand potential

Statistical mechanics

Thermodynamics Kinetic theory
Particle Statistics Spin-Statistics Theorem Identical 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 µVT Open statistical
Models Debye Einstein Ising Potts
Potentials Internal energy Enthalpy Helmholtz free energy Gibbs free energy Grand potential / Landau free energy
Scientists Maxwell Boltzmann Gibbs Einstein Ehrenfest von Neumann Tolman Debye Fermi Bose

The grand potential 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

\Phi_{G} \ \stackrel{\mathrm{def}}{=}\ U - T S - \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

\begin{align} d\Phi_{G} & = dU - TdS - SdT - \mu dN - Nd\mu \\ & = - P dV - S dT - N d\mu \end{align}

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

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 Landau free energy or Landau potential as:^[1]^[2]

\Omega \ \stackrel{\mathrm{def}}{=}\ F - \mu N = U - T S - \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 $\Omega = -PV \,\;$

Grand potential for homogeneous systems (vs. inhomogeneous systems)

In the case of a scale-invariant type of system (where a system of volume $\lambda V$ has exactly the same set of microstates as $\lambda$ systems of volume $V$ ), then when we grow the system 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: $(\partial \langle P \rangle/\partial V)_{\mu,T} = 0$ , and the particle and all extensive quantities (particle number, energy, entropy, potentials, ...) must grow linearly with volume, e.g., $(\partial \langle N \rangle/\partial V)_{\mu,T} = N/V$ . In this case we have simply $\Phi_{G} = - \langle P\rangle V$ , as well as the familiar relationship $G = \langle N \rangle \mu$ for the Gibbs free energy. The value of $\Phi_{G}$ can be understood as the work we can extract from the system by shrinking it down to nothing (putting all the particles and energy back into the reservoir). The fact that $\Phi_{G} = - \langle P\rangle V$ is negative implies that it takes energy to perform this extraction.

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.^[3] 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), $\Phi_{G} \neq - \langle P\rangle V$ .^[4]

Ideal gas

For an ideal gas,

\Phi_{G} = - k_{B} T \ln(\Xi) = - k_{B} T Z_{1} e^{\beta \mu}

where Ξ is the grand partition function, k_B is Boltzmann constant, Z₁ is the partition function for 1 particle and β = 1/k_BT is the inverse temperature. The factor e^βμ is the Boltzmann factor.

References

↑ Lee, J. Chang (2002). "5". Thermal Physics - Entropy and Free Energies. New Jersey: World Scientific.
↑ Reference on "Landau potential" is found in the book: D. Goodstein. States of Matter. p. 19.
↑ Brachman, M. K. (1954). "Fermi Level, Chemical Potential, and Gibbs Free Energy". The Journal of Chemical Physics 22 (6): 1152–1151. doi:10.1063/1.1740312.
↑ Hill, Terrell L. (2002). Thermodynamics of Small Systems. Courier Dover Publications. ISBN 9780486495095.

External links

Grand Potential (Manchester University)

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