Grand potential

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The grand potential is a quantity used in
statistical mechanics In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scien ...
, 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 :$\Phi_ \ \stackrel\ U - T S - \mu N$ where ''U'' is the
internal energy The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...
, ''T'' is the
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement, measured with a thermometer. Thermometers are calibrated in various Conversion of units of temperature, temp ...
of the system, ''S'' is the
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
, μ is the
chemical potential In thermodynamics, the chemical potential of a Chemical specie, species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potent ...
, and ''N'' is the number of particles in the system. The change in the grand potential is given by :$\begin d\Phi_ & = dU - TdS - SdT - \mu dN - Nd\mu \\ & = - P dV - S dT - N d\mu \end$ where ''P'' is
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
and ''V'' is
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
, using the fundamental thermodynamic relation (combined
first First or 1st is the ordinal form of the number 1 (number), one (#1). First or 1st may also refer to: *World record, specifically the first instance of a particular achievement Arts and media Music * 1\$T, American rapper, singer-songwriter, D ...
and
second The second (symbol: s) is the unit of Time in physics, time in the International System of Units (SI), historically defined as of a day – this factor derived from the division of the day first into 24 hours, then to 60 minutes and finally t ...
thermodynamic laws); :$dU = TdS - PdV + \mu dN$ 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.

## Landau free energy

Some authors refer to the grand potential as the ''Landau free energy'' or Landau potential and write its definition as: :$\Omega \ \stackrel\ F - \mu N = U - T S - \mu N$ named after Russian physicist
Lev Landau Lev Davidovich Landau (russian: Лев Дави́дович Ланда́у; 22 January 1908 – 1 April 1968) was a Soviet Union, Soviet-Azerbaijan, Azerbaijani physicist of Jewish descent who made fundamental contributions to many areas of th ...
, which may be a synonym for the grand potential, depending on system stipulations. For homogeneous systems, one obtains $\Omega = -PV$.

# 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 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: :$\left\left(\frac\right\right)_ = 0,$ and all extensive quantities (particle number, energy, entropy, potentials, ...) must grow linearly with volume, e.g. :$\left\left(\frac\right\right)_ = \frac.$ In this case we simply have $\Phi_ = - \langle P\rangle V$, as well as the familiar relationship $G = \langle N \rangle \mu$ for the
Gibbs free energy In thermodynamics, the Gibbs free energy (or Gibbs energy; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work (physics), work that may be performed by a closed system, thermodynamically closed system a ...
. The value of $\Phi_$ 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 $\Phi_ = - \langle P\rangle V$ 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. 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), $\Phi_ \neq - \langle P\rangle V$.

*
Gibbs energy In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these qua ...
* Helmholtz energy

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