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Instatistical mechanics, thegrand canonical ensemble (also known as themacrocanonical ensemble) is thestatistical ensemble that is used to represent the possible states of a mechanical system of particles that are inthermodynamic equilibrium (thermal and chemical) with a reservoir.^[1] The system is said to be open in the sense that the system can exchange energy and particles with a reservoir, so that various possible states of the system can differ in both their total energy and total number of particles. The system's volume, shape, and other external coordinates are kept the same in all possible states of the system.

The thermodynamic variables of the grand canonical ensemble arechemical potential (symbol:µ) andabsolute temperature (symbol:T). The ensemble is also dependent on mechanical variables such as volume (symbol:V), which influence the nature of the system's internal states. This ensemble is therefore sometimes called theµVT ensemble, as each of these three quantities are constants of the ensemble.

In simple terms, the grand canonical ensemble assigns a probabilityP to each distinctmicrostate given by the following exponential:$${\displaystyle P=e^{(\mathrm{\Omega }+\mu N-E)/(kT)},}$$whereN is the number of particles in the microstate andE is the total energy of the microstate.k is theBoltzmann constant.

The numberΩ is known as thegrand potential and is constant for the ensemble. However, the probabilities andΩ will vary if differentµ,V,T are selected. The grand potentialΩ serves two roles: to provide a normalization factor for the probability distribution (the probabilities, over the complete set of microstates, must add up to one); and, many important ensemble averages can be directly calculated from the functionΩ(µ,V,T).

In the case where more than one kind of particle is allowed to vary in number, the probability expression generalizes to$${\displaystyle P=e^{(\mathrm{\Omega }+{\mu }_1{N}_1+{\mu }_2{N}_2+\cdots +{\mu }_s{N}_s-E)/(kT)},}$$whereµ₁ is the chemical potential for the first kind of particles,N₁ is the number of that kind of particle in the microstate,µ₂ is the chemical potential for the second kind of particles and so on (s is the number of distinct kinds of particles). However, these particle numbers should be defined carefully (see thenote on particle number conservation below).

The distribution of the grand canonical ensemble is calledgeneralized Boltzmann distribution by some authors.^[2]

Grand ensembles are apt for use when describing systems such as theelectrons in aconductor, or thephotons in a cavity, where the shape is fixed but the energy and number of particles can easily fluctuate due to contact with a reservoir (e.g., an electrical ground or adark surface, in these cases). The grand canonical ensemble provides a natural setting for an exact derivation of theFermi–Dirac statistics orBose–Einstein statistics for a system of non-interacting quantum particles (see examples below).

Note on formulation

An alternative formulation for the same concept writes the probability as${\displaystyle P=\frac{1}{\mathcal{Z}}{e}^{(\mu N-E)/(kT)}}$, using thegrand partition function${\displaystyle \mathcal{Z}=e^{-\mathrm{\Omega }/(kT)}}$ rather than the grand potential. The equations in this article (in terms of grand potential) may be restated in terms of the grand partition function by simple mathematical manipulations.

The grand canonical ensemble is the ensemble that describes the possible states of an isolated system that is in thermal and chemical equilibrium with a reservoir (the derivation proceeds along lines analogous to the heat bath derivation of the normalcanonical ensemble, and can be found in Reif^[3]). The grand canonical ensemble applies to systems of any size, small or large; it is only necessary to assume that the reservoir with which it is in contact is much larger (i.e., to take themacroscopic limit).

The condition that the system is isolated is necessary in order to ensure it has well-defined thermodynamic quantities and evolution.^[1] In practice, however, it is desirable to apply the grand canonical ensemble to describe systems that are in direct contact with the reservoir, since it is that contact that ensures the equilibrium. The use of the grand canonical ensemble in these cases is usually justified either 1) by assuming that the contact is weak, or 2) by incorporating a part of the reservoir connection into the system under analysis, so that the connection's influence on the region of interest is correctly modeled. Alternatively, theoretical approaches can be used to model the influence of the connection, yielding an open statistical ensemble.

Another case in which the grand canonical ensemble appears is when considering a system that is large and thermodynamic (a system that is "in equilibrium with itself"). Even if the exact conditions of the system do not actually allow for variations in energy or particle number, the grand canonical ensemble can be used to simplify calculations of some thermodynamic properties. The reason for this is that various thermodynamic ensembles (microcanonical,canonical) become equivalent in some aspects to the grand canonical ensemble, once the system is very large.^{[note 1]} Of course, for small systems, the different ensembles are no longer equivalent even in the mean. As a result, the grand canonical ensemble can be highly inaccurate when applied to small systems of fixed particle number, such as atomic nuclei.^[4]

The partial derivatives of the functionΩ(µ₁, …,µ_s,V,T) give important grand canonical ensemble average quantities:^[1][6]

Exact differential: From the above expressions, it can be seen that the functionΩ has theexact differential$${\displaystyle d\mathrm{\Omega }=-SdT-⟨N_1⟩d{\mu }_1\dots -⟨N_s⟩d{\mu }_s-⟨p⟩dV.}$$

First law of thermodynamics: Substituting the above relationship for⟨E⟩ into the exact differential ofΩ, an equation similar to thefirst law of thermodynamics is found, except with average signs on some of the quantities:^[1]$${\displaystyle d⟨E⟩=TdS+{\mu }_{1}d⟨N_1⟩+\cdots +{\mu }_{s}d⟨N_s⟩-⟨p⟩dV.}$$

Thermodynamic fluctuations: Thevariances in energy and particle numbers are^[7][8]

Correlations in fluctuations: Thecovariances of particle numbers and energy are^[1]

The usefulness of the grand canonical ensemble is illustrated in the examples below. In each case the grand potential is calculated on the basis of the relationship$${\displaystyle \mathrm{\Omega }=-kT\ln \left(\sum _{\text{microstates}}{e}^{(\mu N-E)/(kT)}\right)}$$which is required for the microstates' probabilities to add up to 1.

In the special case of a quantum system of manynon-interacting particles, the thermodynamics are simple to compute.^[9]Since the particles are non-interacting, one can compute a series of single-particlestationary states, each of which represent a separable part that can be included into the total quantum state of the system.For now let us refer to these single-particle stationary states asorbitals (to avoid confusing these "states" with the total many-body state), with the provision that each possible internal particle property (spin orpolarization) counts as a separate orbital.Each orbital may be occupied by a particle (or particles), or may be empty.

Since the particles are non-interacting, we may take the viewpoint thateach orbital forms a separate thermodynamic system.Thus each orbital is a grand canonical ensemble unto itself, one so simple that its statistics can be immediately derived here. Focusing on just one orbital labelledi, the total energy for amicrostate ofN_i particles in this orbital will beN_iϵ_i, whereϵ_i is the characteristic energy level of that orbital. The grand potential for the orbital is given by one of two forms, depending on whether the orbital is bosonic or fermionic:

In each case the value${\displaystyle ⟨N_i⟩=-\frac{\mathrm{\partial }{\mathrm{\Omega }}_i}{\mathrm{\partial }\mu }}$ gives the thermodynamic average number of particles on the orbital: theFermi–Dirac distribution for fermions, and theBose–Einstein distribution for bosons.Considering again the entire system, the total grand potential is found by adding up theΩ_i for all orbitals.

In classical mechanics it is also possible to consider indistinguishable particles (in fact, indistinguishability is a prerequisite for defining a chemical potential in a consistent manner; all particles of a given kind must be interchangeable^[1]). We can consider a region of the single-particle phase space with approximately uniform energyϵ_i to be an "orbital" labelledi.

Two complications arise since this orbital actually encompasses many (infinite) distinct states. Briefly:

• An overcounting correction of1/N_i! is needed since themany-particle phase space containsN_i! copies of the same actual state (formed by the permutation of the particles' different exact states).
• The chosen width of the orbital is arbitrary, thus there is a further proportionality factor that is independent ofN_i .

Due to the1/N_i! overcounting correction, the summation now takes the form of an exponentialpower series,the value${\displaystyle ⟨N_i⟩\propto -\frac{\mathrm{\partial }{\mathrm{\Omega }}_i}{\mathrm{\partial }\mu }}$ corresponding toMaxwell–Boltzmann statistics.

The grand canonical ensemble can be used to predict whether an atom prefers to be in a neutral state or ionized state.An atom is able to exist in ionized states with more or fewer electrons compared to neutral. As shown below, ionized states may be thermodynamically preferred depending on the environment.Consider a simplified model where the atom can be in a neutral state or in one of two ionized states (a detailed calculation also includes excited states and the degeneracy factors of the states^[10][11]):

• charge neutral state, withN₀ electrons and energyE₀.
• anoxidized state (N₀ − 1 electrons) with energyE₀ + ΔE_I +qϕ
• areduced state (N₀ + 1 electrons) with energyE₀ − ΔE_A −qϕ

HereΔE_I andΔE_A are the atom'sionization energy andelectron affinity, respectively;ϕ is the localelectrostatic potential in the vacuum nearby the atom, and−q is theelectron charge.

The grand potential in this case is thus determined byThe quantity−qϕ −µ is critical in this case, for determining the balance between the various states. This value is determined by the environment around the atom.

• If a lone atom is placed in metal box, then−qϕ −µ =W, thework function of the box lining material. If${\displaystyle W>\mathrm{\Delta }{E}_{\mathrm{I}}}$ then the atom prefers to exist as a positive ion. This spontaneoussurface ionization effect has been used as a cesiumion source.^[12]
• If the atom is surrounded by an ideal electron gas with density${\displaystyle n_{\mathrm{e}}}$, then${\displaystyle \exp ((\mu +q\varphi )/kT)=n_{\mathrm{e}}{\lambda }_{\mathrm{t}\mathrm{h}}^3/2}$, and substituting this in yields theSaha ionization equation.^[10]
• Insemiconductors, where the ionization of adopant atom is well described by this ensemble.^[11] In the semiconductor, theconduction band edgeϵ_C plays the role of the vacuum energy level (replacing−qϕ), andµ is known as theFermi level. Of course, the ionization energy and electron affinity of the dopant atom are strongly modified relative to their vacuum values. A typical donor dopant in silicon, phosphorus, hasΔE_I =45 meV.^[13]

In order for a particle number to have an associated chemical potential, it must be conserved during the internal dynamics of the system, and only able to change when the system exchanges particles with an external reservoir.

If the particles can be created out of energy during the dynamics of the system, then an associatedµN term must not appear in the probability expression for the grand canonical ensemble. In effect, this is the same as requiring thatµ = 0 for that kind of particle. Such is the case forphotons in a black cavity, whose number regularly change due to absorption and emission on the cavity walls. (On the other hand, photons in a highly reflective cavity can be conserved and caused to have a nonzeroµ.^[14])

In some cases the number of particles is not conserved and theN represents a more abstract conserved quantity:

• Chemical reactions: Chemical reactions can convert one type of molecule to another; if reactions occur then theN_i must be defined such that they do not change during the chemical reaction.
• High energy particle physics: Ordinary particles can be spawned out of pure energy, if a correspondingantiparticle is created. If this sort of process is allowed, then neither the number of particles nor antiparticles are conserved. Instead,N = (particle number − antiparticle number) is conserved.^{[15][note 3]} As particle energies increase, there are more possibilities to convert between particle types, and so there are fewer numbers that are truly conserved. At the very highest energies, the only conserved numbers areelectric charge,weak isospin, andbaryon–lepton number difference.

On the other hand, in some cases a single kind of particle may have multiple conserved numbers:

• Closed compartments: In a system composed of multiple compartments that share energy but do not share particles, it is possible to set the chemical potentials separately for each compartment. For example, acapacitor is composed of two isolated conductors and is charged by applying a difference inelectron chemical potential.
• Slow equilibration: In some quasi-equilibrium situations it is possible to have two distinct populations of the same kind of particle in the same location, which are each equilibrated internally but not with each other. Though not strictly in equilibrium, it may be useful to name quasi-equilibrium chemical potentials which can differ among the different populations. Examples: (semiconductor physics) distinctquasi-Fermi levels (electron chemical potentials) in theconduction band andvalence band; (spintronics) distinct spin-up and spin-down chemical potentials; (cryogenics) distinctparahydrogen andorthohydrogen chemical potentials.

The precise mathematical expression for statistical ensembles has a distinct form depending on the type of mechanics under consideration (quantum or classical), as the notion of a "microstate" is considerably different. In quantum mechanics, the grand canonical ensemble affords a simple description since diagonalization provides a set of distinctmicrostates of a system, each with well-defined energy and particle number. The classical mechanical case is more complex as it involves not stationary states but instead an integral over canonicalphase space.

A statistical ensemble in quantum mechanics is represented by adensity matrix, denoted by${\displaystyle \hat{\rho }}$. The grand canonical ensemble is the density matrix^{[citation needed]}$${\displaystyle \hat{\rho }=\exp \left(\frac{1}{kT}\left(\mathrm{\Omega }+{\mu }_1{\hat{N}}_1+\cdots +{\mu }_s{\hat{N}}_s-\hat{H}\right)\right),}$$whereĤ is the system's total energy operator (Hamiltonian),N̂₁ is the system's totalparticle number operator for particles of type 1,N̂₂ is the totalparticle number operator for particles of type 2, and so on.exp is thematrix exponential operator. The grand potentialΩ is determined by the probability normalization condition that the density matrix has atrace of one,${\displaystyle Tr\hat{\rho }=1}$:$${\displaystyle e^{-\frac{\mathrm{\Omega }}{kT}}=\mathrm{Tr}\exp \left(\frac{1}{kT}\left({\mu }_1{\hat{N}}_1+\cdots +{\mu }_s{\hat{N}}_s-\hat{H}\right)\right).}$$

Note that for the grand ensemble, the basis states of the operatorsĤ,N̂₁, etc. are all states withmultiple particles inFock space, and the density matrix is defined on the same basis. Since the energy and particle numbers are all separately conserved, these operators are mutually commuting.

The grand canonical ensemble can alternatively be written in a simple form usingbra–ket notation, since it is possible (given the mutually commuting nature of the energy and particle number operators) to find a completebasis of simultaneous eigenstates|ψ_i⟩, indexed byi, whereĤ|ψ_i⟩ =E_i|ψ_i⟩,N̂₁|ψ_i⟩ =N_1,i|ψ_i⟩, and so on. Given such an eigenbasis, the grand canonical ensemble is simply$${\displaystyle \hat{\rho }=\sum _i{e}^{\frac{\mathrm{\Omega }+{\mu }_1{N}_{1,i}+\cdots +{\mu }_s{N}_{s,i}-E_i}{kT}}|{\psi }_i⟩⟨{\psi }_i|}$$$${\displaystyle e^{-\frac{\mathrm{\Omega }}{kT}}=\sum _i{e}^{\frac{{\mu }_1{N}_{1,i}+\cdots +{\mu }_s{N}_{s,i}-E_i}{kT}}.}$$where the sum is over the complete set of states with statei havingE_i total energy,N_1,i particles of type 1,N_2,i particles of type 2, and so on.

In classical mechanics, a grand ensemble is instead represented by ajoint probability density function defined over multiplephase spaces of varying dimensions,ρ(N₁, …N_s,p₁, …p_n,q₁, …q_n), where thep₁, …p_n andq₁, …q_n are thecanonical coordinates (generalized momenta and generalized coordinates) of the system's internal degrees of freedom. The expression for the grand canonical ensemble is somewhat more delicate than thecanonical ensemble since:^[1]

• The number of particles and thus the number of coordinatesn varies between the different phase spaces, and,
• it is vital to consider whether permuting similar particles counts as a distinct state or not.

In a system of particles, the number of degrees of freedomn depends on the number of particles in a way that depends on the physical situation. For example, in a three-dimensional gas of monoatomsn = 3N, however in molecular gases there will also be rotational and vibrational degrees of freedom.

The probability density function for the grand canonical ensemble is:$${\displaystyle \rho =\frac{1}{h^{n}C}{e}^{\frac{\mathrm{\Omega }+{\mu }_1{N}_1+\cdots +{\mu }_s{N}_s-E}{kT}},}$$where

• E is the energy of the system, a function of the phase(N₁, …N_s,p₁, …p_n,q₁, …q_n),
• h is an arbitrary but predetermined constant with the units ofenergy×time, setting the extent of one microstate and providing correct dimensions toρ.^{[note 4]}
• C is an overcounting correction factor (see below), a function ofN₁, …N_s.

Again, the value ofΩ is determined by demanding thatρ is a normalized probability density function:$${\displaystyle e^{-\frac{\mathrm{\Omega }}{kT}}=\sum _{N_1=0}^{\mathrm{\infty }}\cdots \sum _{N_s=0}^{\mathrm{\infty }}\int \cdots \int \frac{1}{h^{n}C}{e}^{\frac{{\mu }_1{N}_1+\cdots +{\mu }_s{N}_s-E}{kT}}d{p}_1\cdots d{q}_n}$$This integral is taken over the entire availablephase space for the given numbers of particles.

A well-known problem in the statistical mechanics of fluids (gases, liquids, plasmas) is how to treat particles that are similar or identical in nature: should they be regarded as distinguishable or not? In the system's equation of motion each particle is forever tracked as a distinguishable entity, and yet there are also valid states of the system where the positions of each particle have simply been swapped: these states are represented at different places in phase space, yet would seem to be equivalent.

If the permutations of similar particles are regarded to count as distinct states, then the factorC above is simplyC = 1. From this point of view, ensembles include every permuted state as a separate microstate. Although appearing benign at first, this leads to a problem of severely non-extensive entropy in the canonical ensemble, known today as theGibbs paradox. In the grand canonical ensemble a further logical inconsistency occurs: the number of distinguishable permutations depends not only on how many particles are in the system, but also on how many particles are in the reservoir (since the system may exchange particles with a reservoir). In this case the entropy and chemical potential are non-extensive but also badly defined, depending on a parameter (reservoir size) that should be irrelevant.

To solve these issues it is necessary that the exchange of two similar particles (within the system, or between the system and reservoir) must not be regarded as giving a distinct state of the system.^{[1][note 5]} In order to incorporate this fact, integrals are still carried over full phase space but the result is divided by$${\displaystyle C=N_1!N_2!\cdots N_s!,}$$which is the number of different permutations possible. The division byC neatly corrects the overcounting that occurs in the integral over all phase space.

It is of course possible to include distinguishabletypes of particles in the grand canonical ensemble—each distinguishable type${\displaystyle i}$ is tracked by a separate particle counter${\displaystyle N_i}$ and chemical potential${\displaystyle {\mu }_i}$. As a result, the only consistent way to include "fully distinguishable" particles in the grand canonical ensemble is to consider every possible distinguishable type of those particles, and to track each and every possible type with a separate particle counter and separate chemical potential.

• Microcanonical ensemble
• Canonical ensemble

• Statistical ensembles

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