In statistical mechanics, the grand canonical ensemble (also known as the macrocanonical ensemble) is the statistical ensemble that is used to represent the possible states of a mechanical system of particles that are in thermodynamic equilibrium (thermal and chemical) with a reservoir. 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 are chemical potential (symbol: ) and absolute temperature (symbol: . The ensemble is also dependent on mechanical variables such as volume (symbol: , which influence the nature of the system's internal states. This ensemble is therefore sometimes called the ensemble, as each of these three quantities are constants of the ensemble.
Basics
In simple terms, the grand canonical ensemble assigns a probability to each distinct microstate given by the following exponential:
<math display="block">P = e^,</math>
where is the number of particles in the microstate and is the total energy of the microstate. is the Boltzmann constant.
The number is known as the grand potential and is constant for the ensemble. However, the probabilities and will vary if different are selected. The grand potential 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 function .
In the case where more than one kind of particle is allowed to vary in number, the probability expression generalizes to
<math display="block">P = e^,</math>
where is the chemical potential for the first kind of particles, is the number of that kind of particle in the microstate, is the chemical potential for the second kind of particles and so on ( is the number of distinct kinds of particles). However, these particle numbers should be defined carefully (see the note on particle number conservation below).
The distribution of the grand canonical ensemble is called generalized Boltzmann distribution by some authors.
Grand ensembles are apt for use when describing systems such as the electrons in a conductor, or the photons 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 a dark surface, in these cases). The grand canonical ensemble provides a natural setting for an exact derivation of the Fermi–Dirac statistics or Bose–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 <math>\textstyle P = \frac{1}{\mathcal Z} e^{(\mu N-E)/(k T)}</math>, using the grand partition function <math>\textstyle \mathcal Z = e^{-\Omega/(k T)}</math> 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.
Applicability
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 normal canonical ensemble, and can be found in Reif). 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 the macroscopic limit).
The condition that the system is isolated is necessary in order to ensure it has well-defined thermodynamic quantities and evolution. 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.
Properties
= \operatorname{Tr} \exp\left(\tfrac{1}{kT}\left(\mu_1 \hat N_1 + \dots + \mu_s \hat N_s - \hat H\right)\right).</math>
Note that for the grand ensemble, the basis states of the operators , , etc. are all states with multiple particles in Fock 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 using bra–ket notation, since it is possible (given the mutually commuting nature of the energy and particle number operators) to find a complete basis of simultaneous eigenstates , indexed by , where , , and so on. Given such an eigenbasis, the grand canonical ensemble is simply
<math display="block">\hat \rho = \sum_i e^{\frac{\Omega + \mu_1 N_{1,i} + \dots + \mu_s N_{s,i} - E_i}{k T |\psi_i\rangle \langle \psi_i | </math>
<math display="block">e^{-\frac{\Omega}{k T = \sum_i e^{\frac{\mu_1 N_{1,i} + \dots + \mu_s N_{s,i} - E_i}{k T.</math>
where the sum is over the complete set of states with state having total energy, particles of type 1, particles of type 2, and so on.
Classical mechanical
In classical mechanics, a grand ensemble is instead represented by a joint probability density function defined over multiple phase spaces of varying dimensions, , where the and are the canonical 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 the canonical ensemble since:
- is an overcounting correction factor (see below), a function of .
Again, the value of is determined by demanding that is a normalized probability density function:
<math display="block">e^{-\frac{\Omega}{k T = \sum_{N_1 = 0}^{\infty} \cdots \sum_{N_s = 0}^{\infty} \int \cdots \int \frac{1}{h^n C} e^{\frac{\mu_1 N_1 + \cdots + \mu_s N_s - E}{k T \, dp_1 \cdots dq_n </math>
This integral is taken over the entire available phase space for the given numbers of particles.
Overcounting correction
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 factor above is simply . 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 the Gibbs 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. In order to incorporate this fact, integrals are still carried over full phase space but the result is divided by
<math display="block">C = N_1! N_2! \cdots N_s!,</math>
which is the number of different permutations possible. The division by neatly corrects the overcounting that occurs in the integral over all phase space.
It is of course possible to include distinguishable types of particles in the grand canonical ensemble—each distinguishable type <math>i</math> is tracked by a separate particle counter <math>N_i</math> and chemical potential <math>\mu_i</math>. 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.
See also
- Microcanonical ensemble
- Canonical ensemble