Grand canonical ensemble: Definition from Answers.com

Statistical mechanics
Microcanonical ensemble
Canonical ensemble
Grand canonical ensemble
Isothermal–isobaric ensemble
Isoenthalpic–isobaric ensemble
edit

In statistical mechanics, a grand canonical ensemble is an imaginary collection of model systems put together to mirror the calculated probability distribution of microscopic states of a given physical system which is being maintained in a given macroscopic state. Assuming such a statistical ensemble consists of an overall collection of N microscopic states, the ensemble is constructed so that the proportion p_i/N of members of the ensemble which are in microscopic state i is proportional to the probability, over time, of finding the real-world system in that microscopic state i. Thus the ensemble is an imaginary static collection of microscopic states created to mirror the statistics of the successive fluctuations of the macroscopic physical system which is being modeled.

The physical system represented by a grand canonical ensemble is in equilibrium with an external reservoir with respect to both particle and energy exchange. This is an extension of the canonical ensemble, in which the physical system being modeled is allowed to exchange energy, but not particles, with its environment. The chemical potential (or fugacity) is introduced to specify the fluctuation of the number of particles, just as temperature is introduced into the canonical ensemble to specify the fluctuation of energy.

It is convenient to use the grand canonical ensemble when the number of particles of the system cannot be easily fixed. Especially in quantum systems, e.g., a collection of bosons or fermions, the number of particles is an intrinsic property (rather than an external parameter) of each quantum state. Moreover, fixing the number of particles will cause certain mathematical inconveniences.

1 The partition function
2 Thermodynamic quantities
3 Statistics of bosons and fermions
4 Quantum mechanical ensemble

The partition function

Classically, the partition function of the grand canonical ensemble is given as a weighted sum of canonical partition functions with different number of particles, $N\,$

$\mathcal{Z}(z, V, T) = \sum_{N=0}^{\infty} z^N \, Z(N, V, T) \, =\sum_{N=0}^{\infty} \sum_i z^N \, \exp(-E_i/ k_B T) \,$

where $z\,$ is defined below, and $Z(N, V, T) \,$ denotes the partition function of the canonical ensemble at temperature $T\,$ , of volume $V\,$ , and with the number of particles fixed at $N\,$ . (In the last step, we have expanded the canonical partition function, and $k_B \,$ is the Boltzmann constant, the second sum is performed over all microscopic states, denoted by $i\,$ with energy $E_i\,$ . )

Quantum mechanically, the situation is even simpler (conceptually). For a system of bosons or fermions, it is often mathematically easier to treat the number of particles of the system as an intrinsic property of each quantum (eigen-)state, $i\,$ . Therefore the partition function can be written as

$\mathcal{Z}(z, V, T) = \sum_i z^{N_i} \, \exp(-E_i/ k_B T) \,$

The parameter $z\,$ is called fugacity (the easiness of adding a new particle into the system). The chemical potential is directly related to the fugacity through

$\mu = k_B T \ln z\,$ .

And the chemical potential is the Gibbs free energy per particle. (We have used fugacity instead of chemical potential in defining the partition function. This is because fugacity is an independent parameter of partition function to control the number of particles, as temperature to control the energy. On the other hand, the chemical potential itself contains temperature dependence, which may lead to some confusion. )

Thermodynamic quantities

The average number of particles of the ensemble is obtained as

$\langle N \rangle = z\frac{\partial} {\partial z} \ln \mathcal{Z}(z, V, T).$

And the average internal energy is

$\langle E \rangle = k_B T^2 \frac{\partial} {\partial T} \ln \mathcal{Z}(z, V, T).$

The partition function itself is the product between pressure $P\,$ and volume, divided by $k_B T\,$

$P V = k_B T \ln \mathcal{Z}$

Other thermodynamic potentials can be obtained through linear combination of above quantites. For example, the Helmholtz free energy $F\,$ (some people use $A\,$ ) can be obtained as

$F= N \mu - PV = - k_B T \ln( \mathcal{Z}/z^N).$

Statistics of bosons and fermions

For a quantum mechanical system, the eigenvalues (energies) and the corresponding eigenvectors (eigenstates) of the Hamiltonian (the energy function) completely describe the system. For a macroscopic system, the number of eigenstates (microscopic states) is enormous. Statistical mechanics provides a way to average all microscopic states to obtain meaningful macroscopic quantities.

The task of summing over states (calculating the partition function) appears to be simpler if we do not fix the total number of particles of the system because, for a noninteracting system, the partition function of grand canonical ensemble can be converted to a product of the partition functions of individual modes. This conversion makes the evaluation much easier. (However this conversion cannot be done in canonical ensemble, where the total number of particles is fixed. )

Each mode is a spatial configuration for an individual particle. There may be none or some particles in each mode. In quantum mechanics, all particles are either bosons or fermions. For fermions, no two particles can share a same mode. But there is no such constraint for bosons. Therefore the partition function (of grand canonical ensemble) for each mode can be written as

$\sum_n z^n \exp(- n \epsilon/ k_B T) \, = (1 \pm z \exp(- \epsilon/ k_B T))^{\pm1}$

The $\epsilon \,$ is the energy of the mode. For fermions, $n\,$ can be 0 or 1 (no particle or one particle in the mode). For bosons, $n=0,1,2,...\,$ . The upper (lower) sign is for fermions (bosons) in the last step. The total partition function is then a product of the ones for individual modes.

Quantum mechanical ensemble

An ensemble of quantum mechanical systems is described by a density matrix. In a suitable representation, a density matrix ρ takes the form

$\rho = \sum_k p_k |\psi_k \rangle \langle \psi_k|$

where p_k is the probability of a system chosen at random from the ensemble will be in the microstate

$|\psi_k \rangle.$

So the trace of ρ, denoted by Tr(ρ), is 1. This is the quantum mechanical analogue of the fact that the accessible region of the classical phase space has total probability 1.

It is also assumed that the ensemble in question is stationary, i.e. it does not change in time. Therefore, by Liouville's theorem, [ρ, H] = 0, i.e. ρH = Hρ where H is the Hamiltonian of the system. Thus the density matrix describing ρ is diagonal in the energy representation.

Suppose

$H = \sum_n E_i |\psi_i \rangle \langle \psi_i|$

where E_i is the energy of the i-th energy eigenstate. If a system i-th energy eigenstate has n_i number of particles, the corresponding observable, the number operator, is given by

$N = \sum_n n_i |\psi_i \rangle \langle \psi_i|.$

From classical considerations, we know that the state

$|\psi_i \rangle$

has (unnormalized) probability

$p_i = e^{-\beta (E_i - \mu n_i)} \,.$

Thus the grand canonical ensemble is the mixed state

$\rho = \sum_i p_i |\psi_i \rangle \langle \psi_i| = \sum_i e^{-\beta (E_i - \mu n_i)} |\psi_i \rangle \langle \psi_i| = e^{- \beta (H - \mu N)}.$

The grand partition, the normalizing constant for Tr(ρ) to be 1, is

${\mathcal Z} =\mathbf{Tr} [ e^{- \beta (H - \mu N)} ].$

v • d • e Statistical mechanics

Statistical ensembles	Microcanonical • Canonical • Grand canonical • Isothermal–isobaric • Isoenthalpic–isobaric

Statistical thermodynamics	Characteristic state functions

Partition functions	Translational • Vibrational

Equations of state	Dieterici • Van der Waals • Ideal gas law • Birch–Murnaghan

Entropy	Sackur–Tetrode equation • Nonextensive entropy

Particle statistics	Maxwell–Boltzmann statistics • Fermi–Dirac statistics • Bose–Einstein statistics

Statistical field theory	Conformal field theory • Osterwalder–Schrader axioms

See also	Probability distribution • Structureless particles

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

Donate to Wikimedia

	I would like to know who the Canon of Florence Cathedral was as shown touring Kevin McCloud around Florence is his program Kevin McCloud's Grand Tour? Read answer...
	What is canonicity? Read answer...
	Is the Clarinet a solo or ensemble instrument? Read answer...

	How far is the grand canon from vegas?
	How wide is the grand canon?
	How far is it from milwaukeewisconsin to the grand canon?

		Sci-Tech Dictionary. McGraw-Hill Dictionary of Scientific and Technical Terms. Copyright © 2003, 1994, 1989, 1984, 1978, 1976, 1974 by McGraw-Hill Companies, Inc. All rights reserved. Read more
		Wikipedia. This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article "Grand canonical ensemble". Read more

Grand canonical ensemble

Contents

The partition function

Thermodynamic quantities

Statistics of bosons and fermions

Quantum mechanical ensemble