In statistical mechanics, a **grand canonical ensemble** (or **macrocanonical ensemble**) is a theoretical 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, but instead the grand canonical ensemble being modeled is allowed to exchange energy and particles with its environment. The chemical potential (or fugacity) is introduced to specify the fluctuation of the number of particles as chemical potential and particle numbers are thermodynamic conjugates. This is substitution is analogous to temperature being 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.

Read more about Grand Canonical Ensemble: The Partition Function, Thermodynamic Quantities, Statistics of Bosons and Fermions, Quantum Mechanical Ensemble

### Other articles related to "grand canonical ensemble, canonical ensemble, grand canonical, grand, ensemble, canonical":

**Grand Canonical Ensemble**

... of particles N is maintained small and finite, the orthodicity of the

**canonical ensemble**is exploited to define a finite quantum

**grand canonical**partition function for a finite quantum

**grand canonical ensemble**... The

**canonical ensemble**is natively orthodic so such properties is granted for the finite N

**grand canonical ensemble**by construction ... Consequently one defines the finite quantum

**grand**partition function for identical parti- cles from where and ...

**Grand Canonical Ensemble**- Quantum Mechanical Ensemble

... An

**ensemble**of quantum mechanical systems is described by a density matrix ... of a system chosen at random from the

**ensemble**will be in the microstate So the trace of ρ, denoted by Tr(ρ), is 1 ... It is also assumed that the

**ensemble**in question is stationary, i.e ...

**Grand Canonical Ensemble**

... In

**grand canonical ensemble**V, T and chemical potential are fixed ... chemical potentials, μj, j = 1...n and replace the

**canonical**partition function with the

**grand canonical**partition function where Nij is the number of jth species particles in the ith ... Let's rework everything using a

**grand canonical ensemble**this time ...

**Grand Canonical Ensemble**- Basic Field-theoretic Representation of Grand Canonical Partition Function

... To derive the

**grand canonical**partition function, we use its standard thermodynamic relation to the

**canonical**partition function, given by where is the ... the field-theoretic representation of the

**grand canonical**partition function, where is the

**grand canonical**action with defined by Eq ...

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