# Grand Canonical Ensemble

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 pi/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.

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

Partition Function (statistical Mechanics) - Grand Canonical Partition Function - Finite Quantum 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 ...
Statistical Mechanics - Statistical Ensembles - 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 ...
Polymer Field Theory - 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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