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.
Read more about Grand Canonical Ensemble: The Partition Function, Thermodynamic Quantities, Statistics of Bosons and Fermions, Quantum Mechanical Ensemble
Famous quotes containing the words grand and/or canonical:
“The allurement that women hold out to men is precisely the allurement that Cape Hatteras holds out to sailors: they are enormously dangerous and hence enormously fascinating. To the average man, doomed to some banal drudgery all his life long, they offer the only grand hazard that he ever encounters. Take them away, and his existence would be as flat and secure as that of a moo-cow.”
—H.L. (Henry Lewis)
“If God bestowed immortality on every man then when he made him, and he made many to whom he never purposed to give his saving grace, what did his Lordship think that God gave any man immortality with purpose only to make him capable of immortal torments? It is a hard saying, and I think cannot piously be believed. I am sure it can never be proved by the canonical Scripture.”
—Thomas Hobbes (1579–1688)