In statistical mechanics, the grand canonical ensemble is a statistical ensemble, that means a set of identically prepared systems, each of which is in equilibrium with an external bath with respect to particle and energy exchange. Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... In physics, a statistical ensemble is a very large set of similar systems, considered all at once. ... A particle is Look up Particle in Wiktionary, the free dictionary In particle physics, a basic unit of matter or energy. ...
The grand canonical ensemble frequently provides the most convenient avenue for calculations.
Partition function
The partition function of the grand canonical ensemble is In statistical mechanics, the partition function Z is an important quantity that encodes the statistical properties of a system in thermodynamic equilibrium. ...
Here μ is the chemical potential, β the inverse temperature, sometimes also adorned with the inverse of the Boltzmann constant. is the Hamiltonian of the system class considered, the operator that counts the total number of particles in one system. The chemical potential of a thermodynamic system is the amount by which the energy of the system would change if an additional particle were introduced, with the entropy and volume held fixed. ... The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ... The Hamiltonian, denoted H, has two distinct but closely related meanings. ...
If the canonicalensemble is envisioned as a volume, a microcanonical ensemble may be envisioned as a sheet (a surface for the surface ensemble, or an exceedingly thin volume for the quantum microcanonical ensemble) cut out from the canonicalensemble.
Equivalently, the canonicalensemble may be envisioned as being constructed from microcanonical ensembles the way an onion is constructed from concentric shells of onionskin, each layer of onion corresponding to those states of the canonicalensemble which have a particular value of the energy.
The grandcanonical, isobaric-isothermal, and isodynamic-polythermal ensembles differ from the canonicalensemble in the variables that are held fixed, and in the statistical weights assigned to different states.
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