Another case in which the grand canonical ensemble appears is when considering a system that is large and thermodynamic ( a system that is " in equilibrium with itself " ).
12.
If the particles can be created out of energy during the dynamics of the system, then an associated term must not appear in the probability expression for the grand canonical ensemble.
13.
In quantum mechanics, the grand canonical ensemble affords a simple description since diagonalization provides a set of distinct microstates of a system, each with well-defined energy and particle number.
14.
Even if the exact conditions of the system do not actually allow for variations in energy or particle number, the grand canonical ensemble can be used to simplify calculations of some thermodynamic properties.
15.
The grand canonical ensemble provides a natural setting for an exact derivation of the Fermi Dirac statistics or Bose Einstein statistics for a system of non-interacting quantum particles ( see examples below ).
16.
In practice, however, it is desirable to apply the grand canonical ensemble to describe systems that are in direct contact with the reservoir, since it is that contact that ensures the equilibrium.
17.
The hard hexagon model occurs within the framework of the grand canonical ensemble, where the total number of particles ( the " hexagons " ) is allowed to vary naturally, and is fixed by a chemical potential.
18.
Systems in thermal equilibrium with their environment have uncertainty in energy, and are instead described by the canonical ensemble or the grand canonical ensemble, the latter if the system is also in equilibrium with its environment in respect to particle exchange.
19.
The grand canonical ensemble applies to systems of any size, small or large; it is only necessary to assume that the reservoir with which it is in contact is much larger ( i . e ., to take the macroscopic limit ).
20.
In statistical mechanics, a "'grand canonical ensemble "'is the statistical ensemble that is used to represent the possible states of a mechanical system of particles that is being maintained in thermodynamic equilibrium ( thermal and chemical ) with a reservoir.
