Based on the above definition, the microcanonical ensemble can be visualized as an infinitesimally thin shell in phase space, centered on a constant-energy surface.
12.
When the total energy is fixed but the internal state of the system is otherwise unknown, the appropriate description is not the canonical ensemble but the microcanonical ensemble.
13.
This topic was investigated to completion by Josiah Willard Gibbs who developed the generalized statistical mechanics for arbitrary mechanical systems, and defined the microcanonical ensemble described in this article.
14.
In comparison, the justification of the Boltzmann distribution from the microcanonical ensemble only applies for systems with a large number of parts ( that is, in the thermodynamic limit ).
15.
This definition can be derived from the microcanonical ensemble, which is a system of a constant number of particles, a constant volume and that does not exchange energy with its environment.
16.
In statistical mechanics, a "'microcanonical ensemble "'is the statistical ensemble that is used to represent the possible states of a mechanical system which has an exactly specified total energy.
17.
The microcanonical ensemble is defined by taking the limit of the density matrix as the energy width goes to zero, however a problematic situation occurs once the energy width becomes smaller than the spacing between energy levels.
18.
A commonly cited counter-example where energy is " not " shared among its various forms and where equipartition does " not " hold in the microcanonical ensemble is a system of coupled harmonic oscillators.
19.
However, it has been shown that for an observable satisfying the ETH, quantum fluctuations in its expectation value will typically be of the same order of magnitude as the thermal fluctuations which would be predicted in a traditional microcanonical ensemble.
20.
The general equipartition theorem holds in both the microcanonical ensemble, when the total energy of the system is constant, and also in the canonical ensemble, when the system is coupled to a heat bath with which it can exchange energy.
