Editing Microcanonical ensemble (section)

{{Use American English|date=January 2019}}
{{Use mdy dates|date=January 2019}}
{{Short description|Ensemble of states with an exactly specified total energy}}
{{Statistical mechanics|cTopic=Ensembles}}

In [[statistical mechanics]], the '''microcanonical ensemble''' is a [[statistical ensemble (mathematical physics)|statistical ensemble]] that represents the possible states of a mechanical system whose total energy is exactly specified.<ref name="gibbs">{{cite book |last=Gibbs |first=Josiah Willard |author-link=Josiah Willard Gibbs |title=Elementary Principles in Statistical Mechanics |year=1902 |publisher=[[Charles Scribner's Sons]] |location=New York|title-link=Elementary Principles in Statistical Mechanics }}</ref> The system is assumed to be isolated in the sense that it cannot exchange energy or particles with its environment, so that (by [[conservation of energy]]) the energy of the system does not change with time.

The primary macroscopic variables of the microcanonical ensemble are the total number of particles in the system (symbol: {{math|''N''}}), the system's volume (symbol: {{math|''V''}}), as well as the total energy in the system (symbol: {{math|''E''}}). Each of these is assumed to be constant in the ensemble. For this reason, the microcanonical ensemble is sometimes called the '''{{math|''NVE''}} ensemble'''.

In simple terms, the microcanonical ensemble is defined by assigning an equal probability to every [[microstate (statistical mechanics)|microstate]] whose energy falls within a range centered at {{math|''E''}}. All other microstates are given a probability of zero. Since the probabilities must add up to 1, the probability {{math|''P''}} is the inverse of the number of microstates {{math|''W''}} within the range of energy,
<math display="block">P = 1/W,</math>
The range of energy is then reduced in width until it is [[infinitesimal]]ly narrow, still centered at {{math|''E''}}. In the [[Limit (mathematics)|limit]] of this process, the microcanonical ensemble is obtained.<ref name="gibbs"/>