Microcanonical ensemble

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. The system is assumed to be isolated in the sense that the system cannot exchange energy or particles with its environment, so that (by conservation of energy) the energy of the system remains exactly known as time goes on. The system's energy, composition, volume, and shape are kept the same in all possible states of the system.

The macroscopic variables of the microcanonical ensemble are quantities that influence the nature of the system's internal states such as the total number of particles in the system (symbol: N), the system's volume (symbol: V), as well as the total energy in the system (symbol: E). This ensemble is therefore sometimes called the NVE ensemble, as each of these three quantities is a constant of the ensemble.

In simple terms, the microcanonical ensemble is defined by assigning an equal probability to every microstate whose energy falls within a range centered at E. All other microstates are given a probability of zero. Since the probabilities must add up to 1, the probability P is the inverse of the number of microstates W within the range of energy,

The range of energy is then reduced in width until it is infinitesimally narrow, still centered at E. In the limit of this process, the microcanonical ensemble is obtained.

The microcanonical ensemble is sometimes considered to be the fundamental distribution of statistical thermodynamics, as its form can be justified on elementary grounds such as the principle of indifference: the microcanonical ensemble describes the possible states of an isolated mechanical system when the energy is known exactly, but without any more information about the internal state. Also, in some special systems the evolution is ergodic in which case the microcanonical ensemble is equal to the time-ensemble when starting out with a single state of energy E (a time-ensemble is the ensemble formed of all future states evolved from a single initial state).

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