Microcanonical ensemble

In statistical mechanics, a micro-canonical ensemble is the statistical ensemble that is used to represent the possible states of a mechanical system that 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 the same 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 micro-canonical ensemble are quantities that influence the nature of the system's micro-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 micro-canonical ensemble is defined by assigning an equal probability to every micro-state whose energy falls within a range centered at E. All other micro-states are given a probability of zero. Since the probabilities must add up to 1, the probability P is the inverse of the number of micro-states 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 micro-canonical ensemble is obtained. The micro-canonical 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 micro-canonical 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 micro-canonical 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). In practice, the micro-canonical ensemble does not correspond to an experimentally realistic situation. With a real physical system there is at least some uncertainty in energy, due to uncontrolled factors in the preparation of the system. Besides the difficulty of finding an experimental analogue, it is difficult to carry out calculations that satisfy exactly the requirement of fixed energy since it prevents logically independent parts of the system from being analyzed separately. Moreover, there are ambiguities regarding the appropriate definitions of quantities such as entropy and temperature in the micro-canonical ensemble. 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. Early work in statistical mechanics by Ludwig Boltzmann led to his eponymous entropy equation for a system of a given total energy, S = k log W, where W is the number of distinct states accessible by the system at that energy. Boltzmann did not elaborate too deeply on what exactly constitutes the set of distinct states of a system, besides the special case of an ideal gas. This topic was investigated to completion by Josiah Willard Gibbs who developed the generalized statistical mechanics for arbitrary mechanical systems, and defined the micro-canonical ensemble described in this article. Gibbs investigated carefully the analogies between the micro-canonical ensemble and thermodynamics, especially how they break down in the case of systems of few degrees of freedom. He introduced two further definitions of micro-canonical entropy that do not depend on ω - the volume and surface entropy described above. (Note that the surface entropy differs from the Boltzmann entropy only by an ω-dependent offset.)