|
In statistical mechanics, a canonical ensemble is the statistical ensemble that represents the possible states of a mechanical system in thermal equilibrium with a heat bath at some fixed temperature. The system can exchange energy with the heat bath, so that the states of the system will differ in total energy. The principal thermodynamic variable of the canonical ensemble, determining the probability distribution of states, is the absolute temperature (symbol: ). The ensemble typically also depends on mechanical variables such as the number of particles in the system (symbol: ) and the system's volume (symbol: ), each of which influence the nature of the system's internal states. An ensemble with these three parameters is sometimes called the ensemble. In simple terms, the canonical ensemble assigns a probability to each distinct microstate given by the following exponential: : where is the total energy of the microstate, and is Boltzmann's constant. The number is the free energy (specifically, the Helmholtz free energy) and is a constant for the ensemble. However, the probabilities and will vary if different ''N'', ''V'', ''T'' are selected. The free energy serves two roles: first, it provides a normalization factor for the probability distribution (the probabilities, over the complete set of microstates, must add up to one); second, many important ensemble averages can be directly calculated from the function . An alternative but equivalent formulation for the same concept writes the probability as , using the canonical partition function rather than the free energy. The equations below (in terms of free energy) may be restated in terms of the canonical partition function by simple mathematical manipulations. Historically, the canonical ensemble was first described by Boltzmann (who called it a ''holode'') in 1884 in a relatively unknown paper. It was later reformulated and extensively investigated by Gibbs in 1902.〔 ==Applicability of canonical ensemble== The canonical ensemble is the ensemble that describes the possible states of an isolated system that is in thermal equilibrium with a heat bath (the derivation of this fact can be found in Gibbs〔). The canonical ensemble applies to systems of any size; while it is necessary to assume that the heat bath is very large (i. e., take a macroscopic limit), the system itself may be small or large. The condition that the system is mechanically isolated is necessary in order to ensure it does not exchange energy with any external object besides the heat bath.〔 In general, it is desirable to apply the canonical ensemble to systems that are in direct contact with the heat bath, since it is that contact that ensures the equilibrium. In practical situations, the use of the canonical ensemble is usually justified either 1) by assuming that the contact is mechanically weak, or 2) by incorporating a suitable part of the heat bath connection into the system under analysis, so that the connection's mechanical influence on the system is modeled within the system. 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. For systems where the particle number is variable (due to contact with a particle reservoir), the correct description is the grand canonical ensemble. For large systems (in the thermodynamic limit) these other ensembles become essentially equivalent to the canonical ensemble, at least for average quantities. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「canonical ensemble」の詳細全文を読む スポンサード リンク
|