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In Mathematical Physics , especially as introduced into Statistical Mechanics and Thermodynamics by J. Willard Gibbs in 1878, an ensemble (also '''statistical ensemble''' or '''thermodynamic ensemble''') is an idealization consisting of a large number of mental copies (sometimes infinitely many) of a System , considered all at once, each of which represents a possible state that the real system might be in. This article treats the notion of ensembles in a mathematically rigorous fashion, although relevant physical aspects will be mentioned.


PHYSICAL CONSIDERATIONS


The ensemble formalises the notion that a physicist repeating an experiment again and again under the same macroscopic conditions, but unable to control the microscopic details, may expect to observe a range of different outcomes.

The notional size of the mental ensembles in thermodynamics, statistical mechanics and Quantum Statistical Mechanics can be very large indeed, to include every possible Microscopic State the system could be in, consistent with its observed Macroscopic properties. But for important physical cases it can be possible to calculate averages directly over the whole of the thermodynamic ensemble, to obtain explicit formulas for many of the thermodynamic quantities of interest, often in terms of the appropriate partition function (see below). Some of these results are presented in the article Statistical Mechanics .


Note on terminology


  • The word ensemble is also sometimes used for smaller sets of possibilities, Sampled from the full set of possible states. Thus for example, an ensemble of walkers in a Markov Chain Monte Carlo iteration.


  • The word ensemble is particularly used in thermodynamics; by some physicists working in Bayesian Probability theory; and by mathematicians whose work in probability theory is heavily influenced by physicists, especially those working on Random Matrices . Most "pure" mathematicians working in Probability Theory do not use the term, preferring to use the terminology of Probability Space s.



ENSEMBLES OF CLASSICAL MECHANICAL SYSTEMS


For an ensemble of a Classical Mechanical System , one considers the phase space of the given system. A collection of elements from the ensemble can be viewed as a swarm of representative points in the phase space. The statistical properties of the ensemble then depend on a chosen Probability Measure on the phase space. If a region ''A'' of the phase space has larger measure than region B, then a system chosen at random from the ensemble is more likely to be in a microstate belonging to ''A'' than ''B''. The choice of this measure is dictated by the specific details of the system and the assumptions one makes about the ensemble in general. For example, the phase space measure of the Microcanonical Ensemble (see below) is different from that of the Canonical Ensemble . The normalizing factor of the probability measure is referred to as the Partition Function of the ensemble. Physically, the partition function encodes the underlying physical structure of the system.

When the measure is time-independent, the ensemble is said to be ''stationary''.


Principal ensembles of statistical thermodynamics


Different macroscopic environmental constraints lead to different types of ensembles, with particular statistical characteristics. The following are the most important:

  • Microcanonical Ensemble or NVE ensemble -- an ensemble of systems, each of which is required to have the same total energy (i.e. thermally isolated).


  • Canonical Ensemble or NVT ensemble -- an ensemble of systems, each of which can share its energy with a large heat reservoir or heat bath. The system is allowed to exchange energy with the reservoir, and the heat capacity of the reservoir is assumed to be so large as to maintain a fixed temperature for the coupled system.


  • Grand Canonical Ensemble -- an ensemble of systems, each of which is again in thermal contact with a reservoir. But now in addition to energy, there is also exchange of particles. The temperature is still assumed to be fixed.


The calculations that can be made over each of these ensembles are explored further in the article Statistical Mechanics . The main result for each ensemble however, is its characteristic state function:

Microcanonical: \; \Omega(U,V,N) = e^{\beta TS}

Canonical: \; Z(T,V,N) = e^{- \beta A}

Grand canonical: \; \Xi(T,V,\mu) = e^{\beta P V}

For these ensembles, the choice for the appropriate probability measure is dictated by the expressions above.

Other thermodynamic ensembles can be also defined, corresponding to different physical requirements, for which analogous formulae can often similarly be derived.


Properties of "good" ensembles


The following properties are considered desirable for a classical mechanical ensemble.

  • Representativeness


The chosen probability measure on the phase space should be a Gibbs State of the ensemble, i.e. it should be invariant under time evolution. A standard example of this is the natural measure (locally, it is just the Lebesgue measure) on a constant energy surface for a classical mechanical system. Liouville's Theorem states this measure is invariant under the Hamiltonian Flow .

  • Ergodicity


Once a probability measure μ on the phase space \Lambda is specified, one can define the ''ensemble average'' of an observable, i.e. real-valued function ''f'' defined on \Lambda via this measure by

:\langle f angle = \int _{\Lambda} f d \mu

where we have restricted to those observables which are μ-integrable.

On the other hand, let \; x(0) denote a representative point in the phase space, and \; x(t) be its image under the flow, specified by the system in question, at time ''t''. The ''time average'' of ''f'' is defined to be

:\lim _{T ightarrow \infty} rac{1}{T} \int _0 ^T f(x(t)) d t

, provided that this limit exists μ-almost everywhere and is independent of \; x(0).

The Ergodicity requirement is that the ensemble average coincide with the time average. A sufficient condition for ergodicity is that the time evolution of the system is a Mixing . (See also Ergodic Hypothesis .) Not all systems are ergodic. For instance, it is unknown at this time whether classical mechanical flows on a constant energy surface are ergodic in general. Physically, when a system fails to be ergodic, we may infer that there is more macroscopically discoverable information available about the microscopic state of the system than what we first thought. In turn this may be used to create a better- Conditioned ensemble.


ENSEMBLES IN QUANTUM STATISTICAL MECHANICS

''See main article: Quantum Statistical Mechanics ''

Putting aside for the moment the question of how statistical ensembles are generated Operationally , we should be able to perform the following two operations on ensembles ''A'', ''B'' of the same system:

  • Test whether ''A'', ''B'' are statistically equivalent.


  • If ''p'' is a real number such that 0 < ''p'' < 1, then produce a new ensemble by probabilistic sampling from ''A'' with probability ''p'' and from ''B'' with probability 1- ''p''.


Under certain conditions therefore, Equivalence Class es of statistical ensembles have the structure of a convex set. In quantum physics, a general Model for this convex set is the set of Density Operators on a Hilbert Space . Accordingly, there are two types of ensembles: