Statistical ensemble

In physics, a statistical ensemble is a very large set of macroscopically similar systems, considered all at once. When properly chosen, (see below), the average of a thermodynamic quantity across the members of the ensemble will be the same as the time-average of the quantity for a single system.

Related Topics:
Physics - Systems

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The topic of statistical ensembles is important in thermodynamics, statistical mechanics and quantum physics.

Related Topics:
Thermodynamics - Statistical mechanics - Quantum physics

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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:

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• 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 classes 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:

Related Topics:
Equivalence class - Model - Density operators - Hilbert space

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• Pure ensembles cannot be decomposed as a convex combination of different ensembles. In quantum mechanics, a pure density matrix is one of the form $|phi\; angle\; langle\; phi|$. Accordingly, a ray in a Hilbert space can be used to represent such an ensemble in quantum mechanics. A pure ensemble corresponds to having many copies of the same (up to a global phase) quantum state.
• Mixed ensembles are decomposable into a convex combination of different ensembles. In general, an infinite number of distinct decompositions will be possible.
 
Statistical ensemble: Operational interpretation ►