In physics, a statistical ensemble is a very large set of similar systems, considered all at once. The willingness to question previously held truths and search for new answers resulted in a period of major scientific advancements, now known as the Scientific Revolution. ...

The topic of statistical ensembles is important in thermodynamics, statistical mechanics and quantum physics. 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: Thermodynamics (Greek: thermos = heat and dynamic = change) is the physics of energy, heat, work, entropy and the spontaneity of processes. ... Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... Fig. ... An operational definition of a quantity is a specific process whereby it is measured. ...

• 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: In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x in X | x ~ a } The notion of equivalence classes is useful for constructing sets... An abstract model (or conceptual model) is a theoretical construct that represents physical, biological or social processes, with a set of variables and a set of logical and quantitative relationships between them. ... A density matrix, or density operator, is used in quantum theory to describe the statistical state of a quantum system. ... In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ...

• Pure ensembles cannot be decomposed as a convex combination of different ensembles. In quantum mechanics, a pure density matrix is one of the form . 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.

In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ...

Operational interpretation

Two objections to the above discussion of ensemble are

• It is not clear where this very large set of systems exists (for example, is it a gas of particles inside a container?)

• It is not clear how to physically generate an ensemble.

In this section we attempt to partially answer this question.

Suppose we have a preparation procedure for a system in a physics lab: For example, the procedure might involve a physical apparatus and some protocols for manipulating the apparatus. As a result of this preparation procedure some system is produced and maintained in isolation for some small period of time. By repeating this laboratory preparation procedure we obtain a sequence of systems X₁, X₂, ....,X_k, which in our mathematical idealization, we assume is an infinite sequence of systems. The systems are similar in that they were all produced in the same way. This infinite sequence is an ensemble. Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ...

In a laboratory setting, each one of these prepped systems might be used as input for one subsequent testing procedure. Again, the testing procedure involves a physical apparatus and some protocols; as a result of the testing procedure we obtain a yes or no answer. Given a testing procedure E applied to each prepared system, we obtain a sequence of values Meas(E, X₁), Meas(E, X₂), ...., Meas(E, X_k). Each one of these values is a 0 (or no) or a 1 (yes).

Assume the following time average exists:

For quantum mechanical systems, an important assumption made in the quantum logic approach to quantum mechanics is the identification of yes-no questions to the lattice of closed subspaces of a Hilbert space. With some additional technical assumptions one can then infer that states are given by density operators S so that: In mathematical physics and quantum mechanics, quantum logic can be regarded as a kind of propositional logic suitable for understanding the apparent anomalies regarding quantum measurement, most notably those concerning composition of measurement operations of complementary variables. ...

See also

• density matrix
• Partition function (statistical mechanics)
• microcanonical ensemble
• canonical ensemble
• grand canonical ensemble
• isothermal-isobaric ensemble
• phase space
• Liouville's theorem (Hamiltonian)