mathematical physics

Statistical ensemble (mathematical physics)

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 spaces.

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 rangle = 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 rightarrow infty}frac{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

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 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:

  • 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 rangle 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.

Thus a quantum mechanical ensemble is specified by a mixed state in general. For example, one can specify the density operators describing microcanonical, canonical, and grand canonical ensembles of quantum mechanical systems, in a mathematically rigorous fashion.

The normalization factor required for the density operator to have trace 1 is the quantum mechanical version of the partition function.

We note here that ensembles of quantum mechanical system are sometimes treated by physicists in a semi-classical fashion. Namely, one considers the phase space of the corresponding classical system (e.g. for an ensemble of quantum harmonic oscillators, the phase space of a classical harmonic oscillator is considered). Then, using physical arguments, one derives a suitable "fundamental volume" for the particular system to reflect the fact that quantum mechanical microstates are discretely distributed on the phase space. From the uncertainly principle, it is expected this fundamental volume to be related to the Planck constant, hbar, in some way.

Ensembles in statistics

The formulation of statistical ensembles used in physics has now been widely adopted in other fields, in part because it has been recognized that the Boltzmann distribution or Gibbs measure serves to maximize the entropy of a system, subject to a set of constraints: this is the principle of maximum entropy. This principle has now been widely applied to problems in linguistics, robotics, and the like.

In addition, statistical ensembles in physics are often built on a principle of locality: that all interactions are only between neighboring atoms or nearby molecules. Thus, for example, lattice models, such as the Ising model, model ferromagnetic materials by means of nearest-neighbor interactions between spins. The statistical formulation of the principle of locality is now seen to be a form of the Markov property in the broad sense; nearest neighbors are now Markov blankets. Thus, the general notion of a statistical ensemble with nearest-neighbor interactions leads to Markov random fields, which again find broad applicability; for example in Hopfield networks.

Operational interpretation

In the discussion given so far, while rigorous, we have taken for granted that the notion of an ensemble is valid a priori, as is commonly done in physical context. What has not been shown is that the ensemble itself (not the consequent results) is a precisely defined object mathematically. For instance,

  • 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, ....,Xk, 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.

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, Xk). Each one of these values is a 0 (or no) or a 1 (yes).

Assume the following time average exists:

sigma(E) = lim_{N rightarrow infty} frac{1}{N} sum_{k=1}^N operatorname{Meas}(E, X_k)
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:
sigma(E) = operatorname{Tr}(E S).

We see this reflects the definition of quantum states in general: A quantum state is a mapping from the observables to their expectation values.

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