In physics, specifically statistical mechanics, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual 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. In other words, a statistical ensemble is a set of systems of particles used in statistical mechanics to describe a single

system. The concept of an ensemble was introduced by J. Willard Gibbs in 1902.

A thermodynamic ensemble is a specific variety of statistical ensemble that, among other properties, is in statistical equilibrium (defined below), and is used to derive the properties of thermodynamic systems from the laws of classical or quantum mechanics.

Physical considerations

The ensemble formalises the notion that an experimenter 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 ensembles in thermodynamics, statistical mechanics and quantum statistical mechanics can be very large, including every possible microscopic state the system could be in, consistent with its observed macroscopic properties. For many important physical cases, it is 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.

The concept of an equilibrium or stationary ensemble is crucial to many applications of statistical ensembles. Although a mechanical system certainly evolves over time, the ensemble does not necessarily have to evolve. In fact, the ensemble will not evolve if it contains all past and future phases of the system. Such a statistical ensemble, one that does not change over time, is called stationary and can be said to be in statistical equilibrium.</blockquote>

Three important thermodynamic ensembles were defined by Gibbs:

Equivalence

In thermodynamic limit all ensembles should produce identical observables due to Legendre transforms, deviations to this rule occurs under conditions that state-variables are non-convex, such as small molecular measurements.

Representations

The precise mathematical expression for a statistical ensemble has a distinct form depending on the type of mechanics under consideration (quantum or classical). In the classical case, the ensemble is a probability distribution over the microstates. In quantum mechanics, this notion, due to von Neumann, is a way of assigning a probability distribution over the results of each complete set of commuting observables.

In classical mechanics, the ensemble is instead written as a probability distribution in phase space; the microstates are the result of partitioning phase space into equal-sized units, although the size of these units can be chosen somewhat arbitrarily.

Requirements for representations

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 , then produce a new ensemble by probabilistic sampling from A with probability p and from B with probability .

Under certain conditions, therefore, equivalence classes of statistical ensembles have the structure of a convex set.

Quantum mechanical

A statistical ensemble in quantum mechanics (also known as a mixed state) is most often represented by a density matrix, denoted by <math>\hat\rho</math>. The density matrix provides a fully general tool that can incorporate both quantum uncertainties (present even if the state of the system were completely known) and classical uncertainties (due to a lack of knowledge) in a unified manner. Any physical observable in quantum mechanics can be written as an operator, <math>\hat X</math>. The expectation value of this operator on the statistical ensemble <math>\rho</math> is given by the following trace:

<math display="block">\langle X \rangle = \operatorname{Tr}(\hat X \rho).</math>

This can be used to evaluate averages (operator <math>\hat X</math>), variances (using operator <math>\hat X^2</math>), covariances (using operator <math>\hat X \hat Y</math>), etc. The density matrix must always have a trace of 1: <math>\operatorname{Tr}{\hat\rho} = 1</math> (this essentially is the condition that the probabilities must add up to one).

In general, the ensemble evolves over time according to the von Neumann equation.

Equilibrium ensembles (those that do not evolve over time, <math>d\hat\rho / dt = 0</math>) can be written solely as a function of conserved variables. For example, the microcanonical ensemble and canonical ensemble are strictly functions of the total energy, which is measured by the total energy operator <math>\hat H</math> (Hamiltonian). The grand canonical ensemble is additionally a function of the particle number, measured by the total particle number operator <math>\hat N</math>. Such equilibrium ensembles are a diagonal matrix in the orthogonal basis of states that simultaneously diagonalize each conserved variable. In bra–ket notation, the density matrix is

<math display="block">\hat\rho = \sum_i P_i |\psi_i\rangle \langle\psi_i|,</math>

where the , indexed by , are the elements of a complete and orthogonal basis. (Note that in other bases, the density matrix is not necessarily diagonal.)

Classical mechanical

frame|Evolution of an ensemble of [[Hamiltonian mechanics|classical systems in phase space (top). Each system consists of one massive particle in a one-dimensional potential well (red curve, lower figure). The initially compact ensemble becomes swirled up over time.]]

In classical mechanics, an ensemble is represented by a probability density function defined over the system's phase space. In particular, the probability density function in phase space, , is related to the probability distribution over microstates, by a factor

<math display="block">\rho = \frac{1}{h^n C} P,</math>

where

  • is an arbitrary but predetermined constant with the units of , setting the extent of the microstate and providing correct dimensions to .
  • is an overcounting correction factor (see below), generally dependent on the number of particles and similar concerns.

Since can be chosen arbitrarily, the notional size of a microstate is also arbitrary. Still, the value of influences the offsets of quantities such as entropy and chemical potential, and so it is important to be consistent with the value of when comparing different systems.

Correcting overcounting in phase space

Typically, the phase space contains duplicates of the same physical state in multiple distinct locations. This is a consequence of the way that a physical state is encoded into mathematical coordinates; the simplest choice of coordinate system often allows a state to be encoded in multiple ways. An example of this is a gas of identical particles whose state is written in terms of the particles' individual positions and momenta: when two particles are exchanged, the resulting point in phase space is different, and yet it corresponds to an identical physical state of the system. It is important in statistical mechanics (a theory about physical states) to recognize that the phase space is just a mathematical construction, and to not naively overcount actual physical states when integrating over phase space. Overcounting can cause serious problems:

  • Dependence of derived quantities (such as entropy and chemical potential) on the choice of coordinate system, since one coordinate system might show more or less overcounting than another.
  • Erroneous conclusions that are inconsistent with physical experience, as in the mixing paradox. so overcounting can be corrected simply by integrating over the full range of canonical coordinates, then dividing the result by the overcounting factor. However, does vary strongly with discrete variables such as numbers of particles, and so it must be applied before summing over particle numbers.

As mentioned above, the classic example of this overcounting is for a fluid system containing various kinds of particles, where any two particles of the same kind are indistinguishable and exchangeable. When the state is written in terms of the particles' individual positions and momenta, then the overcounting related to the exchange of identical particles is corrected by using

Classical statistical mechanics

For a classical system in thermal equilibrium with its environment, the ensemble average takes the form of an integral over the phase space of the system:

<math display="block">\bar{A} = \frac{ \displaystyle

\int{A\exp\left[-\beta H(q_1, q_2, \dots, q_M, p_1, p_2, \dots, p_N)\right] \, d\tau}

}{ \displaystyle

\int{\exp \left[-\beta H(q_1, q_2, \dots, q_M, p_1, p_2, \dots, p_N)\right] \, d\tau}

},</math>

where

  • <math>\bar{A}</math> is the ensemble average of the system property ,
  • <math>\beta</math> is <math>\frac{1}{kT}</math>, known as thermodynamic beta,
  • is the Hamiltonian of the classical system in terms of the set of coordinates <math>q_i</math> and their conjugate generalized momenta <math>p_i</math>,
  • <math>d\tau</math> is the volume element of the classical phase space of interest.

The denominator in this expression is known as the partition function and is denoted by the letter Z.

Quantum statistical mechanics

In quantum statistical mechanics, when a system is in thermal equilibrium, thermal averages are taken over the system’s discrete energy eigenstates, because the quantum Hamiltonian has a quantized spectrum rather than a continuous phase space.<math display="block">\bar{A} = \frac{\sum_i A_ie^{-\beta E_i{\sum_i e^{-\beta E_i.</math>

Canonical ensemble average

The generalized version of the partition function provides the complete framework for working with ensemble averages in thermodynamics, information theory, statistical mechanics and quantum mechanics.

The microcanonical ensemble represents an isolated system in which energy (E), volume (V) and the number of particles (N) are all constant. The canonical ensemble represents a closed system which can exchange energy (E) with its surroundings (usually a heat bath), but the volume (V) and the number of particles (N) are all constant. The grand canonical ensemble represents an open system which can exchange energy (E) and particles (N) with its surroundings, but the volume (V) is kept constant.

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 X<sub>1</sub>, X<sub>2</sub>, ...,X<sub>k</sub>, 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, X<sub>1</sub>), Meas (E, X<sub>2</sub>), ..., Meas (E, X<sub>k</sub>). Each one of these values is a 0 (or no) or a 1 (yes).

Assume the following time average exists:

<math display="block"> \sigma(E) = \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{k=1}^N \operatorname{Meas}(E, X_k) </math>

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:

<math display="block"> \sigma(E) = \operatorname{Tr}(E S). </math>

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

See also

Notes

References

  • Monte Carlo applet applied in statistical physics problems.