Ensemble (mathematical physics)

• The word "ensemble" is also used for a smaller set of possibilities sampled from the full set of possible states. For example, a collection of walkers in a Markov chain Monte Carlo iteration is called an ensemble in some of the literature.
• The term "ensemble" is often used in physics and the physics-influenced literature. In probability theory, the term probability space is more prevalent.

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.

A statistical ensemble in quantum mechanics (also known as a mixed state) is most often represented by a density matrix, denoted by ${\displaystyle {\hat {\rho }}}$. 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 X in quantum mechanics can be written as an operator, X̂. The expectation value of this operator on the statistical ensemble ${\displaystyle \rho }$ is given by the following trace:

${\displaystyle \langle X\rangle =\operatorname {Tr} ({\hat {X}}\rho ).}$

This can be used to evaluate averages (operator X̂), variances (using operator X̂ ²), covariances (using operator X̂Ŷ), etc. The density matrix must always have a trace of 1: ${\displaystyle \operatorname {Tr} {\hat {\rho }}=1}$ (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, ${\displaystyle d{\hat {\rho }}/dt=0}$) 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 Ĥ (Hamiltonian). The grand canonical ensemble is additionally a function of the particle number, measured by the total particle number operator N̂. 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

${\displaystyle {\hat {\rho }}=\sum _{i}P_{i}|\psi _{i}\rangle \langle \psi _{i}|}$

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

Evolution of an ensemble of 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.[2] While an individual system evolves according to Hamilton's equations, the density function (the ensemble) evolves over time according to Liouville's equation.

In a mechanical system with a defined number of parts, the phase space has n generalized coordinates called q₁, ... q_n, and n associated canonical momenta called p₁, ... p_n. The ensemble is then represented by a joint probability density function ρ(p₁, ... p_n, q₁, ... q_n).

If the number of parts in the system is allowed to vary among the systems in the ensemble (as in a grand ensemble where the number of particles is a random quantity), then it is a probability distribution over an extended phase space that includes further variables such as particle numbers N₁ (first kind of particle), N₂ (second kind of particle), and so on up to N_s (the last kind of particle; s is how many different kinds of particles there are). The ensemble is then represented by a joint probability density function ρ(N₁, ... N_s, p₁, ... p_n, q₁, ... q_n). The number of coordinates n varies with the numbers of particles.

Any mechanical quantity X can be written as a function of the system's phase. The expectation value of any such quantity is given by an integral over the entire phase space of this quantity weighted by ρ:

${\displaystyle \langle X\rangle =\sum _{N_{1}=0}^{\infty }\ldots \sum _{N_{s}=0}^{\infty }\int \ldots \int \rho X\,dp_{1}\ldots dq_{n}.}$

The condition of probability normalization applies, requiring

${\displaystyle \sum _{N_{1}=0}^{\infty }\ldots \sum _{N_{s}=0}^{\infty }\int \ldots \int \rho \,dp_{1}\ldots dq_{n}=1.}$

Phase space is a continuous space containing an infinite number of distinct physical states within any small region. In order to connect the probability density in phase space to a probability distribution over microstates, it is necessary to somehow partition the phase space into blocks that are distributed representing the different states of the system in a fair way. It turns out that the correct way to do this simply results in equal-sized blocks of canonical phase space, and so a microstate in classical mechanics is an extended region in the phase space of canonical coordinates that has a particular volume.[note 1] In particular, the probability density function in phase space, ρ, is related to the probability distribution over microstates, P by a factor

${\displaystyle \rho ={\frac {1}{h^{n}C}}P,}$

where

• h is an arbitrary but predetermined constant with the units of energy×time, setting the extent of the microstate and providing correct dimensions to ρ.[note 2]
• C is an overcounting correction factor (see below), generally dependent on the number of particles and similar concerns.

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

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.[note 3]
• Erroneous conclusions that are inconsistent with physical experience, as in the mixing paradox.[2]
• Foundational issues in defining the chemical potential and the grand canonical ensemble.[2]

It is in general difficult to find a coordinate system that uniquely encodes each physical state. As a result, it is usually necessary to use a coordinate system with multiple copies of each state, and then to recognize and remove the overcounting.

A crude way to remove the overcounting would be to manually define a subregion of phase space that includes each physical state only once and then exclude all other parts of phase space. In a gas, for example, one could include only those phases where the particles' x coordinates are sorted in ascending order. While this would solve the problem, the resulting integral over phase space would be tedious to perform due to its unusual boundary shape. (In this case, the factor C introduced above would be set to C = 1, and the integral would be restricted to the selected subregion of phase space.)

A simpler way to correct the overcounting is to integrate over all of phase space but to reduce the weight of each phase in order to exactly compensate the overcounting. This is accomplished by the factor C introduced above, which is a whole number that represents how many ways a physical state can be represented in phase space. Its value does not vary with the continuous canonical coordinates,[note 4] so overcounting can be corrected simply by integrating over the full range of canonical coordinates, then dividing the result by the overcounting factor. However, C 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[2]

${\displaystyle C=N_{1}!N_{2}!\ldots N_{s}!.}$

This is known as "correct Boltzmann counting".

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:

${\displaystyle {\bar {A}}={\frac {\int {Ae^{-\beta H(q_{1},q_{2},...q_{M},p_{1},p_{2},...p_{N})}d\tau }}{\int {e^{-\beta H(q_{1},q_{2},...q_{M},p_{1},p_{2},...p_{N})}d\tau }}}}$

where:

${\displaystyle {\bar {A}}}$ is the ensemble average of the system property A,

${\displaystyle \beta }$ is ${\displaystyle {\frac {1}{kT}}}$, known as thermodynamic beta,

H is the Hamiltonian of the classical system in terms of the set of coordinates ${\displaystyle q_{i}}$ and their conjugate generalized momenta ${\displaystyle p_{i}}$, and

${\displaystyle d\tau }$ 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.

In quantum statistical mechanics, for a quantum system in thermal equilibrium with its environment, the weighted average takes the form of a sum over quantum energy states, rather than a continuous integral:

${\displaystyle {\bar {A}}={\frac {\sum _{i}{A_{i}e^{-\beta E_{i}}}}{\sum _{i}{e^{-\beta E_{i}}}}}}$

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) as well as particles with its surroundings but the volume (V) is kept constant.