Canonical ensemble

In statistical mechanics, a canonical ensemble is the statistical ensemble that represents the possible states of a mechanical system in thermal equilibrium with a heat bath at a fixed temperature.<ref name="gibbs">Template:Cite book</ref> The system can exchange energy with the heat bath, so that the states of the system will differ in total energy.

The principal thermodynamic variable of the canonical ensemble, determining the probability distribution of states, is the absolute temperature (symbol: Template:Math). The ensemble typically also depends on mechanical variables such as the number of particles in the system (symbol: Template:Math) and the system's volume (symbol: Template:Math), each of which influence the nature of the system's internal states. An ensemble with these three parameters, which are assumed constant for the ensemble to be considered canonical, is sometimes called the Template:Math ensemble.

The canonical ensemble assigns a probability Template:Math to each distinct microstate given by the following exponential:

<math>P = e^{(F - E)/(k T)},</math>

where Template:Math is the total energy of the microstate, and Template:Math is the Boltzmann constant.

The number Template:Math is the free energy (specifically, the Helmholtz free energy) and is assumed to be a constant for a specific ensemble to be considered canonical. However, the probabilities and Template:Math will vary if different N, V, T are selected. The free energy Template:Math serves two roles: first, it provides a normalization factor for the probability distribution (the probabilities, over the complete set of microstates, must add up to one); second, many important ensemble averages can be directly calculated from the function Template:Math.

An alternative but equivalent formulation for the same concept writes the probability as

<math>\textstyle P = \frac{1}{Z} e^{-E/(k T)},</math>

using the canonical partition function

<math>\textstyle Z = e^{-F/(k T)}</math>

rather than the free energy. The equations below (in terms of free energy) may be restated in terms of the canonical partition function by simple mathematical manipulations.

Historically, the canonical ensemble was first described by Boltzmann (who called it a holode) in 1884 in a relatively unknown paper.<ref>Template:Cite book</ref> It was later reformulated and extensively investigated by Gibbs in 1902.<ref name="gibbs"/>

Applicability of canonical ensemble

The canonical ensemble is the ensemble that describes the possible states of a system that is in thermal equilibrium with a heat bath (the derivation of this fact can be found in Gibbs<ref name="gibbs"/>).

The canonical ensemble applies to systems of any size; while it is necessary to assume that the heat bath is very large (i.e., take a macroscopic limit), the system itself may be small or large.

The condition that the system is mechanically isolated is necessary in order to ensure it does not exchange energy with any external object besides the heat bath.<ref name="gibbs"/> In general, it is desirable to apply the canonical ensemble to systems that are in direct contact with the heat bath, since it is that contact that ensures the equilibrium. In practical situations, the use of the canonical ensemble is usually justified either 1) by assuming that the contact is mechanically weak, or 2) by incorporating a suitable part of the heat bath connection into the system under analysis, so that the connection's mechanical influence on the system is modeled within the system.

When the total energy is fixed but the internal state of the system is otherwise unknown, the appropriate description is not the canonical ensemble but the microcanonical ensemble. For systems where the particle number is variable (due to contact with a particle reservoir), the correct description is the grand canonical ensemble. In statistical physics textbooks for interacting particle systems the three ensembles are assumed to be thermodynamically equivalent: the fluctuations of macroscopic quantities around their average value become small and, as the number of particles tends to infinity, they tend to vanish. In the latter limit, called the thermodynamic limit, the average constraints effectively become hard constraints. The assumption of ensemble equivalence dates back to Gibbs and has been verified for some models of physical systems with short-range interactions and subject to a small number of macroscopic constraints. Despite the fact that many textbooks still convey the message that ensemble equivalence holds for all physical systems, over the last decades various examples of physical systems have been found for which breaking of ensemble equivalence occurs.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref><ref>Template:Cite journal</ref><ref>Template:Cite journal</ref><ref>Template:Cite journal</ref><ref>Template:Cite journal</ref>

Free energy, ensemble averages, and exact differentials

• The partial derivatives of the function Template:Math give important canonical ensemble average quantities:
  • the average pressure is<ref name="gibbs"/> <math display="block"> \langle p \rangle = -\frac{\partial F} {\partial V}, </math>
  • the Gibbs entropy is<ref name="gibbs"/> <math display="block"> S = -k \langle \log P \rangle = - \frac{\partial F} {\partial T}, </math>
  • the partial derivative Template:Math is approximately related to chemical potential, although the concept of chemical equilibrium does not exactly apply to canonical ensembles of small systems.<ref group=note>Since Template:Math is an integer, this "derivative" actually refers to a finite difference expression such as Template:Math, or Template:Math, or Template:Math. These finite difference expressions are equivalent only in the thermodynamic limit (very large Template:Math).</ref>
  • and the average energy is<ref name="gibbs"/> <math display="block"> \langle E \rangle = F + ST.</math>
• Exact differential: From the above expressions, it can be seen that the function Template:Math, for a given Template:Math, has the exact differential<ref name="gibbs"/> <math display="block"> dF = - S \, dT - \langle p\rangle \, dV .</math>
• First law of thermodynamics: Substituting the above relationship for Template:Math into the exact differential of Template:Math, an equation similar to the first law of thermodynamics is found, except with average signs on some of the quantities:<ref name="gibbs"/> <math display="block"> d\langle E \rangle = T \, dS - \langle p\rangle \, dV .</math>
• Energy fluctuations: The energy in the system has uncertainty in the canonical ensemble. The variance of the energy is<ref name="gibbs"/> <math display="block"> \langle E^2 \rangle - \langle E \rangle^2 = k T^2 \frac{\partial \langle E \rangle}{\partial T}.</math>

Example ensembles

"We may imagine a great number of systems of the same nature, but differing in the configurations and velocities which they have at a given instant, and differing in not merely infinitesimally, but it may be so as to embrace every conceivable combination of configuration and velocities..." J. W. Gibbs (1903)<ref>Template:Cite book</ref>

Boltzmann distribution (separable system)

If a system described by a canonical ensemble can be separated into independent parts (this happens if the different parts do not interact), and each of those parts has a fixed material composition, then each part can be seen as a system unto itself and is described by a canonical ensemble having the same temperature as the whole. Moreover, if the system is made up of multiple similar parts, then each part has exactly the same distribution as the other parts.

In this way, the canonical ensemble provides exactly the Boltzmann distribution (also known as Maxwell–Boltzmann statistics) for systems of any number of particles. In comparison, the justification of the Boltzmann distribution from the microcanonical ensemble only applies for systems with a large number of parts (that is, in the thermodynamic limit).

The Boltzmann distribution itself is one of the most important tools in applying statistical mechanics to real systems, as it massively simplifies the study of systems that can be separated into independent parts (e.g., particles in a gas, electromagnetic modes in a cavity, molecular bonds in a polymer).

Ising model (strongly interacting system)

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

In a system composed of pieces that interact with each other, it is usually not possible to find a way to separate the system into independent subsystems as done in the Boltzmann distribution. In these systems it is necessary to resort to using the full expression of the canonical ensemble in order to describe the thermodynamics of the system when it is thermostatted to a heat bath. The canonical ensemble is generally the most straightforward framework for studies of statistical mechanics and even allows one to obtain exact solutions in some interacting model systems.<ref>Template:Cite book</ref>

A classic example of this is the Ising model, which is a widely discussed toy model for the phenomena of ferromagnetism and of self-assembled monolayer formation, and is one of the simplest models that shows a phase transition. Lars Onsager famously calculated exactly the free energy of an infinite-sized square-lattice Ising model at zero magnetic field, in the canonical ensemble.<ref>Template:Cite journal</ref>

Precise expressions for the ensemble

The precise mathematical expression for a statistical ensemble depends on the kind of mechanics under consideration—quantum or classical—since the notion of a "microstate" is considerably different in these two cases. In quantum mechanics, the canonical ensemble affords a simple description since diagonalization provides a discrete set of microstates with specific energies. The classical mechanical case is more complex as it involves instead an integral over canonical phase space, and the size of microstates in phase space can be chosen somewhat arbitrarily.

Quantum mechanical

A statistical ensemble in quantum mechanics is represented by a density matrix, denoted by <math>\hat \rho</math>. In basis-free notation, the canonical ensemble is the density matrixTemplate:Citation needed

<math>\hat \rho = \exp\left(\tfrac{1}{kT}(F - \hat H)\right),</math>

where Template:Math is the system's total energy operator (Hamiltonian), and Template:Math is the matrix exponential operator. The free energy Template:Math is determined by the probability normalization condition that the density matrix has a trace of one, <math>\operatorname{Tr} \hat \rho=1</math>:

<math>e^{-\frac{F}{k T}} = \operatorname{Tr} \exp\left(-\tfrac{1}{kT} \hat H\right).</math>

The canonical ensemble can alternatively be written in a simple form using bra–ket notation, if the system's energy eigenstates and energy eigenvalues are known. Given a complete basis of energy eigenstates Template:Math, indexed by Template:Math, the canonical ensemble is:

<math>\hat \rho = \sum_i e^{\frac{F - E_i}{k T}} |\psi_i\rangle \langle \psi_i | </math>

<math>e^{-\frac{F}{k T}} = \sum_i e^{\frac{- E_i}{k T}}.</math>

where the Template:Math are the energy eigenvalues determined by Template:Math. In other words, a set of microstates in quantum mechanics is given by a complete set of stationary states. The density matrix is diagonal in this basis, with the diagonal entries each directly giving a probability.

Classical mechanical

In classical mechanics, a statistical ensemble is instead represented by a joint probability density function in the system's phase space, Template:Math, where the Template:Math and Template:Math are the canonical coordinates (generalized momenta and generalized coordinates) of the system's internal degrees of freedom. In a system of particles, the number of degrees of freedom Template:Math depends on the number of particles Template:Math in a way that depends on the physical situation. For a three-dimensional monoatomic gas (not molecules), Template:Math. In diatomic gases there will also be rotational and vibrational degrees of freedom.

The probability density function for the canonical ensemble is:

<math>\rho = \frac{1}{h^n C} e^{\frac{F - E}{k T}},</math>

where

• Template:Math is the energy of the system, a function of the phase Template:Math,
• Template:Math is an arbitrary but predetermined constant with the units of Template:Math, setting the extent of one microstate and providing correct dimensions to Template:Math.<ref group=note>(Historical note) Gibbs' original ensemble effectively set Template:Math, leading to unit-dependence in the values of some thermodynamic quantities like entropy and chemical potential. Since the advent of quantum mechanics, Template:Math is often taken to be equal to the Planck constant in order to obtain a semiclassical correspondence with quantum mechanics.</ref>
• Template:Math is an overcounting correction factor, often used for particle systems where identical particles are able to change place with each other.<ref group=note>In a system of Template:Math identical particles, Template:Math (factorial of Template:Math). This factor corrects the overcounting in phase space due to identical physical states being found in multiple locations. See the statistical ensemble article for more information on this overcounting.</ref>
• Template:Math provides a normalizing factor and is also the characteristic state function, the free energy.

Again, the value of Template:Math is determined by demanding that Template:Math is a normalized probability density function:

<math>e^{-\frac{F}{k T}} = \int \ldots \int \frac{1}{h^n C} e^{\frac{- E}{k T}} \, dp_1 \ldots dq_n </math>

This integral is taken over the entire phase space.

In other words, a microstate in classical mechanics is a phase space region, and this region has volume Template:Math. This means that each microstate spans a range of energy, however this range can be made arbitrarily narrow by choosing Template:Math to be very small. The phase space integral can be converted into a summation over microstates, once phase space has been finely divided to a sufficient degree.

See also

• Microcanonical ensemble
• Grand canonical ensemble