Canonical ensemble

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).

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.[11]

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.[12]

Example of canonical ensemble for a quantum system consisting of one particle in a potential well.

Plot of all possible states of this system. The available stationary states displayed as horizontal bars of varying darkness according to |ψ_i(x)|².

A canonical ensemble for this system, for the temperature shown. The states are weighted exponentially in energy.

The particle's Hamiltonian is Schrödinger-type, Ĥ = U(x) + p²/2m (the potential U(x) is plotted as a red curve). Each panel shows an energy-position plot with the various stationary states, along with a side plot showing the distribution of states in energy.

A statistical ensemble in quantum mechanics is represented by a density matrix, denoted by ${\displaystyle {\hat {\rho }}}$. In basis-free notation, the canonical ensemble is the density matrix

${\displaystyle {\hat {\rho }}=\exp \left({\tfrac {1}{kT}}(F-{\hat {H}})\right),}$

where Ĥ is the system's total energy operator (Hamiltonian), and exp() is the matrix exponential operator. The free energy F is determined by the probability normalization condition that the density matrix has a trace of one, ${\displaystyle \operatorname {Tr} {\hat {\rho }}=1}$:

${\displaystyle e^{-{\frac {F}{kT}}}=\operatorname {Tr} \exp \left(-{\tfrac {1}{kT}}{\hat {H}}\right).}$

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 |ψ_i⟩, indexed by i, the canonical ensemble is:

${\displaystyle {\hat {\rho }}=\sum _{i}e^{\frac {F-E_{i}}{kT}}|\psi _{i}\rangle \langle \psi _{i}|}$

${\displaystyle e^{-{\frac {F}{kT}}}=\sum _{i}e^{\frac {-E_{i}}{kT}}.}$

where the E_i are the energy eigenvalues determined by Ĥ|ψ_i⟩ = E_i|ψ_i⟩. 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.

Example of canonical ensemble for a classical system consisting of one particle in a potential well.

Plot of all possible states of this system. The available physical states are evenly distributed in phase space, but with an uneven distribution in energy; the side-plot displays dv/dE.

A canonical ensemble for this system, for the temperature shown. The states are weighted exponentially in energy.

Each panel shows phase space (upper graph) and energy-position space (lower graph). The particle's Hamiltonian is H = U(x) + p²/2m, with the potential U(x) shown as a red curve. The side plot shows the distribution of states in energy.

In classical mechanics, a statistical ensemble is instead represented by a joint probability density function in the system's phase space, ρ(p₁, … p_n, q₁, … q_n), where the p₁, … p_n and q₁, … q_n 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 n depends on the number of particles N in a way that depends on the physical situation. For a three-dimensional gas of monoatoms (not molecules), n = 3N. In diatomic gases there will also be rotational and vibrational degrees of freedom.

The probability density function for the canonical ensemble is:

${\displaystyle \rho ={\frac {1}{h^{n}C}}e^{\frac {F-E}{kT}},}$

where

• E is the energy of the system, a function of the phase (p₁, … q_n),
• h is an arbitrary but predetermined constant with the units of energy×time, setting the extent of one microstate and providing correct dimensions to ρ.[note 3]
• C is an overcounting correction factor, often used for particle systems where identical particles are able to change place with each other.[note 4]
• F provides a normalizing factor and is also the characteristic state function, the free energy.

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

${\displaystyle e^{-{\frac {F}{kT}}}=\int \ldots \int {\frac {1}{h^{n}C}}e^{\frac {-E}{kT}}\,dp_{1}\ldots dq_{n}}$

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 hⁿC. This means that each microstate spans a range of energy, however this range can be made arbitrarily narrow by choosing h 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.