Canonical ensemble

Statistical mechanics
Thermodynamics Kinetic theory
Particle statistics Spin–statistics theorem Indistinguishable particles Maxwell–Boltzmann Bose–Einstein Fermi–Dirac Parastatistics Anyonic statistics Braid statistics
Thermodynamic ensembles NVE Microcanonical NVT Canonical µVT Grand canonical NPH Isoenthalpic–isobaric NPT Isothermal–isobaric
Models Debye Einstein Ising Potts
Potentials Internal energy Enthalpy Helmholtz free energy Gibbs free energy Grand potential / Landau free energy
Scientists Maxwell Boltzmann Bose Gibbs Einstein Ehrenfest von Neumann Tolman Debye Fermi
v t e

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.^[1] 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: T). The ensemble typically also depends on mechanical variables such as the number of particles in the system (symbol: N) and the system's volume (symbol: V), 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 NVT ensemble.

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

${\displaystyle P=e^{(F-E)/(kT)},}$

where E is the total energy of the microstate, and k is the Boltzmann constant.

The number F 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 F will vary if different N, V, T are selected. The free energy F 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 F(N, V, T).

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

${\displaystyle \textstyle P={\frac {1}{Z}}e^{-E/(kT)},}$

using the canonical partition function

${\displaystyle \textstyle Z=e^{-F/(kT)}}$

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.^[2] It was later reformulated and extensively investigated by Gibbs in 1902.^[1]