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. 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: ). The ensemble typically also depends on mechanical variables such as the number of particles in the system (symbol: ) and the system's volume (symbol: ), 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 ensemble.
The canonical ensemble assigns a probability to each distinct microstate given by the following exponential:
:<math>P = e^{(F - E)/(k T)},</math>
where is the total energy of the microstate, and is the Boltzmann constant.
The number 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 will vary if different N, V, T are selected. The free energy 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 .
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. It was later reformulated and extensively investigated by Gibbs in 1902.
Properties
= \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 , indexed by , 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 are the energy eigenvalues determined by . 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,
, where the and 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 depends on the number of particles in a way that depends on the physical situation. For a three-dimensional monoatomic gas (not molecules), . 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
- is the energy of the system, a function of the phase ,
- is an arbitrary but predetermined constant with the units of , setting the extent of one microstate and providing correct dimensions to .
- is an overcounting correction factor, often used for particle systems where identical particles are able to change place with each other.
- provides a normalizing factor and is also the characteristic state function, the free energy.
Again, the value of is determined by demanding that 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 . This means that each microstate spans a range of energy, however this range can be made arbitrarily narrow by choosing 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