The principal thermodynamic variable of the canonical ensemble, determining the probability distribution of states, is the absolute temperature (symbol: T).
An alternative but equivalent formulation for the same concept writes the probability as using the canonical partition function rather than the free energy.
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.
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.
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).
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.
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] 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.
In classical mechanics, a statistical ensemble is instead represented by a joint probability density function in the system's phase space, ρ(p1, … pn, q1, … qn), where the p1, … pn and q1, … qn are the canonical coordinates (generalized momenta and generalized coordinates) of the system's internal degrees of freedom.