The Gibbs measure of an infinite system is not necessarily unique, in contrast to the canonical ensemble of a finite system, which is unique.
32.
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.
33.
Lars Onsager famously calculated exactly the free energy of an infinite-sized square-lattice Ising model at zero magnetic field, in the canonical ensemble.
34.
Although appearing benign at first, this leads to a problem of severely non-extensive entropy in the canonical ensemble, known today as the Gibbs paradox.
35.
In the canonical ensemble, the entropic force \ mathbf F associated to a macrostate partition \ { \ mathbf { X } \ } is given by:
36.
For example, in the canonical ensemble the number of particles inside the system is held fixed, whereas particle number can fluctuate in the grand canonical ensemble.
37.
For example, in the canonical ensemble the number of particles inside the system is held fixed, whereas particle number can fluctuate in the grand canonical ensemble.
38.
The partition function can be written as the weighted sum of the partition function of canonical ensemble, Z ( N, V, T ) \,.
39.
The reason for this is that various thermodynamic ensembles ( canonical ) become equivalent in some aspects to the grand canonical ensemble, once the system is very large.
40.
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.
