On the superiority of the canonical ensemble
by Jonathan Mooney
Senior contributing writer
Statistical thermodynamics is a subject in physics and chemistry which allows the calculation and understanding of macroscopic, thermodynamic properties based on microscopic, quantum behaviour. Statistical mechanics relies on the concept of an ensemble; that is, an enormous collection of systems in the same macroscopic state. For example, a collection of 1023 systems, each of which is composed of a liter of helium gas at a specified temperature and pressure and constitutes an ensemble. While in an ensemble the macroscopic properties of each system are the same, the microscopic properties differ. For example, the positions and momenta of the individual helium atoms will vary from system to system, even if all systems have the same temperature and pressure.
There are three basic ensembles used in statistical mechanics: the microcanonical ensemble, the canonical ensemble, and the grand canonical ensemble. In the microcanonical ensemble, the number of particles, the energy, and the volume are fixed. While certainly useful for calculating certain quantities such as entropy and while indeed fundamental to very basic statistical mechanics, the microcanonical ensemble is sadly not very useful in dealing with most systems since the energy must be fixed. Nevertheless, its useful properties certainly make the microcanonical ensemble the second-best of the major ensembles.
In the canonical ensemble, the number of particles, the temperature, and the volume are fixed. Since the system is able to exchange energy, the canonical ensemble is the best representative for real systems. Indeed, the partition function of the canonical ensemble is the one used to calculate thermodynamic quantities in the majority of cases. Clearly, the canonical ensemble is best.
The grand canonical ensemble, despite its excellent name, is sadly the least useful of all ensembles. In this ensemble, the volume and temperature are fixed but the number of particles is allowed to vary. The grand canonical ensemble does not represent most real systems, and it lacks the beauty, simplicity, and importance to statistical mechanics of the microcanonical ensemble. It is the worst ensemble.