# The Basic Theory of Ensemble Theory

In physics, specifically statistical physics, the idea of the statistical ensemble is used as a method to study the thermodynamics and behaviours of a system. The notion of an ensemble was first introduced by J. Willard Gibbs in 1902 (1). Rather than attempting to study the individual microscopic states that a system can have, ensemble theory involves considering many copies of the system that represent all possible states that it may be in. We can then average over the collection of systems, the ensemble, to study the behaviour of it. This is done using some density function.

The microcanonical ensemble is used to describe a system where-by there is no change in energy, volume, or number of particles with the surroundings. These are called the macroscopic intrinsic variables of the system. Quantities such as pressure are described as extrinsic as they can fluctuate. The thermodynamics of such systems can be found mathematically using a multiplicity function, which describes the total number of accessible microstates for the system. From it, we can derive the entropy, energy, heat capacity, pressure, and many more useful thermodynamic quantities of the system.

The canonical ensemble is used when the energy of the system is no longer held constant. The usual picture for such schemes is that of a system in contact with a heat bath or heat reservoir at a fixed temperature. Now the system can exchange energy with this external source. Here, the intrinsic variables are the volume, number of particles, and the temperature of the system. The partition function is the analogous of the multiplicity function from the microcanonical ensemble, in that it is the mathematical tool used here to study the thermodynamic behaviour of the system, however it does not represent the same thing on a base level. Essentially it is the probability for a system to have a certain energy, compared to the mircocanical description of the number of particles with that energy. (2)

For consideration of systems where both the energy and number of particles can vary, we use the grand-canonical ensemble. As before, the simple picture of the systems emersed in a heat bath is useful and there is a corresponding grand canonical partition function used to mathematically derive our thermodynamic quantities.

But what kind of systems can we study? The incredible thing about statistical physics and ensemble theory is that it can be used to describe both classical and quantum mechanical systems. In fact, the mathematics of the quantum mechanical systems is easier! Phase transitions, which are the change of state of a system- such as water to ice, can be studied in mathematical detail using this formalism. This includes interesting examples such as Bose Einstein condensation of a gas of boson particles (particles with integer spin values), where the particles when cooled to a near zero temperature, form together into an almost single atom state. With ensemble theory and statistical physics, we can find the exact temperature this may occur at!

Bose Einstein Condensation as temperature decreases from left to right. NASA/JPL-Caltech. https://www.jpl.nasa.gov/images/pia22561-bose-einstein-condensate-graph

References:

1. Gibbs JW. Elementary Principles in Statistical Mechanics: Developed with Especial Reference to the Rational Foundation of Thermodynamics. Cambridge University Press; 2010.

2. Pathria, R.K. and Beale, P.D. (2011) Statistical Mechanics. 3rd Edition, Elsevier, Singapore, 583-586.
http://dx.doi.org/10.1016/B978-0-12-382188-1.00015-3

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