# What is statistical equilibrium mechanics

## b. What is statistical mechanics?

One might ask what statistics have to do with phases and phase transitions. In everyday life, statistics appear to us as something unnatural that was invented by some bureaucrats to put people and things into categories. But that is not correct. In reality statistics are very natural and they are very helpful in explaining many physical phenomena that appear in many particle systems. In this sense basedPhases and phase transitions on the statistics.

For example, if you want to describe a gas or a liquid and calculate its further development, you don't have the slightest chance of describing each individual particle separately. Because it is very strongly influenced by the surrounding particles at every moment. However, one can make very good statistical statements about the whole system.

For example, the temperature of an ideal gas in thermal equilibrium is defined as the mean kinetic energy of the particles in the gas divided by the so-called Boltzmann constant, named after Ludwig Boltzmann, one of the founders of statistical mechanics.

The bracket around the energy variable indicates that this is the mean kinetic energy of all particles in the gas. So it is very easy to determine the mean kinetic energy of the particles, but it is almost impossible to determine the speed of each individual particle. In addition, it is also not desirable to determine the speed of each individual particle, as there is not much that can be done with this information, because its energy changes permanently due to collisions with the other particles, while the mean kinetic energy, and thus the temperature, remains constant.

One can extend this concept to many other parameters such as pressure and viscosity. This concept is so universal that it has also been applied to more abstract properties such as magnetism and superconductivity and the associated phase transitions.

In summary, statistical mechanics is the basis for describing physical phenomena in many particle systems. Since phases, in this sense, only appear in many particle systems, statistical mechanics is required to be able to make predictions about which phase, under which conditions, exists and which external parameters have to be changed in order to cause a phase transition.

Statistical mechanics as introduced here is a classical theory. That means it does not include quantum effects. However, quantum effects are a necessary prerequisite for Bose Einstein condensation. Therefore, the next section addresses the question:

What is quantum statistics?