Particle statistics


Particle statistics is a particular description of multiple particles in statistical mechanics. A key prerequisite concept is that of a statistical ensemble that emphasizes properties of a large system as a whole at the expense of knowledge about parameters of separate particles. When an ensemble describes a system of particles with similar properties, their number is called the particle number and usually denoted by N.

Classical statistics

In classical mechanics, all particles in the system are considered distinguishable. This means that individual particles in a system can be tracked. As a consequence, changing the position of any two particles in the system leads to a completely different configuration of the entire system. Furthermore, there is no restriction on placing more than one particle in any given state accessible to the system. These characteristics of classical positions are called Maxwell–Boltzmann statistics.

Quantum statistics

The fundamental feature of quantum mechanics that distinguishes it from classical mechanics is that particles of a particular type are indistinguishable from one another. This means that in an assembly consisting of similar particles, interchanging any two particles does not lead to a new configuration of the system. In the case of a system consisting of particles of different kinds, the wave function of the system is invariant up to a phase separately for both assemblies of particles.
The applicable definition of a particle does not require it to be elementary or even "microscopic", but it requires that all its degrees of freedom that are relevant to the physical problem considered shall be known. All quantum particles, such as leptons and baryons, in the universe have three translational motion degrees of freedom and one discrete degree of freedom, known as spin. Progressively more "complex" particles obtain progressively more internal freedoms, and when the number of internal states, that "identical" particles in an ensemble can occupy, dwarfs their count, then effects of quantum statistics become negligible. That's why quantum statistics is useful when one considers, say, helium liquid or ammonia gas, but is useless applied to systems constructed of macromolecules.
While this difference between classical and quantum descriptions of systems is fundamental to all of quantum statistics, quantum particles are divided into two further classes on the basis of the symmetry of the system. The spin–statistics theorem binds two particular kinds of combinatorial symmetry with two particular kinds of spin symmetry, namely bosons and fermions.