Quantum Statistics & Statistical Distributions

Quantum Statistics refers to situations where statistical mechanics is applied to quantum mechanical systems. A “quantum” ensemble applies when the particles of the set become indistinguishable.

There is a fundamental feature of quantum mechanics that distinguishes it from classical mechanics, 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 any new configuration of the system (that is, the wave function of the system is invariant up to a phase with respect to the interchange of the constituent particles). In the case of a system consisting of particles of different kinds (for example, electrons and protons), 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 (represented with the wave function) and one discrete degree of freedom, known as spin. Progressively more "complex" particles obtain progressively more internal freedoms (such as various quantum numbers in an atom), and when the number of internal states, that "identical" particles in an ensemble can occupy, reduces their count (the particle number), then effects of quantum statistics become negligible.

Let’s say we have two points of collection and only two cells to be occupied.

1) For Maxwell-Boltzmann statistics, in classical mechanics, each particle has a recognizable individuality, hence the particles are distinguishable and each particle is as likely to be in one cell as the other. This means that each or both of the particles can occupy any one of the two cells. So we have four possible arrangements, each of which is counted in order to assign a probability to the distribution.

2) In Bose-Einstein statistics, the particles loose their individuality, and become indistinguishable. So, we must concentrate our attention on the cell rather than the individual articles. Each cell may have any number of particles, so in that case, we have only three possible arrangement.

3) In Fermi-Dirac statistics, since particles are indistinguishable, again we have to concentrate our attention on the cells rather than on individual particles. But in this case, the particles obey Pauli’s exclusion principle according to which it is not possible to have more than one representative point in any one cell. So, each cell will contain at the most one particle. Therefore, in this case, only one arrangement is possible.

Bose-Einstein and Fermi-Dirac are the two quantum statistics.

Now if we have to develop a quantum statistical theory for a system containing a large number of identical and indistinguishable particles, we need to consider a wave function for many body state. For particles having spin zero or integral multiples of the unit h/2π, the wave functions are symmetric(no change of sign on interchanging any pair of identical particles), and for particles with spin that is odd half integral multiples of h/2π, the wave functions are anti-symmetric(Sign changes on interchanging any pair of identical particles). Particles which obey the Bose-Einstein statistics are called Bosons, and those which follow the Fermi-Dirac statistics are called Fermions.

The difference between the three types of statistics

These three statistics concern when we speak about how particles occupy a system which consists of several energy levels (and each energy level could also have several energy states). A particle in this system can be in one of those energy levels depending on the energy particle has. It’s impossible to have just one particle in a system since in real life it needs numerous particle to constitute a system. They occupy the levels under a statistics rule. There are three statistics:

• Particles which are regulated by Maxwell-Boltzmann Statistics have to be distinguishable each other and one energy state can be occupied by two or more particles. Distinguishable means that if we have 2 particles, let say A and B, also two states, 1 and 2, and we put A to state 1 and B to state 2, it will be different with the distribution A to state 2 and B to state 1. It means that A and B are distinct.
• Particles which are regulated by Bose-Einstein Statistics have to be indistinguishable each other and one energy state can be occupied by two or more particles. So instead of saying it as particle A or B, we call it as just “particle” since they are the same thing.
• Particles which are regulated by Fermi-Dirac Statistics have to be indistinguishable each other and one energy state can be occupied by only one particle. So we have to fill it to another state when a state has just been occupied by another particle.

You may ask at when do they apply? Actually it depends on the system you are dealing with. In physics there are a lot of system that use those systems. For instance classical gas satisfies Maxwell-Boltzmann Statistics, electron system satisfies Bose-Einstein Statistics, photon system satisfies Fermi-Dirac Statistics, and so on.