Question

1. Describe the Bose-Einstein, Fermi-Dirac, Maxwell-Boltzmann distribution functions and

compare.

Answer 1

The Bose-Einstein functions:

The Bose-Einstein distribution describes a state of matter (also called the fifth state of matter) which is typically formed when a gas of bosons at low densities is cooled to temperatures very close to absolute zero. It describes the statistical behavior of integer spin particles (bosons). At low temperatures, bosons can behave very differently than fermions because an unlimited number of them can collect into the same energy state, a phenomenon called "condensation" and is given by:

Two examples of materials containing Bose-Einstein condensates are superconductors and superfluids. Superconductors conduct electricity with virtually zero electrical resistance: Once a current is started, it flows indefinitely. The liquid in a superfluid also flows forever.

The Fermi-Dirac distribution function:

The Fermi-Dirac distribution function, also called Fermi function, provides the probability of occupancy of energy levels by Fermions. Fermions are half-integer spin particles, which obey the Pauli exclusion principle. The Pauli exclusion principle postulates that only one Fermion can occupy a single quantum state. It applies to fermions, particles with half-integer spin which must obey the Pauli exclusion principle. Each type of distribution function has a normalization term multiplying the exponential in the denominator which may be temperature dependent. For the Fermi-Dirac case, that term is usually written:

The significance of the Fermi energy is most clearly seen by setting T=0. At absolute zero, the probability is equal to 1 for energies less than the Fermi energy and zero for energies greater than the Fermi energy. We picture all the levels up to the Fermi energy as filled, but no particle has a greater energy. This is entirely consistent with the Pauli exclusion principle where each quantum state can have one but only one particle.

Maxwell-Boltzmann distribution function:

It is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann. It  is the classical distribution function for distribution of an amount of energy between identical but distinguishable particles.

Besides the presumption of distinguishability, classical statistical physics postulates further that:

• There is no restriction on the number of particles which can occupy a given state.
• At thermal equilibrium, the distribution of particles among the available energy states will take the most probable distribution consistent with the total available energy and total number of particles.
• Every specific state of the system has equal probability.

One of the general ideas contained in these postulates is that it is unlikely that any one particle will get an energy far above the average (i.e., far more than its share). Energies lower than the average are favored because there are more ways to get them. If one particle gets an energy of 10 times the average, for example, then it reduces the number of possibilities for the distribution of the remainder of the energy. Therefore it is unlikely because the probability of occupying a given state is proportional to the number of ways it can be obtained.

Comparison:

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) 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.

b) 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.

c) 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.