Bose-Einstein Distribution Function

Statistical mechanics is that branch of Physics that considers how the overall behavior of a system of many particles is related to the properties of particles themselves. Justifying its name, statistical mechanics is not concerned with the actual motion or interaction of individual particles. Rather, it is concerned with what is most likely to happen. And, this is where distribution functions come to existence. The 27th article of Equations of month series is dedicated to one such distribution function, The Bose-Einstein Distribution Function. So, let's begin!

Bose - Einstein distribution function :

The Bose-Einstein Distribution Function basically tells about the probability of finding a boson in a particular energy state as a function of temperature of the system.

The Bose-Einstein Distribution Function is one of most important distributions used to study the quantum mechanical systems. As we know, Our quantum world has two types of particles. One, which obey Pauli's exclusion principle and two, which don't don't have anything to do with the exclusion principle. The first category of particles are called fermions and second ones are known as bosons. And, Bose-Einstein distribution function holds for the latter.

Origin of Bose-Einstein Statistics:

In 1924, Indian physicist S.N Bose derived Planck's radiation formula on the basis of the quantum theory of light with indistinguishable photons. Thus these statistics were introduced for photons in 1924 by Bose. But, his paper was rejected by a leading British journal. Later, he sent his paper to Einstein. Einstein extended Bose's treatment to material particles whose number is conserved. Moreover, Einstein translated Bose's paper to German and submitted it to a German journal where it got published. In this way, both Bose and Einstein got their associated with these statistics.

Different terms in Bose-Einstein Distribution function:

In Bose Einstein distribution function, f(E) represents the probability of a boson occupying an energy state E. T represents the temperature of the system. And, K is the Boltzmann's constant. The quantity A depends on the properties of the particular system. A can also be a function of T. No matter what the values of A, E, or T are, f(E) can never exceed 1. However, if E >> KT, then f(E) approaches the Maxwell Boltzmann statistics, which are obeyed by classical particles such as gas molecules.

Bose-Einstein Condensation:

The greater the number of different ways in which particles can be arranged among the available states to yield a particular distribution of energies, the more probable is the distribution. At very low temperatures, the probability of finding bosons in a particular state increases. Thus, at very low temperatures, bosons condense into a single energy state. This explains the phenomenon of Bose Einstein Condensation.

A comparison of different statistical distributions

Author's message:

Statistical mechanics is one of the most powerful tools for a theoretical physicist. And, the distribution functions are the backbone of statistical physics. So, if you are intending to pursue your career in theoretical physics, you must prepare yourself to encounter such distributions frequently. Not only this, Bose Einstein distribution has also played fundamental roles in development of different complex systems, including the World Wide Web.

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