STATISTICAL MECHANICS | Applications and problems of BE- statistics

Applications and problems of BE- statistics

We shall now consider some applications (some of them, we previously know) of quantum statistics, that focus in this section on the Bose-Einstein distribution.

1. Planck’s law of black- body radiation

2. Wien’s Law

3. Rayleigh- Jeans law

4. Bose-Einstein condensation

5. Specific heat of solids ; Einstein – Debye theory

1. Planck’s radiation law


Let a black body chamber of volume V at absolute temperature T be filled will photons of energy hν. Again, photon chamber is considered as a “gas” whose particles are photons having unit spin angular  momentum having two modes of vibrations.

Hence, they are ” bosons” and we can apply BE statistics for the studies of black body radiation.

The number of photons having energy range E and E+dE

N(E)dE = f(E)g(E)dE …………………….. 1

Where f(E) = BE distribution function

g(E)dE – number of quantum states of photos having energy between E and E+dE

Number of quantum states of photons having momentum range p and p+dp ,

  …………….. 2

( Here 2 is multiplied as photons have two mode of vibration )

Again, momentum of photon having frequency ν

Thus , number of quantum states of photon having frequency  range ν and ν+dν

……………… 3

In a black-body chamber at constant temperature, photons of different energies are absorbed and re- emited by the walls of the enclosure, hence  Σ𝛿Ni = 0  is not maintained.

Thus Lagrange’s undetermined multiplier α = 0

From equation 1

………….. 4

Thus , number of photons having frequency range ν and ν+dν

……………….. 5

Since, energy of each photon of frequency ν is hν, hence energy per unit volume of the enclosure within a frequency ν and ν+dν i.e. energy density

which is known as Planck’s law.


2. Wien’s law

On terms of frequency, The Planck’s radiation law ,

In terms of wavelength to :

…………….. 1

For short wavelength (or high frequency) and low temperature, λT = small.



Which is Wien’s radiation law.


3. Rayleigh – Jeans law

The Rayleigh–Jeans law agrees with experimental results at the large wavelengths (i.e. low frequencies) but strongly disagrees at the short wavelengths ( i.e. high frequencies). This inconsistency between observations and the predictions of classical physics is very popular in physics and known as the ultraviolet catastrophe.

For long wavelength (or short frequency) and high temperature,

hc/λkT is small.

which is Rayleigh – Jeans law.


4. Bose – Einstein condensation

Bose first sent a paper to Einstein on the quantum statistics of light quanta., namely photons, in which he derived Planck’s quantum radiation law without any reference to classical physics. Einstein was extremely impressed and translated the paper himself from English to German and submitted it for Bose to the Zeitschrift für Physik, which published it in 1924. Einstein then extended Bose’s ideas to matter in two other papers. The result of their efforts is the concept of a Bose gas, governed by Bose–Einstein statistics, that describes the statistical distribution of identical particles with integer spin, known as bosons. Bosons particles that include the photon as well as atoms such as ( helium-4 ), are allowed to share a quantum state. Einstein proposed that cooling bosonic atoms to a very low temperature would cause them to  “condense” into the lowest accessible quantum state, resulting in a new form of matter.

The quest to produce a Bose– Einstein condensate in the laboratory was stimulated by a paper published in 1976 by two Program Directors at the National Science Foundation (William Stwalley and Lewis Nosanow). This led to the immediate pursuit of the idea by four independent research groups

Average number of bosons lying between energy range E and E+dE,

Where f(E) = BE distribution function –

number of quantum states in the energy range E and E+dE,

Replacing the summation by integration,

So, Particle density –

    …………………. 1

Let us vary the temperature of the gas keeping particle density is constant .To keep N/V constant, R.H.S. of equation 1 must be constant ; This means when T is lowered α  decreases.

Let at temperature

,  α  is zero.


Put ,

…………….. 2

……………. 3

TB is called Base temperature ; which depends on particle mass( m) and particle density (N/V) .

What will happen if the temperature is further reduced ?

α can’t be positive or negative again ; hence the only possibility is for a to remain equal to zero when the temperature is decreased further from TB up to 0.

The variation of α of the ideal ‘Boson gas’ is shown in figure. There is a difficulty attains in this stage. The ground state term with E = 0 and α=0

  becomes infinity.

That is, for Boson gas, there is no limitation on the number of particles that can belong to a single state.

To remove this contraction, we isolate the population of the ground state N in a separate term No and we can write then,

using equation 1

Taking α = 0 and at temperature T the integral is as like equation 2

Substituting the value of v from equation, 3, we get –


……………………… 4

At T → 0, No→ N, all the Bosons at T =o are in the ground state. This macroscopic occupation of the zero momentum ground state is called Bose – Einstein condensation.

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