STATISTICAL MECHANICS | Bose - Einstein Statistics | Thermodynamic probability | Fermi Dirac Statistics | Comparison among three distribution curve

Bose – Einstein Statistics or BE Statistics

Basic postulates of BE statistics

i) Particles are identical and hence indistinguishable.

ii) The particles must have integral spin i.e. 0,1, 2, 3,…………….. i.e. spin angular momentum 0, ħ, 2ħ, ……………….

iii) There is no restriction imposed on the number of particles having same energy. (i.e. no priory restriction in this regard)

iv) Pauli’s exclusion principle is not applicable to the particles.

v) Heisenberg’s uncertainly principle is applicable to the particles .

Moreover, if the system is isolated,

a) Total number of particles is constant i.e.

N = ΣNi = Constant.

𝛿N = Σ𝛿Ni = 0 —————– 1

b) Total energy of the system is constant i.e.

E = ΣNiEi = constant.

𝛿E = ΣEi𝛿Ni = 0 —————– 2

Note : The particles which obey BE statistics, called Bosons.

Example of Bosons : Photons with spin 1,

Photons with spin 0,

deuterons with spin 1,

α particles with spin 0,

π- mesons wilt spin 0.

Derivation of BE – distribution functions

To determine BE-distribution function we divide the problem into two parts —

i) First, the distinct ways of forming groups of N1, N2, N3 ——- particles with energy E1, E2, E3 —- respectively out of total number of particles N ; denoted by G.

ii) Second, for each group of Ni particles, the number of ways of

distributing the Ni particles among gi sub shells ; denoted by G’

Thermodynamic probability :

could be chosen

i) Since the particles are indistinguishable, the number of ways in which the group of particles N1, N2, ……… from N particles, G = 1

ii) Consider the following figure ; where a line denotes the quantum state ( gi = 7 here) and a dot denotes the particles (Ni = 12 here).

In the array, there are (gi+Ni) quantum states plus particles i.e. (qi+Ni) objects.

Keeping the first quantum state ( first line ) fixed, the remaining (Ni + gi -1) objects can be permuted in (Ni + gi -1)! possible ways. Again Ni particles can be permuted among themselves in Ni ! ways and (gi – 1) quantum states can be permuted among themselves in (gi – 1) ! do not affect the distributions .

Thus possible distinguishable different ways / arrangements of Ni particles in gi quantum states is

Considering all groups, total number of ways of distribution all the N particles in different states,

Thermodynamic probability, w = GG’

(b) Most probable distribution function :

Taking logarithm of both sides of equation 3,

using Stirling approximation , ln x ! = xin x-x

For most probable distribution; entropy of the system , s = k lnw

must be maximum i.e. ðs = 0

According to Lagrange’s method of undetermined multiplier, multiplying equation 1 by -α and equation by “-β” and adding with equation 5, we get –

Since ; is independent and above equation holds for any value of i,

Thus, BE – distribution function,

Again,

Thus,

Under what condition do the BE-distribution function tend to MB distribution function :

At very high temperature the particles are distinguishable in wide energy range. The number of particles in any range be much smaller than the number of quantum states in that range,

So,

BE distribution function

Taking log ,

which is MB distribution function

Fermi Dirac Statistics

Basic postulates of info FD Statistics

i) The particles are identical and hence indistinguishable.

ii) The particles must have half integral spin i.e. 1/2, 3/2, …………. spin angular momentum ħ/2, 3ħ/2, ………….

iii) There is a restriction imposed on the number of particles having same energy i.e. a definite priory restriction on the number of particles in a quantum state.

iv) Pauli’s exclusion principle is applicable to the particles

v) Heisenberg’s uncertainty principles is applicable to the particles.

Moreover, if the system is isolated,

a) total number of particles is constant i.e.

N = ΣNi = Constant.

𝛿N = Σ𝛿Ni = 0 —————– 1

b) total energy of the system is constant i.e.

E = ΣNiEi = constant.

𝛿E = ΣEi𝛿Ni = 0 —————– 2

Note: The particles which obey FD statistics, called Fermions.

Example of Fermions:

Nucleons ( protons and neutrons) —- spin 1/2

Electron and Positron ——- spin 1/2

Neutrino, μ-meson He3 ———- spin 1/2

Derivation of FD-distribution function

To derive FD-distribution function, we divide the problem into two parts –

i) First, the distinct ways of forming groups of N1, N2, N3 ——- particles with energies E1, E2, E3 —- respectively out of a N particles ; denoted by G

ii) Second, for each group of Ni particles, the number of ways of distributing among gi sub shells ; denoted by G’

Thermodynamic probability :

i) Since the particles are indistinguishable ; G = 1

i) Let Ni particles are distributed among gi sub shells. Pauli’s exclusion principle imposes the restriction that each quantum state can be occupied by at most one particle.

Since Ni particles are to be distributed among gi quantum states ( gi ≥ Ni) having same energy Ei ; Ni stales will be filled up and (gi – Ni) will remain vacant.

The gi shells /states can be arranged in gi ! ways. Again Ni particles can be arranged themselves in Ni ! ways which is irrelevant as the particles are indistinguishable . Also, (gi – Ni) vacant states can be arranged in (gi – Ni) ! ways which is irrelevant as the states are not occupied.

Thus, number of distinguishable arrangements of Nį particle among gi quantum states in the i th energy level is –

Considering all energy levels , number of arrangements of N particles-

Thermodynamic probability, w = GG’

(b) Most probable distribution :

Taking logarithm of bolt sides of equation 3 ,

using Stirling approximation ln x ! = xin x-x

For most probable distribution; entropy of the system s = k lnw

must be maximum i.e. ðs = 0

According to Lagrange’s method of undermined multiplier, multiplying equation 1 by -α and equation 2 by “-β” and adding with equation 5, we get –

Since ðNi is independent and above equation holds for any value of i,

Thus ,

So, FD-distribution function –

Again,

and the quantity is usually expressed in the form where EF is called Fermi energy of the state.

Note- If the energy levels are closely spaced the

where, g(E) is the number of states in unit energy interval around E.

N(E) is the number of particles in unit energy interval around E.

f(E) is the probability of occupancy in unit energy interval around E.

Under what condition FD -statistics reduces to classical MB-statistics –

We have FD -distribution functions,

At very high temperature the particles are distinguishable over a wide energy range i.e. number of particles in any range would be much smaller than the number of the quantum states, i.e.