2. INTRODUCTION
Definition: Statistical mechanics deals with the
properties of macroscopic bodies which are in
thermal equilibrium it provides special
probability laws for the distribution of atoms.
The study of statistical mechanics can be
classified as:
• A)classical statistics
• B)Quantum statistics.
3. The quantum statistics developed by Bose,
Einstein, Fermi and Dirac , can again be put into
two categories:
• I) Bose-Einstein (B-E) statistics, and
• Ii)Fermi-Dirac (F-D) statistics.
4. • The classical statistics is based on the classical
results of Maxwell-Boltzmann distribution of
velocities of an assembly in equilibrium . this
successfully explained the observed
phenomenon such as pressure , temperature,
energy etc.
• But fails to account for several observed
phenomenon like black body radiation,
specific heat at low temperature ,
photoelectric effect etc.
5. • In order to deduce the planks radiation law ,
Bose , in 1924 formulated bcertain
fundamental assumption which were different
from classical statistics . On the basis of these
assumption he derived radiation law . in the
same year Einstein utilised practically the
same principle and explained the kinetic
theory of gases.
• In 1926 fermi-dirac modified Bose Einstein
statistics using some additional principle
suggested by Pauli in connection with
electromagnetic structure.
6. • According to Pauli principle no two electronic
having the same set of quantum numbers may be
in the same state .
7. Now we consider the fallowing three kinds of identical
particles:
a) Identical particles of any spin which are
seperated in the assembly and can be distinguished
from one another .the molecules of the gas are
particles of this kinds
b)Identical particles of zero or integral spin which
can not be distinguished from one another .thease
particles are known as Bosons .they do not obey
pauli ‘s exclusion principles .photons , alpha
particles etc.
c)Identical particles of half integral spin which
cannot be distinguished from one another these
particles obey Pauli's exclusion principle s.
8. Bose- Einstein statistics:
• In B-E satistics the particles
are indistnguisable so the
interchange of two particles
b/n two energy states will
not produce any new state.
• For ex.let us distribute four
particles (a,b,c and d)
• among two cells x and y such
that there are three particles
in cell x while one particles in
cell y.
• There will 4 states in M- B
statistics & one in B-E
statistics as shown in fig.
9. • Now , further suppose that each cell is divided
into four states .in case the distribution is
shown in fig . in this case there will be 20
possible distrubution for 3 particles in cell x
and 4 possible distribution of one particles in
cell y .
• There will be 20x4 =80 possible distribution.
10. Thus in Bose-Einstein statistics the conditions
are:
• 1) the particles are indistinguishable from
each other so that there is no distinction
between the different ways in which no
𝑛𝑖particles can be chosen .
• Each cell or sublevel of 𝑖 𝑡ℎ
quantum state
0,1,2.....,in identical particles
• The sum of energies of all the particles in the
different quantum groups, taken together
constitutes the total energy of the system.
11. Most probable distribution of the elements
among various energy levels for a system
obeying Bose-Einstein statistics is .
𝑛𝑖
𝑔𝑖
=
1
[𝑒α+βε 𝑖 − 1]
12. Fermi-Dirac statistics:
• In M-B,or B-E statistics there is no restrictions o n
the particles to present in any energy state . but
in case of fermi- Dirac statistics, applicable to
particles like electron and obeying PEP
• (no two electron in an atom have same energy
state)
• Only one particles can occupy a single energy
state
• The distribution of four particles (a,b,c&d) among
two cells x &y each having 4 energy state
13. • Such that there are three particles in cell x while
one particles in cell y is as shown in fi
• In this case there will 4x4 = 16 possible
distribution
• Most probable distribution according to fermi-
dirac statistics is
𝑛 𝑖
𝑔 𝑖
=
1
[𝑒α+βε 𝑖+1]
14. Maxwell-Boltzmann statistics
• Here the classical assumption that equal
region in the phase space are a priori equally
probable is replaced by the assumption that
each quantum state is priori equally pribable
• Consider a system having n distinguisale
particle . let these particle be divided into
quantum groups such that n1,n2....ni partcles
lie in groupsb having energies e1,e2....ei
respectively
15. The conditions in Maxwell- Boltzmann statistics:
• 1)particles are distinguishable i.e, there are no
symmetry restrictions
• 2) each eigen state 𝑖 𝑡ℎ
quantum group may
contain 0,1,2....𝑛𝑖 particles.
• Most probable distribution is given by
𝑛 𝑖
𝑔 𝑖
=
1
[𝑒α+βε 𝑖]
16. Comparison of M-B,B-E &F-D
STATISTICS
M-B statistics
• Particles are distinguishable
and only particles are taken
into consideration
• There are no restriction on the
number of particles in a given
state.
• Applicable to ideal gas
molecule.
• Volume in six dimensional
space is not known.
• The most probable distribution
is given by
𝑛 𝑖
𝑔 𝑖
=
1
[𝑒α+βε 𝑖]
F-D statistics
• Particles are indistinguishable
and quantum states are taken
into consideration
• Only one particles may be in a
given quantum state
• Applicable to electrons and
elementary particles.
• Volume in phase space is
known (h^3).
• The most probable distribution
is given by is
𝑛 𝑖
𝑔 𝑖
=
1
[𝑒α+βε 𝑖+1]
17. B-E statistics
• Particles are indistinguishable and quantum
states are taken into consideration.
• No restriction on number of particles in a given
quantum state.
• Applicable to photons and symmetrical particles.
• Volume in phase space is known (h^3).
• The most probable distribution is given by
𝑛 𝑖
𝑔 𝑖
=
1
[𝑒α+βε 𝑖 −1]
