1. STATISTICAL MECHANICS
UNIT-III
ELEMENTS OF
STATISTICAL MECHANICS

2. Statistical mechanics can be applied to systems such as
The subject which deals with the relationship between the overall behavior of the system and the properties of the particles is called ‘Statistical Mechanics’.
Statistical mechanics can be applied to systems such as
molecules in a gas
photons in a cavity
free electrons in a metal

3. Macro state
Any state of a system as described by actual or hypothetical observations of its macroscopic statistical properties is known as‘ Macro state ’
and it is specified by ‘ ( N, V and E ) ’.
NOTE
For ‘ N ’ particle system , there may be always possible ‘N+1’ Macro states.

4. Micro state
The state of system as specified by the actual properties of each individual, elemental components, in the ultimate detail permitted by the uncertainty principle is known as
‘ Microstate ’.
NOTE
For ‘ N ’ particle system , there may be always possible ‘2n’ Micro states.

5. Small volume in a position space dV = dx dy dz
Phase space
The three dimensional space in which the location of a particle is completely specified by the three position co-ordinates, is known as ‘Position space’.
Small volume in a position space dV = dx dy dz

6. Small volume in a momentum space
The three dimensional space in which the momentum of a particle is completely specified by the three momentum co-ordinates Px Py and Pz
is known as ‘Momentum space’.
Small volume in a momentum space
dГ = dpx dpy dpz

7. Small volume in a phase space dτ = dV dГ
The combination of the position space and momentum space is known as
‘Phase space’.
Small volume in a phase space dτ = dV dГ

8. Phase space volume Consider Let ‘ pm’ be the maximum value of the
momentum of the particles in the system.
Let px , py , pz represents the three mutually
perpendicular axes in the momentum space
as shown in figure.

9. Draw a sphere with an origin ‘ O ’ as centre pz px py pm
Draw a sphere with an origin ‘ O ’ as centre
and the maximum momentum ‘ pm’ as radius.

10. All the points within this sphere will have their
momentum lying between ‘ 0 ’ and ‘ pm’.
The momentum space volume =
volume of the sphere of radius ‘ pm’ .
Momentum volume is given by
Г = 4/3 π pm3

11. Similarly phase space volume is given by
τ = 4/3 π pm3 V
where,
V – position space volume

12. ENSEMBLE
The set of possible states for a system of ‘N’ particles is referred as ensemble in statistical mechanics.
(OR)

13. A collection of large number of microscopically
identical but essentially independent systems is
called ensemble.
NOTE:
An ensemble satisfies the same macroscopic
condition.
Example:
In an ensemble the systems play the role of as
the non-interactive molecules do in a gas.

14. There are 3-types of ensembles, those are
1) MICRO CANONICAL ENSEMBLE
2) CANONICAL ENSEMBLE
3) GRAND CANONICAL ENSEMBLE

15. 1) MICRO CANONICAL ENSEMBLE
It is the collection of a large number of essentially
independent systems having the same energy (E),
volume(V) and the number of particles(N).
NOTE:
We assume that all the particles are identical and the individual
systems of micro canonical ensemble are separated by rigid,
well insulated walls such that the values of E, V & N for
a particular system are not affected by the presence of
other systems.

16. E,V,N
E,V,N
E,V,N
E,V,N
E,V,N
E,V,N
E,V,N
E,V,N
E,V,N

17. 2) CANONICAL ENSEMBLE NOTE:
It is the collection of a large number of essentially
independent systems having the same temperature (T),volume(V) and the same number of identical particles(N).
NOTE:
The equality of temperature of all the systems can be
achieved by bringing each in thermal contact at temperature(T).
The individual systems of a canonical ensemble are separated by
rigid, impermeable but conducting walls as a result all the
systems will arrive at the common temperature(T).

18. T,V,N
T,V,N
T,V,N
T,V,N
T,V,N
T,V,N
T,V,N
T,V,N
T,V,N

19. 3) GRAND CANONICAL ENSEMBLE
It is the collection of a large number of essentially
independent systems having the same temperature (T),volume(V) and the chemical potential (µ).
NOTE:
The individual systems of a grand canonical ensemble are
separated by rigid & conducting walls.
Since the separating walls are conducting, the exchange of heat
energy takes place between the systems of particles such a way that all the systems arrive at a common temperature (T) & chemical potential (µ).

20. T,V,μ
T,V,μ
T,V,μ
T,V,μ
T,V,N
T,V,μ
T,V,μ
T,V,μ
T,V,μ

21. N(E) = g(E) f(E) Statistical distribution
Statistical mechanics determines the most probable way of distribution of total energy ‘ E ’ among the ‘ N ’ particles of a system in thermal equilibrium at absolute temperature ‘ T ’.
In statistical mechanics one finds the number of ways
‘ W ’ in which the ‘ N ’ number of particles of energy ‘ E ‘ can be arranged among the available states is given by.
N(E) = g(E) f(E)

22. g(E) is the number of states of energy E
Where
g(E) is the number of states of energy E
And f(E) is the probability of occupancy of each state of energy E .

23. a) Maxwell - Boltzmann distribution
We have two different types of statistical functions.
1. Classical Statistics
a) Maxwell - Boltzmann distribution
2. Quantum Statistics
a) Bose - Einstein distribution
b) Fermi - Dirac distribution

24. Maxwell – Boltzmann statistics
(Classical statistics)
Let us consider a system consisting of molecules of an ideal gas under ordinary conditions of temperature and pressure.
Assumptions:
The particles are identical and distinguishable.

25. The volume of each phase space cell chosen is
extremely small and hence chosen volume has
very large number of cells.
Since cells are extremely small, each cell can have either one particle or no. of particles though there
is no limit on the number of particles which can occupy a phase space cell.

26. The system is isolated which means
that both the total number of particles
of the system and their total energy
remain constant.
The state of each particle is specified
instantaneous position and momentum
co-ordinates.
Energy levels are continuous.

27. MAXWELL - BOLTZMANN DISTRIBUTION
This distribution is applied to a macroscopic system consisting of a large number ‘n’ of identical but distinguishable particles, such as gas molecules in a container.
This distribution tells us the way of distribution of total energy ‘E’ of the system among the various identical particles.
Let us consider that the entire system is divided into groups of particles, such that in every group the particles have nearly the same energy.

28. Let the number of particles in the 1st, 2nd , 3rd ,……. ,ith,……
Let the number of particles in the 1st, 2nd , 3rd ,…….,ith,……. groups be n1,n2,n3,…..ni,….. Respectively.
Also assume that the energies of each particle in the 1st group is E1, in the 2nd group is E2 and so on.
Let the degeneracy parameter denoted by ‘g’ [or the number of electron states] in the 1st, 2nd ,3rd,……,ith,… groups be g1,g2,g3,….gi,…. and so on respectively.
In a given system the total number of particles is constant.

29. Hence it’s derivative
The total energy of the particles present in different groups is equal to the energy of the system(E).
Hence it’s derivative

30. The probability of given distribution W is given by the product of two factors.
The first factor is, the number of ways in which the groups of n1,n2,n3,…ni,… particles can be chosen.
To obtain this, first we choose n1 particles which are to be placed in the first group. This is done in
The remaining total number of particles is (n-n1). Now we arrange n2 particles in the second group. This is done in

31. The number of ways in which the particles in all groups are chosen is
is the multiplication factor

32. The second factor is the distribution of particles over the different states and is independent of each other.
Of the ni particles in the ith group the first particle can occupy any one of the gi states. So there are gi ways , and each of the subsequent particles can also occupy the remaining states in gi ways. So, the total number of the ways the ni particles are distributed among the gi states is gini ways.
The probability distribution or the total number of ways in which n particles can be distributed among the various energy states is W2

33. The number of different ways by which n particles of the system are to be distributed among the available electron states is

34. Where πi represents the multiplication factor.
Taking natural logarithms on both sides of equation in above.

35. For the most probable distribution, W is maximum provided n and E are constants.
Differentiate equation and equate to zero for the maximum value of W.

36. Multiplying equation (1) with -α and equation (2) with –β and adding to the (3) equation and there by

37. The above equation is called Maxwell-Boltzmann law
The above equation is called Maxwell-Boltzmann law. The value of β has been extracted separately and is equal to 1/kBT.
Where KB is called Boltzmann constant and T is called absolute temperature.

38. 2) Quantum Statistics Bose - Einstein distribution
According to quantum statistics the particles of the system are indistinguishable, their wave functions do overlap and such system of particles fall into two categories
Bose - Einstein distribution
Fermi - Dirac distribution.

39. Bose – Einstein statistics
Sir J.Bose
A . Einstein
Bose – Einstein statistics
According to Bose-Einstein statistics the particles of any physical system are identical, indistinguishable and have integral spin, and further those are called as Bosons.
Assumptions:
The Bosons of the system are identical and indistinguishable.

40. The Bosons have integral spin angular
momentum in units of h/2π.
Bosons obey uncertainty principle.
Any number of bosons can occupy a single
cell in phase space.
Bosons do not obey the Pauli's
exclusion principle
Energy states are discrete.

41. Wave functions representing the bosons
are symmetric
i.e.  Ψ(1,2) = Ψ(2,1)
The probability of Boson occupies a state
of energy E is given by
This is called Bose – Einstein distribution function.
The quantity ‘ ά ’ is a constant and depends on the property of the system and temperature T.

42. Bose – Einstein distribution functions
Let us divided a box into gi sections & the particles
are distributed among these sections.
Once this has been done, the remaining (gi -1) compartments
& ni particles, the number of ways doing this will equal to
(ni + gi -1)!

43. gi(ni + gi -1)!/gi! ni! (OR) (ni + gi -1)!/ni! (gi -1)!
Thus the total number of ways realizing the distribution
will be gi (ni + gi – 1)! (1)
There are ni permutations which corresponds to the
same conservative function we thus obtain the
required number of ways as
gi(ni + gi -1)!/gi! ni!
(OR)
(ni + gi -1)!/ni! (gi -1)!

44. G = (n1 + g1-1)!/n1! (g1-1)! . (n2+ g2-1)!/n2! (g2-1)!..... (ni + gi-1)!/ni! (gi-1)!
= πi (ni + gi-1)!/ni! (gi-1)! (2)
We have the probability ‘W’ of the system for occurring
with the specified distribution to the total number of
Eigen states.
W = πi (ni + gi -1)!
ni !(gi -1)!
X constant (3)

45. Taking the log of eq(3), we have
log W = Σ log (ni+gi-1)! – log ni! – log (gi -1)! + constant (4)
Using the stirling approximation eq(4) becomes as
log W = Σ (ni+gi) log( ni + gi ) –ni log ni – gi log gi constant – (5)
δ log w = Σi δ (ni+gi) log(ni + gi) – ni logni – gi log gi)
Where,
ni , gi >> 1. Hence 1 is neglected

46. Σ log ni / (ni + gi ) δni = 0 ---- (7)
δ log w = Σi δni log(ni + gi) + (ni + gi ) δni – δni log ni) - ni/ni δ ni
( ni + gi )
δ log w = Σi δni log(ni + gi) – δni log ni)
δ log w = -Σi log ni / (ni + gi ) δni ---- (6)
The condition for maximum probability gives as
Σ log ni / (ni + gi ) δni = (7)

47. δn = Σ δni = 0 --- (8) δE = Σ Ei δni = 0 --- (9) (OR)
The auxillary condition to be satisfied
(OR)
δn = Σ δni = (8)
δE = Σ Ei δni = (9)
Multiplying eq(8) by α & eq(9) by β & adding
the resultant expression to eq(7) , we get

48. Therefore 1 + gi / ni = e α + β Ei gi / ni = e α + β Ei - 1
Σ log ni / (ni + gi ) + α + β Ei δni = (10)
δni is independent of each other.
Therefore
log ni / (ni + gi ) + α + β Ei = 0
1 + gi / ni = e α + β Ei
gi / ni = e α + β Ei - 1

49. ni = gi / (eα+βEi - 1) ---- (11)
Therefore,
ni = gi / (eα+βEi - 1) ---- (11)
This represents the most probable distribution
of the elements for a system obeying
Bose – Einstein statistics.

50. Fermi – Dirac statistics
According to Fermi - Dirac statistics the particles of any physical system are indistinguishable and have half integral spin. These particles are known as Fermions.
Assumptions:
Fermions are identical and indistinguishable.
They obey Pauli’ s exclusion principle

51. Fermions have half integral spin.
Wave function representing fermions
are anti symmetric
i.e.
Uncertainty principle is applicable.
Energy states are discrete.

52. The distribution function & Fermi level is valid
only in equilibrium.
The distribution function changes only with
temperature.
It is valid for all fermions.
Electrons & holes follow the Fermi – Dirac statistics and hence they are called Fermions.

53. Fe(E) = Pe(E) = 1 + e (E –EF) / KBT Ph(E) = 1 - Pe(E) =
The value of F(E) never exceeds unity.
The probability finding energy for electron is
Fe(E) = Pe(E) = 1
1 + e (E –EF) / KBT
The probability finding energy for hole is
Ph(E) = 1 - Pe(E) =
1 + e (EF –E) / KBT 1

54. Explanation(F –D statistics)
Where,
E3 > E2 > E1 > EF
T3 > T2 > T1 > T E
P(E) E O At
T = 0 K EF
Figure-1
1.0 E3 E2 E1 At
T = 0 K EF T1 T2 T3
P(E) O
1.0
Figure-2

55. From figure the following points are noted….
When the material is at a temperature higher than
0 K, it receives the thermal energy from surroundings
and they excited.
As a result, they move into higher energy levels
which were unoccupied at 0 K. The occupation
obeys a statistical distribution called Fermi – Dirac
distribution law.

56. Where, --------- (1) F(E) = P(E) = 1 + e (E –EF) / KBT
According to this law, the probability F(E) or P(E)
that a given energy state ‘ E ’ is occupied at a
temperature ‘ T ’ is given by
F(E) = P(E) = 1
1 + e (E –EF) / KBT
(1)
Where,
F(E) = P(E) = Fermi – Dirac probability function
KB = Boltzmann constant

57. And ‘ Ef is called Fermi energy and is a constant for a given system.
The maximum energy possessed by an electron at absolute temperature is known as Fermi energy( EF ).

58. CASE:1 From equation (1) we may discuss the following 2 cases.
If E > EF ,
then the exponential term becomes infinite and F(E) = 0
i.e. there is no probability of finding an occupied state of
energy greater than EF at absolute zero.
Hence, Fermi energy is the maximum energy that any
electron may occupy at 0 K
As temperature increases the electrons are occupied
the higher energy states which are unoccupied at 0 K
as shown in figure(2).

59. CASE:2 If E = EF , then the F(E) = P(E) = ½ at any temperature.
Hence, Fermi level represents the energy state with
50% probability of being filled.
Electrons in a crystal(metal) obey the Fermi – Dirac
(F-D) distribution (statistics)

60. Fermi – Dirac distribution functions
Let E1, E2, E3… Ei be the energy of the particles,
g1, g2, g3---- gi be the energy states & n1, n2, ---- ni .
The number ways in which ni particles can
be put in gi number of states are given by
gi(gi -1)(gi -2)(gi -3)-----(gi – ni +1)
(OR)
gi !/(gi – ni)! (1)

61. G = Σi gi!/ni !( gi – ni )! ---- (3)
Dividing the eq(1) with ni ! , we get number of Eigen
energy states
gi! / ni! (gi – ni)! (2)
Thus, for the whole system, the total number of Eigen
energy states can be written as
G = Σi gi!/ni !( gi – ni )! (3)
(OR)
The probability W of a specific state is
proportional to the total number of energy (G)

62. Σi W = Σi log log W = W α G W = CG --- (4) gi! ni! (gi -ni)! gi!
Where,
C -- constant
W = CG --- (4)
Substituting the eq(3) in eq(4), we get,
gi!
ni! (gi -ni)!
X constant (5) Σi
W =
Taking log on both sides of eq(5), we get,
gi!
ni! (gi -ni)!
+ constant
Σi log
log W =

63. log W = Σi log gi! – log (ni! (gi -ni)! + log C
log W = Σi log gi! – log ni! + log (gi-ni)! log C
log W = Σi log(g1, g2,-- gi ) – log(n1 , n2 -- ni ) + log (g1-n1)----
---log (gi –ni ) log C
log W = Σi gi log gi – ni log ni – (gi – ni ) log (gi – ni) + log c
log W = Σi (ni – gi ) log (gi – ni ) + gi log gi
– ni log ni log c --- (6)

64. Differentiating eq(6) w.r.t. ni , we get,
δ (log w) = Σi (δni – 0) log( gi - ni) + (ni - gi ) (0 –δni )
+ 0 - ni/ni δ ni – log ni δ ni
(gi – ni )
δ (log w) = Σi (δni log( gi - ni) + (ni - gi ) δni - δni
– log ni δ ni
ni – gi
δ (log w) = Σi δni log( gi – ni) – log ni δ ni

65. - Σi Hence, eq(7) becomes as Therefore,
δ (log w) = - Σi log ni - log( gi – ni ) δ ni
log ni
( gi – ni )
- Σi
δ (log w) =
δ ni --- (7)
For maximum probability,
δ (log w) = 0
Hence, eq(7) becomes as

66. δn = Σ δni = 0 --- (9) δE = Σ Ei δni = 0 --- (10) Σi 1----- 2-----
( gi – ni ) Σi
δ ni = (8)
log ni
To evaluate the distribution function, two auxillary conditions
has to be introduce in eq(8)
δn = Σ δni = (9)
1-----
δE = Σ Ei δni = (10)
2-----
Multiplying eq(9) by α & eq(10) by β & adding
the resultant expression to eq(8) , we get

67. Σi (OR) Σi Therefore, ( gi – ni ) δ ni = 0 log ni δ ni + α δni + β Ei
Since, δni is arbitrary it is ≠ 0
Since,
Therefore,
( gi – ni )
= 0
log ni
+ α + β Ei

68. (OR) = e-(α + βEi ) 1 = e(α + βEi ) = e(α + βEi ) (OR) log ni
( gi – ni )
log ni
= - (α + β Ei )
( gi – ni ) ni
= e-(α + βEi )
- ni ( 1 - gi / ni ) ni
e(α + βEi ) 1 =
gi / ni - 1 1
e(α + βEi ) =
(OR)

69. eα + βEi 1 1 + F (E) = ni/gi = = e(α + βEi ) (OR) = 1 + e(α + βEi )
gi / ni - 1 =
e(α + βEi )
(OR)
gi / ni =
1 + e(α + βEi )
Therefore, the Fermi –Dirac function can be written as
F (E) = ni/gi = 1
1 +
eα + βEi