Statistical Mechanics

1. ELEMENTS OF STATISTICAL MECHANICS
Definition: The Subject which deals with the relationship between the overall behavior of the
system and the properties of the particles is called statistical mechanics.
Macro state: Each compartment distribution of a system of particles is known as a macrostate.
Ex: Distribution of x, y, z particles in 2 compartments.
Compartment Possible distributions of particles
1 0 1 2 3
2 3 2 1 0
Micro State: Each distinct arrangement is known as the micro state of the system.
Ex: Distinct arrangement of particles x, y, z in 2 compartments.
Macro
State
Compartment
1
Compartment
2
No of
Micro states
0, 3 0 X Y Z 1
1, 2
X
Y
Z
Y Z
X Z
X Y
3
2, 1
X Y
Y Z
X Z
Z
X
Y
3
3, 0 X Y Z 0 1
Position space: The three dimensional space in which the location of a particle is completely
specified by the three position coordinates is known as position space.
A small volume element in position space dv is given by dv = dxdydz
Momentum Space: The three dimensional space in which the momentum of particle is
completely specified by the three momentum coordinates px, py and pz is known as momentum
space.
A small volume element in momentum space dτ = dpxdpydpz
Phase space: A combination of the position space and momentum space is known as phase
space.
 A point phase space is completely specified by six coordinates x, y, z,px, py,pz
 If there are N particles. 6N coordinates provide complete information regarding the
position and momentum of all N particles in the phase space in a dynamic system.
 A small volume in phase space dT is given by

2. dT = dxdydzdpxdpydpz
dT = dvdτ
Thus a volume element dT in phase space is the product of a volume element dv in position
space and Volume element dτ in momentum space.
∴ 𝑇ℎ𝑒 𝑇𝑜𝑡𝑎𝑙 phase space volume T = ∫𝑑𝑇 = ∫𝑑𝑣 ∫ 𝑑𝜏
T =Ѵ τ
 The small volume dT in phase space is called a cell.

𝑇𝑜𝑡𝑎𝑙 𝑎𝑣𝑎𝑖𝑙𝑎𝑏𝑙𝑒 𝑉𝑜𝑙𝑢𝑚𝑒 𝑖𝑛 𝑝ℎ𝑎𝑠𝑒 𝑠𝑝𝑎𝑐𝑒 (Ѵ τ)
𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑂𝑛𝑒 𝑐𝑒𝑙𝑙 𝑑𝑇
= 𝑇ℎ𝑒 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑒𝑙𝑙𝑠 𝑖𝑛 𝑝ℎ𝑎𝑠𝑒 𝑠𝑝𝑎𝑐𝑒
 Each spherical shell is called an energy compartment.
 A cell is a sub-compartment.
 Each compartment is divided into a very large number of cells in such a way that each
cell is of the same size. Hence all the cells have the same a priori probability of a
particle going into a cell.
 In classical mechanics the volume of a cell in phase space can even approach zero(dT =
dvdτ→ 0).
 In quantum mechanics the volume of the cell in phase space cannot be less than h3.
 Three kinds of distributions are possible corresponding to 3 different kinds of particles.
Classical Statistics – Maxwell Boltzmann Distribution:
 The main assumptions of M B Statistics.
1. The particles are identical and distinguishable.
2. The volume of each phase space cell chosen is extremely small and hence chosen
volume has very large number of cells.
3. Since cell are extremely small, each cell can have either one particle or no particle
though there is no limit on the number of particles which can occupy a phase space cell.
4. The systemis isolated which means that both the total number of particles of the
system and their total energy remain constant.
5. The state of each particle is specified either by its cell number in phase space or
instantaneous position and momentum co-ordinates.
6. Energy levels are continuous.
7. 𝑛𝑖(𝐸𝑖) = 𝑒−∝
𝑒
−
𝐸𝑖
𝐾𝑇
= 𝑓𝑀𝐵(𝐸𝑖)
is the Maxwell – Boltzmann distribution function. It gives the
average number of particles ‘n’ with energy. E in equilibrium at a given temperature for
a system of classical particles. Since ‘n’ is the average number of particles, it need not
be an integer.
 𝑤ℎ𝑒𝑟𝑒 ∝ =
𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑎𝑛𝑑 𝑑𝑒𝑝𝑒𝑛𝑑𝑠 𝑜𝑛 𝑡ℎ𝑒 𝑝𝑟𝑜𝑝𝑒𝑟𝑡𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑦𝑠𝑡𝑒𝑚 𝑎𝑛𝑑 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑇
 Ei = ith energy state
 ni = number of particles at ith energy state.

3.  K = Boltzmann constant = 1.38 x 10-23j/k.
Limitation of classical Statistics.
1. The observed energy distribution of electrons in metals.
2. The observed energy distribution of photons inside an enclosure.
3. The behavior of helium at low temperatures.
Quantum Statistics:
Bose EinsteinStatistics: Assumptions:
1. In Bose Einstein particles are known as Bosons.
2. The Bosons of the system are identical and indistinguishable.
3. The Bosons have integral spin angular momentum in units of h/2π.
4. Bosons obey uncertainty principle.
5. Any number of Bosons can occupy a single cell in phase space.
6. Bosons do not obey the exclusion principle.
7. The number of phase space cells is comparable with the number of Bosons.
8. Wave functions representing the bosons are symmetric i.e. 𝜓(1,2) = 𝜓(2,1).
9. The wave functions of bosons do overlap slightly i.e. weak interaction exists.
10. Energy states are discrete.
11. The probability f(E) that a boson occupies a state of energy E is given by
𝐅𝐁𝐄(𝐄) =
𝟏
𝐞
(𝛂+
𝐄
𝐊𝐓
)
− 𝟏
This is called Bose – Einstein distribution function. The quantity α is a constant and
depends on the property of the system and temperature T.
If ‘n’ is the number of Bosons with total energy E when in equilibrium at temperature T, the
most probable distribution of bosons among the various energy levels is given by
𝐧𝐢 =
𝐠𝐢
𝐞
(𝛂+
𝐄
𝐊𝐓
)
− 𝟏
Where gi is the number of quantum states (phase space cells) available for the same
energy Ei.
 The quantity gi is called degeneracy.
Examples of Bosons are photons, He4, π and K – mesons.
Fermi Dirac Statistics: Assumptions:
1. Half integral spin particles are known as fermions.
2. Fermions are identical and indistinguishable.
3. They obey Pauli’s exclusion principle i.e. there cannot be more than one particle in a single
cell in phase space.
4. Wave function representing Fermions are antisymmetric. i.e. 𝜓(1,2) = −𝜓(2,1).
5. Weak interaction exists between the particels.
6. Uncertainty principle is applicable.
7. Energy states are discrete.
8. The probability f(E) that a Fermion occupies a state of energy E is given by

4. 𝐅𝐅𝐃(𝐄) =
𝟏
𝐞
(𝛂+
𝐄
𝐊𝐓
)
+ 𝟏
This is called Fermi – Dirac distribution function. The quantity α is a constant and depends on
the property of the system and Temperature ‘T’.
 Let the total energy of the electrons be E, this energy is distributed among all the
‘n’ electrons. According to the Fermi – Dirac distribution law.
𝐧𝒊 =
𝒈𝒊
𝐞
(𝛂+
𝐄
𝐊𝐓
)
+ 𝟏
Where gi is the number of phase space cells with energy Ei.
 Examples of Fermions: Free electrons in conductor, protons, neutrons and He3
atoms.
Fermi energy: Fermi – Dirac Distribution function.
𝐧(𝑬) =
𝒈(𝑬)
𝐞
(𝛂+
𝐄
𝐊𝐓
)
+ 𝟏
But α=
−µ
KT
𝐧(𝑬) =
𝒈(𝑬)
𝐞
(
−µ
KT
+
𝐄
𝐊𝐓
)
+ 𝟏
=
𝒈(𝑬)
𝐞
(
E−µ
KT
)
+ 𝟏
Since electrons have spin 1/2, there are two states with spin projection +1/2 and -1/2
with the same energy.
∴ 𝑔(𝐸) = 2 𝑓𝑜𝑟 𝑎𝑙𝑙 𝐸
Then the average number of particles in a state with energy ‘E’ or the probability of
occupation of a state of energy E is given by
𝒇(𝑬) =
𝟏
𝐞
(
E−𝐸𝑓
KT
)
+ 𝟏
 This is known as Fermi function. We have replaced the Chemical potential µ by
𝐸𝑓 which is known as Fermi Energy.
 The meaning of Fermi Energy becomes clear from the following considerations.
If T=0, then
E − Ef
KT
= −∞, if E ≺ Ef and
E − Ef
KT
= ∞, if E > Ef
∴ 𝑒
E−𝐸𝑓
KT + 1 = {
1 𝑖𝑓 E < Ef
0 𝑖𝑓 E > Ef
𝑎𝑠 𝑎 𝑟𝑒𝑠𝑢𝑙𝑡
𝑓(𝐸) = {
1 𝑖𝑓 E < Ef
0 𝑖𝑓 E > Ef
The probability of occupation of a state of energy E is one upto E=EF and also is zero for E > Ef.
That means all states with energy upto EF are occupied and all states with energy higher than EF
are empty at T=0. This is the meaning of EF at T=0.

5. If T > 0
 For T > 0 the exponential function 𝑒
E−𝐸𝑓
KT is well behaved for all E. As a result 0<f(E)<1 for
all E.
 That means the probability of occupation is not one even if E < Ef and the probability of
occupation of no state is zero even if E > Ef.
 Further if E = Eff(E) = ½. Therefore we interpret Ef as the energy at which the
probability of occupation is ½ for T > 0.
PHOTON GAS:
 Radiation enclosed in a container which is at thermal equilibrium with the walls of the
container is called Black Body Radiation.
 In thermal equilibrium with the walls of the container means there is a constant
exchange of energy between the radiation and the walls of the container so that the
temperature of the walls is the same as the temperature of radiation.
 This radiation contains electromagnetic waves of all wavelengths (frequencies) from
zero to infinity. Such a radiation can be considered to be a collection of photons of all
wavelengths from zero to infinity at thermal equilibrium.
 The spin of these particles (photons) is known to be 1. Because of this they have
polarization index +1 or -1. Therefore there can be two photons of the same frequency
one with polarization index +1 and another with -1.
 Within the container the number of photons of a given frequency ν is not constant. A
photon of certain energy can split itself into two or more photons of lower energy or
two or more photons can fuse together to form a photon higher energy.
 This process goes on all time. As a result the number of photons of a given frequency is
not a constant in time.
 i.e. without spending or gaining energy one can increase or decrease the number of
photons of a particular energy.
∴ ∑ 𝑛𝑖 = 𝑁 𝑖𝑠 𝑚𝑒𝑎𝑛𝑖𝑛𝑔𝑙𝑒𝑠𝑠
𝑖
This we take into account by making α = 0 Thus the chemical potential of a photon gas is
zero.
Black body radiationand the Planck radiationlaw
 The momentum of photons = p =
ℎ𝜈
𝑐
 The Volume of each allowed cell in the phase space can be dГ = dxdydzdpxdpydpz = h3
 Now the phase space is split into the position space and momentum space. If the
range of position coordinates is continuously increased till they embrace the whole
volume ‘V’ of the enclosure then the particles can be found anywhere in the
enclosure.
 So according to Heisenberg’s uncertainty relation an element of volume in the
momentum space can be written as = 𝜎𝑝 =
ℎ3
𝑉

6.  𝐴𝑠 𝜎𝑝Denotes the size of an elementary cell in the momentum space only a single
value of momentum can be recognized within a cell.
 At any instant all photons having their momenta between p and p+dp will be within a
spherical shell described in momentum space with radii p and p+dp.
 Therefore the volume of this shell is 4πp2dp
Therefore the total number of energy states between momenta p and p+dp is given by
g(p)dp = [
4πp2
𝑉
ℎ3 ]𝑑𝑝 ------ 1
 For a photon p =
ℎ𝜈
𝑐
𝑜𝑟 𝑑p =
ℎ𝑑𝜈
𝑐
Substituting these values in above equation the total number of energy states
between frequencies ν and ν+dν is given by
g(ν)dν =[
4πV𝜈2
𝑐3 ]𝑑𝜈 ---- 2
Taking into account the doubling of the states due to polarization of photons the total
number of Eigen states (Energy States) available for the photons in the frequencies
range ν and ν+dν
g(ν)dν =[
8πV𝜈2
𝑐3 ]𝑑𝜈 ---- 3
Introducing this result in Bose Einstein’s distribution law
𝑑𝑛 =
g(ν)dν
𝑒𝛼 +𝛽𝐸−1
------- 4
In this case α = 0, β=1/KT, E=hν
∴ 𝑑𝑛 =
g(ν)dν
𝑒𝛼 +𝛽𝐸−1
=
8πV𝜈2
𝑐3
1
𝑒
hν
𝐾𝑇−1
𝑑𝜈 -----5
𝑑𝑛
𝑉
=
8π𝜈2
𝑐3
1
𝑒
hν
𝐾𝑇 −1
𝑑𝜈 ------6
This equation represents the number of photons per unit volume lying in the frequency range ν
and ν+dν.
The energy density of radiation of frequencies between ν and ν+dν can now be found by
multiplying above equation by the energy of the photon hν.
Therefore if Eνdν represents the energy density of radiation within the specified frequency
range then the energy distribution can be written as
Eνdν =
𝑑𝑛
𝑉
hν =
8π𝜈2
𝑐3
hν
𝑒
hν
𝐾𝑇 − 1
𝑑𝜈
𝐄𝛎𝐝𝛎 =
𝟖𝛑𝐡𝝂𝟑
𝒄𝟑
𝟏
𝒆
𝐡𝛎
𝑲𝑻−𝟏
𝒅𝝂----- 7
This is well known Plank’s law of radiation in terms of frequency.
V=nλ ---c= νλ -ν = c/λ and dν =
𝑐
𝜆2 𝑑𝜆

7. E(λ)dλ =
8πh𝑐3
𝑐3𝜆3
1
𝑒
hc
𝜆𝐾𝑇 − 1
𝑐
𝜆2
𝑑𝜆
∴ 𝐄(𝛌)𝐝𝛌 =
𝟖𝛑𝐡𝐜
𝝀𝟓
𝟏
𝒆
𝐡𝐜
𝝀𝑲𝑻−𝟏
𝒅𝝀-----8
This is Plank’s law of radiation in terms of wavelength ‘λ’
 Wien’s Distribution Law for shorter wavelength range.
In the shorter wavelength range e
hc
λKT ≫ 1 hence we neglect 1 in the denominator of the
equation
∴ 𝐄(𝛌)𝐝𝛌 =
𝟖𝛑𝐡𝐜
𝝀𝟓
𝟏
𝒆
𝐡𝐜
𝝀𝑲𝑻
𝒅𝝀 ------ 9
This equation is known as Wien’s Distribution law.
 Rayleigh – Jeans law for Larger Wavelength range.
In the longer wavelengths range e
hc
λKT ≪ 1 ≈ 1 +
hc
λKT
– 1 =
hc
λKT
∴ E(λ)dλ =
8πhc
𝜆5
𝑑𝜆
hc
λKT
∴ 𝐄(𝛌)𝐝𝛌 =
𝟖𝛑𝐊𝐓
𝝀𝟒
𝒅𝝀 ------10
This equation is called Rayleigh – Jeans law.