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S tatiStical

MechanicS

75. Statistical Mechanics I

76. Statistical Mechanics II

P A R T – IX

928 MODERN PHYSICS

If the system is isolated, the total energy E must be constant. According to the basic postulate

of statistical mechanics, the greater the number W of different ways in which the particles can

be arranged among the available states to yield

a particular distribution of energies, the more

probable is the distribution. It is assumed that each

state of a certain energy is equally likely to be

occupied.

Therefore, the procedure for determining the

most probable statistical distribution involves two

steps: ( i ) obtaining the number of distinguishable

arrangements ( W ) which give rise to the same

distribution, and ( ii ) maximizing this number

of arrangements ( W ) with respect to different

distributions.

We shall consider systems of three different

kinds of particles.

1. Maxwell-Boltzmann Statistics: This is

applicable to the identical, distinguishable particles

of any spin. The molecules of a gas are the particles

of this kind.

2. Bose-Einstein Statistics: This is applicable

to the identical, indistinguishable particles of zero or integral spin. These particles are called bosons.

The examples of bosons are helium atoms at low temperature and the photons.

3. Fermi-Dirac Statistics: This is applicable to the identical indistinguishable particles of half-

integral spin. These particles obey Pauli exclusion principle and are called fermions. The examples

of fermions are electrons, protons, neutrons, etc.

75.2 Phase Space

To specify the position as well as energy of a molecule inside a gas, we must specify three space

coordinates x , y , z and three momentum coordinates px , py , pz. As a purely mathematical concept, we

may imagine six dimensional space in which coordinates are x , y , z , px , py , pz. This six dimensional

space for a single molecule is called phase space or μ- space. The instantaneous state of a particle

in the phase space is represented by a point known as phase point or representative point. In phase

space we may consider an element of volume dx , dy , dz , dpx , dpy , dpz. Any such element of volume

in six dimensional space is called a cell. Thus the phase space may be divided into a large number

of cells. A cell may contain a large number of phase points. Each cell in phase space has the volume

h

3 .

75.3 Maxwell-Boltzmann Distribution Law

Consider a system of N distingishable molecules of a gas. Suppose N 1 of them have energy

E 1 , n 2 have energy E 2 , ... ni have energy Ei and so on. Thus the entire assembly of molecules can

be divided into different energy states with energies E 1 , E 2 , E 3 , ... Ei and having n 1 , n 2 , n 3 , ... ni

molecules.

(1) The total number of molecules N is constant. Hence

N = n 1 + n 2 + n 3 + ... + n i + ... = constant

or δ N = δ n 1 + δ n 2 + δ n 3 + ... + δ ni + ... = 0

Metallic Phases for Bosons.

STATISTICAL MECHANICS I 929

i.e. , (^) i i

∑ δ n = 0 ...(1)

(2) The total energy E of the gas molecules is constant. Hence

E = E 1 n 1 + E 2 n 2 + E 3 n 3 + ... + Eini + ... = constant

or δ E = E 1 δ n 1 + E 2 δ n 2 + E 3 δ n 3 + ... + Ei δ ni + ... = 0

i.e ., (^) i i i

∑ E δ n = 0 ...(2)

(3) Suppose there are gi cells with the energy Ei. The total number of ways in which ni molecules

can have the energy Ei is ( gi )

ni

. Hence the total number of ways in which N molecules can be

distributed among the various energies is

W 1 = ( g 1 )

n (^1) ( g 2 )

n (^2) ( g 3 )

n (^3) … ( g i )

n i (^) …

The number of ways in which the groups of n 1 , n 2 , n 3 ... ni ... particles can be chosen from N

particles is given by

W 2 =

1 2 3

N

n n n

The number of distinguishable ways in which N molecules can be distributed among the

possible energy levels is

W = 1 2 1 1 2 2 3 3

1 2 3

N n n n W W g g g n n n

The quantity W is called the thermodynamic probability for the system. For the most probable

distribution, W is a maximum subject to the restriction that the total number of particles N and the

total energy E are constants.

The natural logarithm of Eq. (3) is

ln W = ln N! – Σ ln ni! + Σ ni ln gi

By Stirling’s theorem, 1n n! = n ln n – n

ln W = N ln N – N – Σ ni ln ni + Σ ni + Σ ni ln gi

ln W = N ln N – Σ ni ln ni + Σ ni ln gi ...(4)

From Eq. (4), we have for maximum W

max

ln – (^) i i – (ln (^) i ) (^) i (ln (^) i ) (^) i 0 i

W n n n n g n n

δ = Σ δ Σ δ + Σ δ =

or – Σ (ln ni ) δ n i + Σ (ln g i) δ ni = 0 [Q Σ δ ni = 0] ...(5)

Eqs. (1) and (2) can be incorporated into Eq. (5) by making use of Lagrange’s method of

undetermined multipliers. Multiplying Eq. (1) by – α and Eq. (2) by – β and adding to Eq. (5), we get

Σ (– ln ni + ln gi – α – β Ei ) δ ni = 0 ...(6)

or – ln ni + ln gi – α – β Ei = 0

or ni = gi e

- α e - β E i (^) ...(7)

Eq. (7) is called Maxwell-Boltzmann distribution law.

M-B distribution in terms of temperature. It can be shown that β = 1/ kT where

k = Boltzmann’s constant and T = absolute temperature of the gas.

∴ ni = gi e

• α e

–( Ei / kT )

. ...(8)

75.4 Molecular Energies in an Ideal Gas

The M.B. distribution law is

STATISTICAL MECHANICS I 931

or C = (^3) 2

N

kT

π

π

∴ n ( E ) dE = (^3) 2

N E kT E e dE kT

π

π

Eq. (10) is plotted in Fig. 75.2 in terms of kT.

Fig. 75.

The total internal energy of the system is

E =

0 0

E kT E n E dE E e dE kT

∞ ∞

∫ ∫

The value of the definite integral is

kT π kT.

∴ E = 3

2

( )^4

N

kT kT NkT kT

π × π = π

The average energy of an ideal-gas molecule is E / N.

∴ E =

kT. ...(12)

Maxwell-Boltzmann Velocity Distribution Law

Substitute E = 1/2 mv

2 and

dE = mv dv in Eq. (10). Then we

get

3 (^2 )

(^32)

Nm mv kT n v dv v e dv kT

π −

π

This equation represents the

number of molecules with speeds

between v and v + dv in an assem-

bly of ideal gas containing N mol-

ecules at absolute temperature T.

This formula is plotted in Fig. 75.3.

Fig. 75.

932 MODERN PHYSICS

75.5 Bose-Einstein Distribution Law

Consider an assembly of N bosons. They are identical and indistinguishable. No restriction

is imposed as to the number of particles that may occupy a given cell. Let us now consider a box

divided into gi sections by ( gi – 1) partitions and ni indistinguishable particles to be distributed

among these sections. The permutations of ni particles and ( gi – 1) partitions simultaneously is given

by ( ni + gi – 1)!. But this includes also the permutations of ni particles among themselves and also

( gi – 1) partitions among themselves, as both these groups are internally indistinguishable. Hence

the actual number of ways in which ni particles are to be distributed in gi sublevels is

i i

i i

n g

n g

Therefore, the total number of distinguishable and distinct ways of arranging N particles in all

the variable energy states is given by

W =

i i

i i

n g

n g

ni and gi are large numbers. Hence we may neglect 1 in the above expression.

∴ W =

i i

i i

n g

n g

ln W = Σ [ln ( ni + gi )! – ln ni! – ln gi !]

As ni and gi are large numbers, we can use Stirling approximation.

ln W = Σ ( ni + gi ) ln ( ni + gi ) – ni ln ni – gi ln gi ...(3)

Here, gi is not subject to variation and ni varies continuously.

For most probable distribution, δ ln W max = 0.

Hence, if the W of Eq. (3) represents a maximum,

δ ln Wmax = Σ [ln ( ni + gi ) – ln ni ] δ ni = 0 ...(4)

The total number of particles and total energy are constants.

∴ Σ δ ni = 0 ...(5)

Σ Ei δ ni = 0 ...(6)

Multiplying Eq. (5) by – α, Eq. (6) by – β and adding to Eq. (4),

Σ [ln ( ni + gi ) – ln ni – α – β Ei ] δ ni = 0

The variations δ ni are independent of each other. Hence we get

ln – –

i i i i

n g E n

  α^ β  

or ni = ( i ) –

i E

g

e

α +β

or ni = ( i ) –

i E kT

g

e e

α

This is known as Bose-Einstein distribution law.

75.6 Fermi-Dirac Distribution Law

F-D statistics is obeyed by indistinguishable particles of half-integral spin that have

antisymmetric wave functions and obey Pauli exclusion principle. Consider N ferminons with the

934 MODERN PHYSICS

fBE ( Ei ) =

( i ) –

E kT e e

α

fFD ( Ei ) =

( i ) 1

E kT e e

α

When E >> kT , fBE and fFD approach fMB. In general,

f ( E ) =

exp[ α + E ( kT )]+ δ

is the probability of occupation of a single state at energy E and δ is +1 for F-D, 0 for M-B, and

• 1 for B-E statistics. We will refer to f ( E ) as the occupation probability or the distribution function.

Comparison of MB, BE, and FD Distribution Functions

MB BE FD

(1) Holds for distinguishable

particles; approximations

of BE and FD distribu-

tions at E >> kT.

(2) f ( E ) = Ae

–E/kT

(3) Applies to common gases

at normal temperatures.

Holds for indistinguishable

particles not obeying Pauli’s

exclusion principle

f ( E ) = /

E kT e e

α −

Applies to photon gas, phonon

gas.

Holds for indistinguishable

particles obeying the exclusion

principle.

f ( E ) = ( )/

F 1

E E kT e

−

Applies to electron gas in met-

als.

75.8 Black-Body Radiation

The first indication of the

inadequacy of classical ideas to explain

the properties of matter, occurred

in black-body radiation. A black-

body is a body which absorbs all the

radiation incident on it, and hence is

the perfect absorber. Consideration

of equilibrium of different bodies at

the same temperature implies that it

is also the best emitter of radiation

energy. A blackbody may be idealized

by a small hole drilled in a cavity.

If the radiation from a black-body

is analysed by a spectrometer, it is

found that the intensity distribution

as a function of frequency has a well-

defined shape. The spectrum of blackbody radiation is shown in Fig. 75.4 for two temperatures. The

spectral distribution of energy in the radiation depends only on the temperature of the body. What is

most significant is that, for a given temperature, it is a universal curve independent of the properties

of the walls of the cavity.

Black-Body Radiation.

STATISTICAL MECHANICS I 935

0 2 4 6 8 10 12

Visible light

Frequency, ν X 10 Hz

14

Spectral Energy Density, E( ) d

ν

ν

6000 K

3000 K

Fig. 75.

75.9 Rayleigh-Jeans Formula

Rayleigh’s formula for the distribution of energy in the black-body spectrum is based on

the principle of equipartition of energy for all the possible modes of free vibration which might

be assigned to radiation. Thus they considered average energy of an oscillator ( i.e. , per mode of

vibration) as

e = kT (classical result).

The number of independent standing waves per unit volume in a cavity in the frequency range

ν and ν + d ν is given by G (ν) d ν

2

3

d. c

πν ν

The energy u (ν) d ν per unit volume in the cavity within frequency range ν and ν + d ν is,

u (ν) d ν =

2 2

3 3

kT d G dv kT d c c

πν^ πν^ ν ε ν = × ν =

Rayleigh Jean’s law agrees well with the experimental results at low frequencies but near

the maximum in the spectrum and at higher frequencies it is in violent disagreement (Fig. 75.5).

According to this law, the energy density will continuously increase with increase in frequency ν and

approaches ∞ when ν → ∞.

STATISTICAL MECHANICS I 937

The energy-density of radiation in the frequency range ν to ν + d ν is

u (ν) d ν = h ν G (ν) f (ν) d ν

3

3

hv kT

h d

c (^) e

π ^ ν^ ν       

This is Planck radiation formula. Planck’s formula agrees with experimental curves.

75.11 Wien’s Displacement Law

Eq. (4) can be expressed in terms of wavelength of radiation as follows:

u l d l = 5

hc kT

ch d e

λ

π ^    λ   λ (^)  

or u l =

• 5 8 ( ) exp. –

hc hc kT

π λ (^)    (^) λ   ^  

Let lmax be the wavelength whose energy density is the greatest. Then,

du

d

λ

λ

Hence we get, lmax T =

• 3 2.898 10 mK.

hc

k

= ×

This is Wien’s displacement law.

The peak in the blackbody spectrum shifts to progressively shorter wavelengths (higher

frequencies) as the temperature is increased (Fig. 75.4).

75.12 Stefan-Boltzmann Law from Planck’s Formula

Planck’s radiation formula is

u (ν) d ν =

3

3

hv kT

h d

c e

π ν ν

The total energy density u over all frequencies is

u =

3 4 3 0 0

hv kT

h d u d aT c e

∞ ∞ π ν ν ν ν = = ∫ ∫

where a is a constant.

The energy R radiated by an object per second per unit area is

R = e σ T

4

σ = Stefan’s constant = 5.670 × 10

• 8 Wm - 2 K - 4 .

The emissivity e depends on the nature of the radiating surface.

This is the Stefan-Boltzmann law.

938 MODERN PHYSICS

AT A GLANCE

76.1 Macroscopic and Microscopic Descriptions

76.2 Ensembles

76.3 Probability

76.4 Thermodynamic Probability

76.5 Boltzmann’s Theorem on Entropy and Probability

76.6 Fundamental Postulates of Statistical Mechanics

76.7 Statistical Equilibrium

76.8 Quantum Statistics

76.9 Electron Gas

76.1 Macroscopic and Microscopic Descriptions

We may mentally isolate any finite portion of matter from its local surroundings. We call such a

portion the system. Everything outside the system which has a direct bearing on its behaviour we

call the environment. The system and its environment taken together are usually referred to as the

universe. Let us consider an example of a system and its surroundings. Consider some dilute gas

of fixed mass enclosed in a cylinder provided with a frictionless piston. The gas is the system. The

walls of the cylinder and the piston form the system boundary. The open atmosphere plays the role

of the surroundings.

We want to determine the behaviour of the system by finding how it interacts with its

environment. A system such as a dilute gas can be studied from ( i ) microscopic point of view

or ( ii ) macroscopic point of view. Here the term ‘ micro ’ stands for small scale and “ macro ” stands

for large scale. Molecules form the ‘ micro ’ systems, while any actual mass of gas is a ‘ macro ’

system. A macroscopic system is thus made up of an assembly of a large number of microscopic

ones. In the microscopic description, we consider quantities that describe the atoms and molecules

that make up the system, their speeds, energies, masses, behaviour during collision etc. These

quantities form the basis for the science of statistical mechanics. The variables used in microscopic

description are ( i ) very large in number ( ii ) not perceptible to our organic senses and ( iii ) not easily

StatiStical MechanicS ii

C H A P T E R

940 MODERN PHYSICS

76.4 Thermodynamic Probability

The thermodynamic probability of a particular macrostate is defined as the number of

microstates corresponding to that macrostate.

It is represented by W.

Consider two cells in phase space represented by i and j and four molecules a , b , c and d. Let

Ni and Nj be the number of molecules in the cells i and j respectively. Then the possible macrostates

are five in number (Fig. 76.1).

Ni 4 3 2 1 0

Nj 0 1 2 3 4

Fig. 76.

In general, to each of these five macrostates there corresponds a different number of microstates.

Let us consider the microstates corresponding to the macrostate Ni = 3, Nj =1. The number of

microstates corresponding to the macrostate Ni = 3, Nj =1 is four, as shown in Fig. 76.2.

Cell i abc abd acd bcd

Cell j d c b a

Fig. 76.

Therefore, the thermodynamic probability for the macrostate, Ni = 3, Nj =1 is 4. That is W = 4.

76.5 Boltzmann’s Theorem on Entropy and Probability

Boltzmann discovered a relation between entropy (a thermodynamical quantity) and probability

(a statistical quantity). Boltzmann started from a simple idea that the equilibrium state of the system

is the state of maximum probability. That is, the probability of the system in equilibrium state is

maximum. But from the thermodynamic point of view, the equilibrium state of the system is the

state of maximum entropy. If the system is not in equilibrium, then changes take place within the

system until the equilibrium state or the state of maximum entropy is reached. Thus, in equilibrium

state both the entropy and thermodynamical probability have their maximum values. As equilibrium

state is the state of maximum entropy and maximum probability, Boltzmann concluded that entropy

is a function of probability. That is,

S = f ( W ) ...(1)

Here, S is entropy and W is the thermodynamical probability of the state. Let us consider two

separate systems having entropies S 1 and S 2 and thermodynamic probabilities W 1 and W 2.

Then S 1 = f ( W 1 ) and S 2 = f ( W 2 ) ...(2)

The total entropy of the two systems is

S 1 + S 2 = f ( W 1 ) + f ( W 2 ) ...(3)

But the thermodynamic probability of the two systems taken together is W 1 W 2.

f ( W 1 W 2 ) = f ( W 1 ) + f ( W 2 ) = S 1 + S 2 ...(4)

If this relation is to be satisfied, f ( W ) must be a logarithmic function of W.

∴ f ( W ) = k log W

or S = k log W. ...(5)

STATISTICAL MECHANICS II 941

76.6 Fundamental Postulates of Statistical Mechanics

1. Any gas may be considered to be composed of molecules that are in motion and behave

like very small elastic spheres.

2. All the cells in the phase space are of equal size.
3. All accessible microstates corresponding to possible macrostates are equally probable.

This is called the postulate of equal a priori probability.

4. The equilibrium state of a gas corresponds to the macrostate of maximum probability.
5. The total number of molecules is constant.

76.7 Statistical Equilibrium

Boltzmann canonical principle is applied to determine the equilibrium state of the system.

According to this principle, Equilibrium state of a system is that which is most probable.

Consider an isolated system composed of a large number N of particles, in which each particle

has available to it several states with energies E 1 , E 2 , E 3 , .... At a particular time the particles are

distributed among the different states, so that n 1 particles have energy E 1 ; n 2 particles have energy

E 2 ; and so on. The total number of particles is

N = 1 2 3 ... (^) i i

n + n + n + = ∑ n ...(1)

We assume that the total number of particles remains constant for all processes occurring in the

system.

The total energy of the system is

E = 1 1 2 2 ... (^) i i i

n E + n E + = ∑ n E ...(2)

If the system is isolated, the total energy E must be constant. However, as a result of their mutual

interactions and collisions, the distribution of the particles among the available energy states may be

changing. For example in a gas, a fast molecule may collide with a slow one; after the collision, the

fast molecule may have slowed down and the slow one may have sped up. Or an excited atom may

collide inelastically with another atom, with a transfer of its excitation energy into kinetic energy of

both atoms. Hence, in both examples, the particles after the collision are in different states. In other

words, the numbers n 1 , n 2 , n 3 , ... which give the distribution of the N particles among the available

energy states, may be changing. It is reasonable to assume that for each macroscopic state of a

system of particles there is a distribution which is more favoured than any other. In other words,

we may say that given the physical conditions of the system of particles , there is a most probable

distribution. When this distribution is achieved, the system is said to be in statistical equilibrium.

76.8 Quantum Statistics

Statistical mechanics can be divided into two main classes.

1. Classical Statistics or Maxwell-Boltzmann Statistics.
2. Quantum Statistics.

Classical statistics interpreted successfully many ordinarily observed phenomena such as

temperature, pressure, energy etc. But it failed to account for several other experimentally observed

phenomena such as black body radiation, photoelectric effect, specific heat capacity at low

temperatures, etc. This failure of classical statistics forced the issue in favour of the new quantum idea

STATISTICAL MECHANICS II 943

empty, then f ( E ) = 0. If the level is certainly full, then f ( E ) = 1. In general f ( E ) has a value between

zero and unity.

The distribution function for electrons at T = 0 K has the form

f ( E ) = 1 when E < Ef

and f ( E ) = 0 when E > Ef ...(2)

That is, all levels below Ef are completely filled and all

levels above Ef are completely empty. This function is plotted

in Fig. 76.4 which shows the discontinuity at the Fermi-

energy. As the temperature rises, f ( E ) changes from 1 to 0

more and more gradually as shown in Fig. 76.4.

For E = Ef , f ( E ) = 0

1 e^2

at all temperatures.

Thus , the probability of finding an electron with energy

equal to the Fermi-energy in a metal is 1/2 at any temperature.

Since the electrons are confined inside the crystal, their

wave properties will limit the energy values which they may

have. Let g ( E ) dE be the number of quantum states available to electrons with energies between E

and E + dE. It can be shown that

g ( E ) dE =

3/

3

Vm dE h

π ...(3)

where m is the mass of the electron and V is the volume of the electron gas.

We can calculate the Fermi-energy EF by filling up the energy states in the metal sample with

the N free electrons it contains in order of increasing energy, starting from E = 0. The highest state

to be filled will then have the energy E = EF by definition. The number of electrons that can have

the same energy E is equal to the number of states that have this energy, since each state is limited

to one electron. Hence,

N =

3/ 1/ 3 0 0

E (^) f EF Vm g E dE E dE h

π

∫ ∫

3/ 3/ 3

F

Vm E h

π

∴ EF =

2/ 2 3

2 8

h N

m V

  π 

The quantity N/V is the density of free electrons.

N/V represents the number of free electrons per unit volume of the metal.

An effective temperature of the electron-gas, known as the Fermi-temperature , is defined by

TF = EF / k

ExamplE 1. An electron gas obeys the Maxwell-Boltzmann statistics. Calculate the average

thermal energy ( in eV ) of an electron in the system at 300 K.

Sol. E =

(1.38 10 ) 300 J

kT

− = × × ×

23

19

−

−

× × ×

× ×

eV = 0.039 eV.

Fig. 76.

944 MODERN PHYSICS

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