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Subject Physics

Paper No and Title P10 Statistical Physics

Module No and Title Module 14Ensemble Theory(classical)-IV(Canonical Ensemble( More Applications), Equipartition Theorem Virial Theorem and Density of States) Module Tag Phy_P10_M14.doc

Content Writer Prof. P.K. Ahluwalia Physics Department Himachal Pradesh University Shimla 171 005

TABLE OF CONTENTS

1. Learning Outcomes

2. Introduction

3. Canonical Ensemble(More Applications):

3.1 A System of Harmonic Oscillators (Classical and Quantum Mechanical Treatment)

3.2 Para-magnetism (Classical and Quantum Mechanical Treatment)

4. Equipartition Theorem

5. Virial Theorem

6. Density of Statesover Energy

7. Summary

Appendices

A1 Expansion of a logarithmic function and 퐜퐨퐭퐡 풙

A2 Spreadsheet for plotting Langevin and Brillouin functions

Physics PAPER No. 10 : Statistical Mechanics MODULE No.14 : Ensemble Theory(classical)-IV (Canonical Ensemble( More Applications), Equipartition Theorem Virial Theorem and Density of States)

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1. Learning Outcomes

After studying this module, you shall be able to

 Apply the approach of canonical ensemble via partition function to two very important prototype systems in physics via partition function o A collection of harmonic oscillators (classical and quantum mechanical treatment) o Para-magnetism (classical and quantum mechanical treatment).  Compare the classical and quantum mechanical results obtained in the above two examples and see under what conditions quantum mechanical results approach classical results.  To appreciate equipartition theorem and virial theorems as classical canonical ensemble 흏푯 averages of the quantity 푿풂 , where 푿풂 and 푿풃 are any two of the phase space co- 흏푿풃 ordinates of the system.  To apply equipartition theorem to derive some well-known results for systems such as monoatomic gas, a diatomic gas, a harmonic oscillator and a crystalline solid.  Derive virial theorem.  Understand the importance of virial in computation of equation of state of a system and paving way for calculating pressure of the system.  Calculate density of states over energy for a three dimensional system of free particles in non-relativistic, relativistic and ultra-relativistic regime.

2. Introduction

In this module we carry forward the applications of canonical ensemble two most important problems of the physics: a collection of harmonic oscillators and classical model of para-magnetism treated as a system of N magnetic dipoles. Also we shall encounter two beautiful theorems called virial theorem of Clausius and equipartition theorem of Boltzmann. These theorems have many applications to arrive at some useful results in thermodynamics and statistical mechanics.

3. Canonical Ensemble (More Applications)

Now we will look at some more interesting applicationsof canonical ensemble and simplicity with results can be obtained using partition function approach.

3.1 A System of Harmonic Oscillators

Let us now examine a collection of N independent identical Linear harmonic oscillators using partition function approach as an application of canonical ensemble which forms a precursor to the study the statistical mechanics of a collection of photons in black body radiation and a collection of phonons in theory of lattice vibrations in solids. We treat the oscillators as first as classical entities and later as quantum mechanical entities to get the thermodynamics of the system and compare the results for any change in going from classical system of harmonic oscillators to quantum mechanical system of harmonic oscillators.

Physics PAPER No. 10 : Statistical Mechanics MODULE No.14 : Ensemble Theory(classical)-IV (Canonical Ensemble( More Applications), Equipartition Theorem Virial Theorem and Density of States)

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The Hamiltonian for a this set of identical linear harmonic oscillators, each having mass m and frequency 흎 is given by

푵 풑ퟐ ퟏ (1) 푯 = 풊 + 풎흎ퟐ풒ퟐ ퟐ풎 ퟐ 풊 풊=ퟏ

(a) Classical Approach

Now we are ready to write a one particle partition function for the recalling the definition in the Module 13.

ퟏ (2) 풁 = 풆−휷푯풊 풒풊,풑풊 풅ퟑ풒 풅ퟑ퐩 ퟏ 풉 풊 퐢 ퟐ 휷풑 ퟏ ퟏ − 풊 − 휷풎흎ퟐ풒ퟐ = 풆 ퟐ풎 풅ퟑ풑 풆 ퟐ 풊 풅ퟑ퐪 풉 풊 퐢 ퟏ ퟏ (3) ퟏ ퟐ풎흅 ퟐ ퟐ흅 ퟐ ퟏ 풁 = = ퟏ 풉 휷 휷풎흎ퟐ 휷ℏ흎

ퟏ Here 휷 = . The partition function for the collection of independent N linear 풌푩푻 harmonic oscillators can be written as

푵 −푵 풁 = 풁ퟏ = 휷ℏ흎 (4)

Now all thermodynamic properties can be derived

Helmholtz free energy (F)

ℏ흎 (5) 푭 = −풌푩푻 퐥퐧 풁 = 푵풌푩푻 퐥퐧 휷ℏ흎 = 푵풌푩푻 퐥퐧 풌푩푻

Entropy (S)

흏푭 ℏ흎 (6) 푺 = − = 푵풌푩 ퟏ − 퐥퐧 흏푻 푵,푽 풌푩푻

Internal Energy (E=F+TS)

푬 = 푵풌푩푻 (7)

Physics PAPER No. 10 : Statistical Mechanics MODULE No.14 : Ensemble Theory(classical)-IV (Canonical Ensemble( More Applications), Equipartition Theorem Virial Theorem and Density of States)

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This result is in accordance with law of equipartition of energy, according to which each quadratic term in the Hamiltonian for each harmonic oscillator contributes a ퟏ term 풌 푻. ퟐ 푩

Pressure (P)

흏푬 (8) 푷 = − = ퟎ 흏푽 푵,푻

Chemical Potential (흁)

흏푭 ℏ흎 (9) 흁 = = 푵풌푩푻 퐥퐧 흏푵 푽,푻 풌푩푻

Specific heat (푪풑 = 푪푽)

흏푬 (10) 푪푷 = 푪푽 = = 풌푩푻 흏푻 푵,푽

(b) Quantum mechanical approach

Quantum mechanically, each linear harmonic oscillator is characterized by energy eigen value given by

ퟏ (11) 흐 = 풏 + ℏ흎 , 풏 = ퟎ, ퟏ, ퟐ, ퟑ … … … … 풏 ퟐ

Once again single oscillator partition function can be written as

ퟏ ∞ ∞ − 휷ℏ흎 (12) ퟏ ퟐ − 휷ℏ흎 풆 풛 = 풆−휷흐풏 = 풆 ퟐ 풆−휷풏ℏ흎 = ퟏ ퟏ − 풆−휷풏ℏ흎 풏=ퟎ 풏=ퟎ ퟏ −ퟏ = ퟐ 퐬퐢퐧퐡 휷ℏ흎 ퟐ

The partition function for the N Oscillator system is given by

Physics PAPER No. 10 : Statistical Mechanics MODULE No.14 : Ensemble Theory(classical)-IV (Canonical Ensemble( More Applications), Equipartition Theorem Virial Theorem and Density of States)

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ퟏ −푵 (13) 풁 = 풁 푵 = ퟐ 퐬퐢퐧퐡 휷ℏ흎 ퟏ ퟐ

Now we can derive all thermodynamic properties

Helmholtz free energy (F)

ퟏ (14) 푭 = −풌 푻 퐥퐧 풁 = 푵풌 푻 퐥퐧 ퟐ 퐬퐢퐧퐡 휷ℏ흎 푩 푩 ퟐ ퟏ = 푵 ℏ흎 + 풌 푻 퐥퐧 ퟏ − 풆−휷ℏ흎 ퟐ 푩

Entropy (S)

흏푭 (15) 푺 = − 흏푻 푵,푽 ퟏ ퟏ = 푵풌 휷ℏ흎 퐜퐨퐭퐡 휷ℏ흎 푩 ퟐ ퟐ ퟏ − 퐥퐧 ퟐ 퐬퐢퐧퐡 휷ℏ흎 ퟐ

Internal Energy (푬 = 푭 + 푻푺)

ퟏ ퟏ ퟏ ℏ흎 (16) 푬 = 푭 + 푻푺 = 푵ℏ흎 퐜퐨퐭퐡 휷ℏ흎 = 푵 ℏ흎 + ퟐ ퟐ ퟐ 풆휷ℏ흎 − ퟏ

This result is in not in accordance with law of equipartition of energy, according to which each quadratic term in the Hamiltonian for each harmonic oscillator ퟏ contributes a term 풌 푻. ퟐ 푩

Pressure (P)

흏푬 (17) 푷 = − = ퟎ 흏푽 푵,푻

Chemical Potential (흁)

흏푭 ퟏ −휷ℏ흎 (18) 흁 = = ℏ흎 + 풌푩푻 퐥퐧 ퟏ − 풆 흏푵 푽,푻 ퟐ

Physics PAPER No. 10 : Statistical Mechanics MODULE No.14 : Ensemble Theory(classical)-IV (Canonical Ensemble( More Applications), Equipartition Theorem Virial Theorem and Density of States)

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Specific heat (푪풑 = 푪푽)

ퟐ (19) 흏푬 ퟏ ퟐ ퟏ 푪푷 = 푪푽 = = 푵풌푩 ℏ흎휷 퐜퐨퐬퐞퐜퐡 휷ℏ흎 흏푻 푵,푽 ퟐ ퟐ

풆휷ℏ흎 = 푵풌 ℏ흎휷 ퟐ 푩 풆휷ℏ흎 − ퟏ ퟐ

All these results from (5) to (9) approach classical results for 풌푩푻 ≫ ℏ흎

3.1 Paramagnetism

Let us now examine a collection of N independent identical spins sitting at different points using partition function approach. Each of these spins can be treated as magnetic dipoles with a magnetic moment 흁. These are distinguishable as they are sitting at different locations. These form a non-interacting system in the sense that each magnetic dipole does not experience the field of the other dipoles. These magnetic dipoles can be orientedin any direction with the application of external magnetic field. Since the spins are localized they have no kinetic energy. The Hamiltonian of such a system consists, therefore, of magnetic potential energy in the presence of magnetic field 횮 only which can be written as

푵 (20) 푯 = − 흁 풊 . 횮 풊=ퟏ

Here 흁풊 is the magnetic moment of the ith magnetic dipole.

We look at the system both classically and quantum mechanically and try to get some well-knownresult such as Curie Law for paramagnetic systems.

(i) A classical Treatment

In this case equation (20) can be written explicitly in terms of orientation 휽풊 of each magnetic dipole as

푵 (21) 푯 = − 흁풊 횮 퐜퐨퐬 휽풊 = −푵흁횮 퐜퐨퐬 휽 풊=ퟏ 휽

Now the single particle partition function can be written, remembering that three dimensional orientation is decided by 휽 and 흓

Physics PAPER No. 10 : Statistical Mechanics MODULE No.14 : Ensemble Theory(classical)-IV (Canonical Ensemble( More Applications), Equipartition Theorem Virial Theorem and Density of States)

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So that single dipole partition function can be written as

ퟐ흅 흅 (22) (휷흁횮 퐜퐨퐬 휽) (휷흁횮 퐜퐨퐬 휽) 풁ퟏ = 풆 = 풆 퐬퐢퐧 휽 풅휽풅흓 휽 ퟎ ퟎ

Putting 퐱 = 퐜퐨퐬 휽 we solve equation (22) to get

ퟐ흅 +ퟏ (23) (휷흁횮 풙) 퐬퐢퐧퐡(휷흁 횮) 풁ퟏ = 풆 풅풙풅흓 = ퟒ흅 ퟎ −ퟏ (훃훍횮)

So the partition function of the system can be written as

퐬퐢퐧퐡(휷흁 횮) 푵 (24) 풁 = 풁 푵 = ퟒ흅 ퟏ 훃훍횮

Now we can calculate the thermodynamic properties using the results derived in module XIII.

Helmholtz free energy (F)

퐬퐢퐧퐡(휷흁 횮) (25) 푭 = −푵 퐤 퐓 퐥퐧 ퟒ흅 퐁 훃훍횮

Magnetization 푴 of the system

흏푭 훛(퐥퐧 퐬퐢퐧퐡(휷흁 횮) − 퐥퐧 훃훍횮 ) (26) 푴 = − = 푵퐤퐁퐓 = 흏횮 푻,푵 흏횮 ퟏ (27) 푴 = 푵 흁 퐜퐨퐭퐡 휷흁횮 − = 푵흁푳 휷흁횮 훃훍횮

The mean magnetization in the direction of the magnetic field is

푴 ퟏ (28) 흁 = = 흁 퐜퐨퐭퐡 휷흁횮 − 푵 훃훍횮

ퟏ Where 푳 풙 = 퐜퐨퐭퐡 풙 − is the Langevin function, here 풙 = 휷흁횮 is a parameter 퐱 giving the ratio of magnetic potential energy (흁횮) to thermal energy (풌푩푻)..

Physics PAPER No. 10 : Statistical Mechanics MODULE No.14 : Ensemble Theory(classical)-IV (Canonical Ensemble( More Applications), Equipartition Theorem Virial Theorem and Density of States)

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퐿(푥)

푥

Figure 1Langevin Function

There are two limiting cases which provide us interesting results

(i) Low temperature or High magnetic field limit (흁퐇 ≫ 풌푩푻):

In this limit Langevin function saturates to 1, figure 1, i.e all magnetic dipoles align in the direction of magnetic field and the net magnetization shall be

푴 = 푵 흁 (29)

(ii) High temperature or Low magnetic field limit (흁퐇 ≪ 풌푩푻):

ퟏ 풙 풙ퟑ In this limit 퐜퐨퐭퐡 풙 = + − + ⋯ and the Langevin function 푳 풙 ≈ 풙/ퟑ 풙 ퟑ ퟒퟓ

Therefore,

흁ퟐ퐇 (30) 푴 = 푵 퐤퐁푻

흏푴 In this limit susceptibility 흌 = becomes 흏푯

흏푴 푵흁ퟐ 푪 (31) 흌 = = = 흏푯 퐤퐁푻 푻

Physics PAPER No. 10 : Statistical Mechanics MODULE No.14 : Ensemble Theory(classical)-IV (Canonical Ensemble( More Applications), Equipartition Theorem Virial Theorem and Density of States)

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This is famous Curie Law of para-magnetism which was experimentally verified to be true for copper-potassium sulphate hexahydrate.

(i) A Quantum Mechanical treatment.

To treat the problem of para-magnetism quantum mechanically, we must note that (i) magnetic moment of the particle involved of the system depends on its angular momentum 푱 (ii) magnetic moment 흁 and its component in the direction of magnetic field does not have arbitrary values but are discrete.

If angular momentum of the system is 푱 its magnetic moment 흁 is given by

(32) 흁 = 흁푩품푱

풆ℏ Where 흁 = is Bohr magneton and 품 is Lande factor given by 푩 ퟐ풎풄

ퟑ 푺 푺 + ퟏ − 푳 푳 + ퟏ (33) 품 = + ퟐ ퟐ푱 푱 + ퟏ

If the particles involved are electrons i.e. 푺 = ퟏ/ퟐ with 푳 = ퟎ the total angular ퟏ momentum 푱 = , 품 = ퟐ. On the other hand if 푺 = ퟎ then 품 is solely due to 푳 and ퟐ 품 = ퟏ.

Now if we take magnetic field in the direction ofz-axis the component of 흁 in this direction 흁풛 is given by:

흁풛 = 품흁푩풎, 풎 = −푱, −푱 + ퟏ, … , 푱 − ퟏ, 푱 (34)

Now we are in a position to write down the single partition function in the presence of magnetic field:

풎=+푱 (35) 품흁푩풎횮훃 풁ퟏ = 풆 풎=−푱

The sum in equation (17) is a geometrical progression with ퟐ푱 + ퟏ terms with first term −품흁푩퐉횮훃 품흁푩횮훃 풆 and common ratio 풆 , therefore, 풁ퟏ is given by

풆−품흁푩푱횮훃 ퟏ − 풆품흁푩횮훃 ퟐ푱+ퟏ (36) 풁ퟏ = ퟏ − 풆품흁푩횮훃

This can be further simplified to give

Physics PAPER No. 10 : Statistical Mechanics MODULE No.14 : Ensemble Theory(classical)-IV (Canonical Ensemble( More Applications), Equipartition Theorem Virial Theorem and Density of States)

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ퟏ ퟏ −품흁 횮휷 푱+ 품흁 횮훃 풋+ ퟏ (37) 풆 푩 ퟐ − 풆 푩 ퟐ 퐬퐢퐧퐡 품흁 횮퐣훃 ퟏ + 푩 ퟐ푱 풁ퟏ = 품흁 횮휷 품흁 횮휷 = 품흁 횮푱휷 − 푩 푩 퐬퐢퐧퐡 푩 풆 ퟐ − 풆 ퟐ ퟐ푱

Putting 품흁푩횮퐣훃 = 풙, equation (36) becomes

ퟏ 퐬퐢퐧퐡 풙 ퟏ + (38) ퟐ푱 풁ퟏ = = 풙 퐬퐢퐧퐡 ퟐ푱

So the total partition function of the system is given by

ퟏ 푵 (39) 퐬퐢퐧퐡 풙 ퟏ + ퟐ푱 풁 = 풙 퐬퐢퐧퐡 ퟐ푱

Now we will be in a position to calculate thermodynamic properties of the system.

Helmholtz Free Energy(푭):

ퟏ 품흁 횮퐣훃 (40) 푭 = −푵풌 푻 퐥퐧 퐬퐢퐧퐡 품흁 횮퐉훃 ퟏ + − 퐥퐧 퐬퐢퐧퐡 푩 푩 푩 ퟐ퐉 ퟐ퐉

흏푭 Magnetization 푴: Since 푴 = − 흏횮 푻,푵

ퟏ ퟏ (41) 푴 = 푵 품흁 푱 ퟏ + 퐜퐨퐭퐡 품흁 횮푱휷 ퟏ + 푩 ퟐ푱 푩 ퟐ퐉 ퟏ 품흁 횮푱휷 − 퐜퐨퐭퐡 푩 ퟐ푱 ퟐ푱

Or

푴 = 푵품흁푩푱 푩푱 품흁푩횮푱휷 (42)

Where 푩푱 품흁푩횮푱휷 is a Brillouin function of order 푱, where 푱 is the relevant quantum number. The plot of the Brillouin function is given in Figure 2 below:

Physics PAPER No. 10 : Statistical Mechanics MODULE No.14 : Ensemble Theory(classical)-IV (Canonical Ensemble( More Applications), Equipartition Theorem Virial Theorem and Density of States)

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푩푱 품흁푩횮푱휷

품흁푩횮푱휷

Figure 2 Brillouin function for sample values of 푱

From Figure 2, it is clear that large values of 품흁푩횮푱휷 ≫ ퟏ i.e for strong field and low temperature, the Brillouin function approaches 1 for all values of 푱, showing state of magnetic saturation. However, for 품흁푩횮푱 ≪ ퟏ i.e.weak field and high temperature, the ퟏ 풙 Brillouin function, knowing that 퐜퐨퐭퐡 풙 ≅ + for 풙 ≪ ퟏ so that Brillouin function can 풙 ퟑ be written as

ퟏ ퟏ ퟏ 품흁푩횮푱휷 ퟏ + ퟏ ퟏ (43) ퟐ퐉 푩푱 풙 ≅ ퟏ + ퟏ + − 품흁 횮푱휷 ퟐ푱 품흁 횮푱휷 ퟏ + ퟑ ퟐ푱 푩 푩 ퟐ퐉 ퟐ풋 ퟏ 품흁 횮푱휷 푩 ퟐ퐉 + ퟑ

Or

ퟐ ퟐ 품흁푩횮푱휷 ퟏ ퟏ 품흁푩횮푱휷 ퟏ (44) 푩 풙 ≅ ퟏ + − = ퟏ + 푱 ퟑ ퟐ푱 ퟐ푱 ퟑ 푱

Therefore,

푵 품흁 ퟐ푱( 푱 + ퟏ) (45) 푴 = 푩 퐇 ퟑ풌푩푻

Once again we get Curie Law:

Physics PAPER No. 10 : Statistical Mechanics MODULE No.14 : Ensemble Theory(classical)-IV (Canonical Ensemble( More Applications), Equipartition Theorem Virial Theorem and Density of States)

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흏푴 푵 품흁 ퟐ푱( 푱 + ퟏ) 푪 (46) 흌 = = 푩 = 흏푯 ퟑ풌푩푻 푻

ퟐ ퟐ 푵 품흁푩 푱( 푱+ퟏ) 푵흁 ퟐ ퟐ Where, now 푪 = = with 흁 = 품흁푩 푱( 푱 + ퟏ), a departure from classical ퟑ풌푩 ퟑ풌푩 result.

In the limit 푱 → ∞, Brillouin function reduces toLangevin function:

ퟏ ퟐ풋 ퟏ ퟏ ퟏ 풙 ퟐ푱 ퟏ ퟏ + 퐜퐨퐭퐡 풙 ퟏ + − 퐜퐨퐭퐡 → 퐜퐨퐭퐡 풙 − = 퐜퐨퐭퐡 풙 − ퟐ푱 ퟐ퐉 ퟐ푱 ퟐ푱 풙 풙

This is expected, 푱 → ∞ amounts to saying that there are tremendously large orientations in which the magnetic dipoles can align, this situation is the same as in the classical case and should yield the classical result.

4. Equipartition Theorem

Equipartition theorem also known as law of equipartition of energy is a classical theorem. According to it in classical limiting case every canonical variable (generalized position and momentum) entering quadratically or harmonically in a Hamiltonian 풌 푻 function(Energy) has a mean thermal energy 푩 . For example, Hamiltonian function in ퟐ three dimensions may be written as

풑ퟐ ퟏ ퟏ 푳ퟐ (푬ퟐ + 푩ퟐ) (47) 푯 = + 풎흎ퟐ풓ퟐ + + ퟐ풎 ퟐ ퟐ 푰 ퟖ흅

This Hamiltonian represents energy possessed by a molecule which is respectively made up of translational kinetic energy, harmonic potential energy, rotational kinetic energy and electromagnetic energy in an electromagnetic field. Each term is quadratic in one of the canonical variables. Therefore, according to law of equipartition of energy, in three ퟑ풌 푻 ퟑ풌 푻 dimensions.Thus thermal energy 푬 is given by 푩 + 푩 + 풌 푻 + ퟐ풌 푻, here we ퟐ ퟐ 푩 푩 have three canonical variables for translational motion, three for vibrational motion, two for rotational motion and two each for electrical energy and magnetic energy. This theorem can be proved easily using canonical distribution function. In this section we prove this result with a slightly sophisticated approach.

Proof: Suppose we have a classical system of 푵 particles with ퟔ푵 generalized coordinates (풒풊, 풑풊), where 풊 goes from 1 to ퟑ푵, giving a total of ퟔ푵 coordinates. We are 흏푯 interested in calculating the mean value of the quantity 풙풊 , where 풙풊and 풙풋 are any of 흏풙풋 the ퟔ푵 coordinates. In canonical ensemble it is given by:

Physics PAPER No. 10 : Statistical Mechanics MODULE No.14 : Ensemble Theory(classical)-IV (Canonical Ensemble( More Applications), Equipartition Theorem Virial Theorem and Density of States)

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흏푯 (48) 풆−휷푯(풒,풑) 풙 풅흎 흏푯 풊 흏풙 풋 풙풊 = −휷푯(풒,풑) 흏풙풋 풆 풅흎

ퟑ푵 ퟑ푵 Where 풅흎 = 풅 풒 풅 퐩. Out of 풅흎 let us pick up integration over 풙풋 in the numerator −휷푯 풒,풑 −휷푯 풒,풑 흏푯 ퟏ 흏풆 and do this integration by parts. First we note that 풆 풙풊 = − 풙풊 so 흏풙풋 휷 흏풙풋 equation (48) can be written as

흏풆−휷푯 풒,풑 (49) 풙 풅풙 퐝훚 흏푯 ퟏ 풊 흏풙 풋 풓풆풎풂풊풏품 풄풐−풐풓풅풊풏풂풕풆풔 풋 풙풊 = − −휷푯(풒,풑) 흏풙풋 휷 풆 풅흎

So integral over 풙풋 becomes

−휷푯 풒,풑 흏풆 풙 흏풙 (50) 풙 풅풙 = 풙 풆−휷푯 풒,풑 풋ퟐ − 풊 풆−휷푯 풒,풑 풅풙 풊 풋 풊 풙 풋 흏풙풋 풋ퟏ 흏풙풋

To take limits over the first term we note that there are two possibilities:

(i) If 풙풋 are momentum co-ordinates, the kinetic energy tends to infinity at extremely large values, making Hamiltonian infinite and 풆−휷푯 풒,풑 → ퟎ. (ii) If 풙풋 happens to be position co-ordinates, the limits mean on the walls of the container in which system is enclosed, where potential energy is infinite and once again Hamiltonian becomes infinite leading to 풆−휷푯 풒,풑 → ퟎ.

Thus we can conclude that first term vanishes. So equation (50) becomes

−휷푯 풒,풑 (51) 흏풆 흏풙풊 −휷푯 풒,풑 풙풊 풅풙풋 = − 풆 풅풙풋 흏풙풋 흏풙풋 −휷푯 풒,풑 = − 휹풊풋 풆 풅풙풋

Hence equation (49) can be written as

흏푯 ퟏ 휹 풆−휷푯 풒,풑 풅풙 퐝훚 (52) 풊풋 풋 풓풆풎풂풊풏품 풄풐−풐풓풅풊풏풂풕풆풔 풙풊 = −휷푯(풒,풑) 흏풙풋 휷 풆 풅흎 −휷푯 풒,풑 휹풊풋 풆 퐝훚 휹풊풋 = = = 휹 풌 푻 휷 풆−휷푯(풒,풑)풅흎 휷 풊풋 푩

This is an extremely general result and is independent of the precise form of the 푯(풒, 풑).

From equation (52) the following results follow immediately:

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흏푯 ퟑ푵 If 풙풊 = 풙풋 = 풑풊 흏푯 풑풊 = 풌푩푻 흏풑풋 풑풊 = ퟑ푵풌푩푻 흏풑풋 풊=ퟏ If 풙풊 = 풙풋 = 풒풊 흏푯 풒풊 = 풌푩푻 흏푯 흏풒 풒풊 = ퟑ푵풌푩푻 흏풒풊

What about the statement of the law of equipartion of energy made earlier that every canonical variable (generalized position and momentum) entering quadratically or 풌 푻 harmonically in a Hamiltonian function(Energy) has a mean thermal energy 푩 ퟐ

This can be checked easily. Let us take a Hamiltonian which is is a quadratic function of co-ordinates, by suitable canonical transformation, it can be converted into a form:

ퟐ ퟐ (53) 푯 = 푨풋푷풋 + 푩풋푸풋 풊 풊

Where, 푸풋and 푷풋 are canonically conjugate transformed coordinates and 푨풋 and 푩풋 are suitable constants.

Then it immediately follows that

흏푯 흏푯 (54) 푷풋 + 푸풋 = ퟐ푯 흏풑풋 흏풒풋 풋

Therefore, from the table it follows that

ퟏ 흏푯 흏푯 ퟏ ퟏ (55) 푯 = 푷풋 + 푸풋 = ퟔ푵풌푩푻 = 풇풌푩푻 ퟐ 흏푷풋 흏푸풋 ퟐ ퟐ 풋

Here 6N are the number of degrees of freedom corresponding to number of non-zero coefficients in the transformed Hamiltonian (53). One can immediately conclude that ퟏ corresponding to each quadratic term in the Hamiltonian there is a contribution of 풌 푻, ퟐ 푩 proving the statement providing an alternative proof for the equipartition theorem.

Applications of law of equipartition:

Law of equipartition theorem provides a back of the stamp method to calculate in classical regime (i.e. at sufficiently high temperature and low density) to calculate internal energy and specific heat. Quantum mechanically a system has discrete energy levels, however at sufficiently high temperatures, the spacing 횫푬 between energy levels is ≪ 풌푩푻. One can then treat discrete energy level structure as a continuum and law of

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equipartition which is valid in classical regime can be applied. In the following we look at application of equipartition theorem in some well-known systems.

A monoatomic gas: Single atom in a monoatomic gas has only translational kinetic energy, so that total Hamiltonian of the 1mole monoatomic system can be written as ퟐ 풑ퟐ 풑 풑ퟐ 푵 ( 풊풙 + 풊풚 + 풊풛 ), where N is Avagadro’s number, so that there are ퟑ푵 quadratic 풊=ퟏ ퟐ풎 ퟐ풎 ퟐ풎 terms in momentum coordinate in it. According to law of equipartition of energy, the energy of this 푵 particle monoatomic gas can be written as

풌 푻 ퟑ (56) 푬 = ퟑ푵 푩 = 푹푻 ퟐ ퟐ

ퟑ The molar specific heat at constant volume 푪 = 푹. Using 푪 − 푪 = 푹, specific heat 풗 ퟐ 풑 풗 ퟓ at constant pressure is then given by 푪 = 푹. The ratio of the two specific heats 풑 ퟐ 푪 ퟓ 휸 = 풑 = , a well-known result. 푪풗 ퟑ

A diatomic gas: Each particle of the gas in thios case is made up of a diatomic molecule, which can be assumed to be a dumb bell shaped rigid rotator, with arotational symmetry about the line joining the nuclei of the two atoms along Z-axis. So the Hamiltonian of the diatomic molecule has three translational kinetic energy terms quadratic in three components of the linear momentum and two terms corresponding to rotational kinetic ퟐ ퟐ 풑ퟐ 풑 풑ퟐ 푱ퟐ 푱 energy quadratic in 푵 풊풙 + 풊풚 + 풊풛 + 풊풙 + 풊풚 where 푱 is angular momentum 풊=ퟏ ퟐ풎 ퟐ풎 ퟐ풎 ퟐ푰 ퟐ푰 and 푰 moment of inertia. Hamiltonian thus has 5N quadratic terms.According to law of equipartition of energy the energy of this N diatomic molecule system can be written as

ퟓ ퟓ (57) 푬 = 푵풌 푻 = 푹푻 ퟐ 푩 ퟐ

Where 푵 is Avagadro’s number.

ퟓ The molar specific heat at constant volume 푪 = 푹. Using 푪 − 푪 = 푹, specific heat 풗 ퟐ 풑 풗 ퟕ at constant pressure is then given by 푪 = 푹. The ratio of the two specific heats 풑 ퟐ 푪 ퟕ (휸 = 풑 = , a well-known result which agree very nicely with experimental results at 푪풗 ퟓ room temperature. It must be mentioned here that if we relax the condition of rigid molecule in this discussion we expect molecule to vibrate as a one dimensional harmonic oscillator having energy which is partly kinetic and partly kinetic contributing two additional terms so that each molecule contributes vibrational kinetic energy of ퟐ × ퟏ 풌 푻, so that ퟐ 푩

ퟓ ퟕ (58) 푬 = 푵풌 푻 + 푵 풌 푻 = 푹푻 ퟐ 푩 푩 ퟐ

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ퟕ ퟗ ퟗ So that 푪 = 푹, 푪 = 푹 and 휸 = . The observed values do not agree with 푽 ퟐ 풑 ퟐ ퟕ experiments indicating that at room temperature diatomic molecule has no vibrational motion. The vibrations get excited only in the case of high temperature limit.

A crystalline solid:

A simple model of crystalline solid consists of a collection of atoms, with each lattice having one atom, where it can not move or rotate but can behave as an independent harmonic oscillator. Each oscillator having 6 quadratic terms three for kinetic energy and ퟐ 풑ퟐ 풑 풑ퟐ ퟏ three for potential energy: 풊풙 + 풊풚 + 풊풛 + 푲(풒ퟐ + 풒ퟐ + 풒ퟐ ) . So according to ퟐ풎 ퟐ풎 ퟐ풎 ퟐ 풊풙 풊풚 풊풛 law of equipartition energy, we have energy of one mole of a solid:

ퟏ (59) 푬 = ퟔ푵 풌 푻 = ퟑ 푹푻 ퟐ 푩

Thus molar specific heat of a solid at constant volume 푪풗 = ퟑ푹, well-known Dulong and Petit’s law. This is found to be the case at high temperature, but is not true at low temperatures and we need to go beyond classical law of equipartition of energy and quantum mechanics invoked as was done by Einstein.

5. Virial Theorem:

흏푯 흏푯 In classical mechanics the products 풑풊 = 풑풊 풒 풊 and 풒풊 = −풒풊 풑 풊 have a special 흏풑풊 흏풒풊 significance. The second expression which is a product of position coordinate and generalized force is called virial and when summed over all i its mean value gives virial theorem of statistical mechanics:

ퟑ푵 ퟑ푵 흏푯 (60) 퓥풊풅 = 풒풊 풑 풊 = − 풒풊 = −ퟑ푵풌푩푻 풊=ퟏ 풊=ퟏ 흏풒풊

Virial has a very interesting relationship with the physical quantities of a system and starting from it equation of state can be derived. Let us take the case of an ideal gas. It is enclosed in a container and walls provide the only external force to keep the gas confinedin the form of external pressure 푷. So here we have a forse −푷풅 푺 on an area element 풅 푺 of the wall. The virial of the ideal gas can be calculated as:

ퟑ푵 ퟑ푵 (61)

퓥풊풅 = 풒풊 풑 풊 = 풒풊 푭풊 = −푷 풓 . 풅풔 풊=ퟏ 풊=ퟏ 풔

Physics PAPER No. 10 : Statistical Mechanics MODULE No.14 : Ensemble Theory(classical)-IV (Canonical Ensemble( More Applications), Equipartition Theorem Virial Theorem and Density of States)

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Where 풓 is the position vector corresponding to the surface element with a particle in its vicinity. Using divergence theorem we have

(62)

퓥풊풅 = −푷 훁 . 풓 풅푽 = −ퟑ푷푽 푽

Applying Virial Theorem, we have ideal equation of state:

푷푽 = 푵푲푻 (63)

ퟑ 퓥 Furthermore, Kinetic energy of an ideal gas say 푬 = 푵풌 푻 = − 풊풅, therefore, 풌 ퟐ 풃 ퟐ

퓥풊풅 = −ퟐ푬풌 (64)

Virial has many interesting applications where it can from the knowledge of two particle potential be used to calculate the equation of state of the system, which becomes very useful in calculation of pressure computationally.

6. Density of State over Energy:

We have till this point focused on the number of microstates in a region of phase space and arrived at a result that for a system with f degrees of freedom, a volume ퟐ흅ℏ 풇 in the phase space contains one microstate. However, in many practical situations, e.g. in condensed matter physics we are interested in finding the number of states which are available to it in a small energy interval 흐and 흐 + 풅흐. This important number can be obtained if we know relation between energy (흐) and momentum (풑), also called dispersion relation. These relations for a free particle system for non-relativistic, ultra relativistic (mass neglected or mass less particles) and relativistic particles are given in the table below:

Table 1 Dispersion relation for a free particle system

System Dispersion relation Non-relativistic 풑ퟐ 흐 = ퟐ풎 Relativistic 흐ퟐ = 풄ퟐ풑ퟐ + 풎ퟐ푪ퟒ Ultra relativistic 흐 = 풄풑

Let us derive for a three dimensional system, the density of states 퓓(흐), the number of states of a particle per unit volume per unit energy range about 흐.

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Let us begin by counting the number of states for a free particle in three dimensions in small region of phase space 풅ퟑ풓 풅ퟑ풑, then since particle has three degrees of freedom we have

풅ퟑ풓풅ퟑ풑 Number of states in phase space volume풅ퟑ풓 풅ퟑ풑 = (65) ퟐ흅ℏ ퟑ

.

If the particles have some internal degrees of freedom, let us denote this by 품 then the abover relation becomes

풅ퟑ풓풅ퟑ풑 Number of states in phase space volume풅ퟑ풓 풅ퟑ풑 = 품 (66) ퟐ흅ℏ ퟑ

Let us assume that system is homogeneous, then this number is independent of the position of the phase space volume element. Then we can integrate over the postion coordinates to obtain

푽풅ퟑ풑 Number of states in momentum space volume풅ퟑ풑 = 품 (67) ퟐ흅ℏ ퟑ

Now if we take system to be isotropic, this number depends only on the magnitude of momentum 풑 , we can then integrate over angular variables 휽 and 흓, so that we get

ퟒ흅푽풑ퟐ풅풑 Number of states between 풑 and 풑 + 풅풑 = 품 (68) ퟐ흅ℏ ퟑ

Now let 퓓(흐) be the number of states of a particle per unit volume per unit energy range about 흐 then

ퟒ흅푽풑ퟐ풅풑 (69) 푽퓓 흐 풅흐 = 품 ퟐ흅ℏ ퟑ

Or

ퟒ흅 풑ퟐ 풅풑 (70) 퓓 흐 = 품 ퟐ흅ℏ ퟑ 풅흐

Now by using dispersion relations given in Table 1, we can get density of states for each of the cases for a three dimensional system as given in Table 2 below

Physics PAPER No. 10 : Statistical Mechanics MODULE No.14 : Ensemble Theory(classical)-IV (Canonical Ensemble( More Applications), Equipartition Theorem Virial Theorem and Density of States)

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Table 2 Density of states over energy for a three dimensional system

System Dispersion relation 풅풑 퓓 흐

풅흐 ퟐ ퟏ ퟑ Non- 풑 ퟑ − 흐 = ퟐ풎 흐 ퟐ 품 ퟐ풎 ퟐ ퟏ relativistic 흐ퟐ ퟐ풎 ퟒ흅ퟐ ℏퟐ ퟐ ퟐ ퟐ ퟐ ퟒ ퟏ ퟏ Relativistic 흐 = 풄 풑 + 풎 풄 ퟐ흐 − ퟐ ퟐ ퟒ 흐ퟐ − 풎ퟐ풄ퟒ ퟐ 품 흐 흐 − 풎 풄 ퟐ 푪 ퟐ흅ퟐ 풄ℏ ퟑ Ultra 흐 = 풄풑 ퟏ 품 흐ퟐ

relativistic 풄 ퟐ흅ퟐ ℏ풄 ퟑ

Similarly density of states for two dimensional systems, one dimensional systems and zero dimensional systems can be obtained.

7. Summary

In this module we have learnt

 How the properties from the knowledge of patition function of the following systems their thermodynamic properties can be obtained:

풑ퟐ ퟏ i. A system of harmonic oscillators: 푯 = 푵 풊 + 풎흎ퟐ풒ퟐ 풊=ퟏ ퟐ풎 ퟐ 풊

Physical Quantity Classical approach Quantum Approach 푵 −푵 −푵 Partition function 풁 = 풁ퟏ = 휷ℏ흎 ퟏ 풁 = ퟐ 퐬퐢퐧퐡 휷ℏ흎 ퟐ Helmholtz free energy ℏ흎 ퟏ −휷ℏ흎 푭 = 푵풌푩푻 퐥퐧 푭 = 푵 ℏ흎 + 풌푩푻 퐥퐧 ퟏ − 풆 풌푩푻 ퟐ Entropy ℏ흎 ퟏ ퟏ 푺 = 푵풌푩 ퟏ − 퐥퐧 푺 = 푵풌푩 휷ℏ흎 퐜퐨퐭퐡 휷ℏ흎 풌푩푻 ퟐ ퟐ ퟏ − 퐥퐧 ퟐ 퐬퐢퐧퐡 휷ℏ흎 ퟐ Internal Energy 푬 = 푵풌 푻 ퟏ ퟏ 푩 푬 = 푵ℏ흎 퐜퐨퐭퐡 휷ℏ흎 = ퟐ ퟐ Pressure 푷 = ퟎ 푷 = ퟎ

Chemical potential ℏ흎 ퟏ −휷ℏ흎 흁 = 푵풌푩푻 퐥퐧 흁 = ℏ흎 + 풌푩푻 퐥퐧 ퟏ − 풆 풌푩푻 ퟐ Specific heat 푪 = 푪 = 풌 푻 풆휷ℏ흎 푷 푽 푩 푪 = 푪 = 푵풌 ℏ흎휷 ퟐ 푷 푽 푩 풆휷ℏ흎 − ퟏ ퟐ

ℏ흎 In the classical limit ≪ ퟏ all quantum mechanical results approach classical results. 풌푩푻

ii. Para-magnetism (Classical and Quantum Mechanical Treatment)

The Hamiltonian for a classical system of magnetic dipoles in a magnetic field is given by

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푵 푵

푯 = − 흁 풊 . 횮 = − 흁풊 횮 퐜퐨퐬 휽풊 = −푵흁횮 퐜퐨퐬 휽 풊=ퟏ 풊=ퟏ 휽

Physical Quantity Classical approach Quantum Approach Partition function 푵 ퟏ 푵 퐬퐢퐧퐡(휷흁 횮) 퐬퐢퐧퐡 풙 ퟏ + 풁 = = ퟒ흅 ퟐ푱 훃훍횮 풁 = 풙 퐬퐢퐧퐡 ퟐ푱 풘풉풆풓풆 풙 = 품흁푩횮퐣휷 Helmholtz free 퐬퐢퐧퐡(휷흁 횮) ퟏ 푭 = −푵 퐤 퐓 퐥퐧 ퟒ흅 푭 = −푵풌 푻 퐥퐧 퐬퐢퐧퐡 풙 ퟏ + energy 퐁 훃훍횮 푩 ퟐ퐉 풙 − 퐥퐧 퐬퐢퐧퐡 ퟐ퐉 Magnetization 흏푭 ퟏ ퟏ 푴 = − = 푵 흁 퐜퐨퐭퐡 휷흁횮 푴 = 푵 품흁푩푱 ퟏ + 퐜퐨퐭퐡 품흁푩횮푱휷 ퟏ + 흏횮 푻,푵 ퟐ푱 ퟐ퐉 ퟏ ퟏ 품흁푩횮푱휷 − − 퐜퐨퐭퐡 훃훍횮 ퟐ푱 ퟐ푱 = 푵흁푳 휷흁횮 = 푵품흁푩푱 푩푱 품흁푩횮푱휷 Magnetization in 푴 = 푵 흁 푴 = 푵 품흁푩푱 High magnetic field or low temperature limit ퟐ ퟐ Magnetization in high 흁 퐇 푵 품흁푩 푱( 푱 + ퟏ) temperature and low 푴 = 푵 푴 = 퐇 퐤퐁푻 ퟑ풌푩푻 magnetic field limit

ퟐ ퟐ ퟐ Susceptibility (in high 흏푴 푵흁 푪 푵흁 푵 품흁푩 푱 푱 + ퟏ 푪 temperature and low 흌 = = = , 푪 = 흌 = = , 흏푯 퐤퐁푻 푻 퐤퐁 ퟑ풌푩푻 푻 magnetic field limit) 푵 품흁 ퟐ푱( 푱 + ퟏ) 푪 = 푩 ퟑ풌푩

 That law of equipartition theorem states that every canonical variable (generalized position and momentum) entering quadratically or harmonically in a 퐤 퐓 Hamiltonian function(Energy) has a mean thermal energy 퐁 ퟐ  How to derive law of equipartition of energy in its most general form 흏푯 풙풊 = 휹풊풋풌푩푻 흏풙풋

Where 풙풊 and 풙풋 are any of the ퟔ푵 coordinates.

 How to apply law of equipartition of energy to get in the classical limit, internal energy 푬, molar specific heat at constant volume 푪풗, molar specific heat at constant pressure 푪풑 and ratio of the specific heat 휸 for a monoatomic gas, for a diatomic gas (with and without vibrations) and a crystalline solid highlighting the limitations of the approach.  How to calculate virial of an ideal gas,

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ퟑ푵 ퟑ푵 흏푯 퓥풊풅 = 풒풊 풑 풊 = − 풒풊 = −ퟑ푵풌푩푻 풊=ퟏ 풊=ퟏ 흏풒풊

And how it can we used to get equation of state of the ideal gas.

 How to calculate density of state over energy of a three dimensional gas of free particles ퟒ흅 풑ퟐ 풅풑  That density of state over energy 퓓 흐 = 품 for non-relativistic, ퟐ흅ℏ ퟑ 풅흐 relativistic and ultra relativistic(mass less )gas is given by

System 퓓 흐 Non- ퟑ 품 ퟐ풎 ퟐ ퟏ relativistic 흐ퟐ ퟒ흅ퟐ ℏퟐ Relativistic ퟏ 품 흐 흐ퟐ − 풎ퟐ풄ퟒ ퟐ

ퟐ흅ퟐ 풄ℏ ퟑ Ultra 품 흐ퟐ

relativistic ퟐ흅ퟐ ℏ풄 ퟑ

Appendices

A1 Expansion of a logarithmic function and 퐜퐨퐭퐡 풙

풙ퟐ 풙ퟑ 풙ퟒ 퐥퐧 ퟏ + 풙 = 풙 − + − + ⋯for풙 ≪ ퟏ. ퟐ ퟑ ퟒ

ퟏ 풙 풙ퟑ 퐜퐨퐭퐡 풙 = + − + ⋯for x<<1 풙 ퟑ ퟒퟓ

A2 Spreadsheet for plotting Langevinand Brillouin functions

Bibliography

1. Pathria R.K. and Beale P. D., Statistical Mechanics, 3rd ed. (Elsevier, 2011). 2. Kubo R., “Statistical Mechanics: An Advanced Course with problems and Solutions,” Amsterdam: North Holland Publishing Company, 1965. 3. Pal P.B., “An Introductory Course of Statistical Mechanics”, New Delhi: Narosa Publishing House Pvt. Ltd., 2008. 4. Panat P.V., “Thermodynamics and Statistical Mechanics,” New Delhi: Narosa Publishing House Pvt. Ltd., 2008

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5. Brush S.G., “Statistical Physics and the Atomic Theory of Matter, from Boyle and Newton to Landau and Onsager”, New Jersey: Princeton University Press, 1983.

Physics PAPER No. 10 : Statistical Mechanics MODULE No.14 : Ensemble Theory(classical)-IV (Canonical Ensemble( More Applications), Equipartition Theorem Virial Theorem and Density of States)