Statistical Mechanics

Teacher

Dr. Nibedita Nayak

Category

Core Courses

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Course Name : Statistical Mechanics

Code(Credit) : CUTM1496(3-2-1)

Course Objectives

  • To expose the students to the relationship between macroscopic (bulk) thermodynamic properties of materials and the underlying microscopic (single particle) picture
  • To the development of the concept of the Blackbody Radiation and the development of quantum theory of energy/ radiation.
  • To expose the students to the temperature dependent (energy-) distribution functions in Classical Mechanical and Quantum Mechanical domains
  • To illustrate the application of theoretical concepts of energy and entropy to real systems like pure & doped bulk materials.

Course Outcomes

COS course outcome mapping cos with pos   (high-3, medium-2, low-1)
CO1 understand the classical statistics of particles po1-3, po2-3, po9-1
CO2 understand and apply classical theory of radiation po2-3, po9-1
CO3 understand and apply quantum theory of radiation po4-1, po5-2, po9-1
CO4 understandbose-einstein statistics po1-3, po2-3, po9-1
CO5 understand fermi-dirac statistics po1-3, po2-3, po3-2

Course Syllabus

Module I

Classical Statistics-1: Macrostate & microstate, Elementary concept of ensemble, Phase space, Entropy and thermodynamic probability, Maxwell-Boltzmann distribution law, Partition function, Thermodynamic functions of an ideal gas.

Practice-1

Plot Maxwell-Boltzmann distribution functions with energy at different temperatures. (Using Scilab / Python) 

Practice-2

Under Maxwell-Boltzmann Statistics compute Most Probable Speed, Mean Speed and RMS Speed and compare their ratios with theoretical values; (Using Scilab / Python).

Module II

Classical Statistics-2: Classical entropy expression, Gibbs paradox, Sackur Tetrode equation, Law of equipartition of energy (with proof) – applications to specific heat and its limitations.

Practice-3

Plot specific heat of solids according to Dulong-Petit law for high temperature and low temperature and compare them for these two cases.

Practice-4

Plot specific heat of solids according to Einstein distribution function for high temperature and low temperature and compare

Practice-5

Plot specific heat of solids according to Debye distribution function for high temperature and low temperature and compare them for these two cases.

Module III

Classical Theory of Radiation-1: Properties of thermal radiation, Black body radiation, Pure temperature dependence, Kirchhoff’s law, Stefan-Boltzmann law: thermodynamic proof. 

Module IV

Classical Theory of Radiation-2: Radiation pressure, Wien’s displacement law, Wien’s distribution law, Saha’s ionisation formula, Rayleigh-Jean’s law, Ultraviolet catastrophe.

Practice-6

Plot Wien’s distribution Law

Practice-7

Plot Rayleigh-Jeans Law

Module V

Quantum Theory of Radiation: Spectral distribution of black body radiation, Planck’s quantum postulates, Planck’s law of black body radiation: experimental verification, Deduction of (1) Wien’s distribution law, (2) Rayleigh-Jeans law, (3) Stefan-Boltzmann law, (4) Wien’s displacement law from Planck’s law. 

Practice-8

Plot Planck’s law for black body radiation and compare it with Raleigh-Jeans law at high temperature and low temperature

Module VI

Bose-Einstein Statistics: B-E distribution law, Thermodynamic functions of a strongly degenerate Bose gas, Bose Einstein condensation, properties of liquid He (qualitative description), Radiation as a photon gas and thermodynamic functions of photon gas. 

Practice-9

Implications of Planck's Blackbody Radiation formula: Wien's Displacement Law & Stefan-Boltzmann law: Numerical Simulation;  (Using Scilab / Python).

Practice-10

Plot Bose-Einstein distribution and Maxwell-Boltzmann Distribution functions with energy at different temperatures. (Using Scilab / Python)

Module VII

Fermi-Dirac Statistics: Fermi-Dirac distribution law, Thermodynamic functions of a completely and strongly degenerate fermi gas, Fermi energy, Electron gas in a metal. 

Practice-11

To  Plot  Fermi-Dirac Distribution and compare with Bose-Einstein distribution and Maxwell-Boltzmann Distribution functions with energy at different temperatures. (Using Scilab / Python)

Text Book:

Statistical Mechanics by R.K. Pathria, Butterworth Heinemann: 2nd Ed., 1996, Oxford University Press. 

e-book

Reference Books:

  1. Statistical Physics, Berkeley Physics Course, F. Reif, 2008, Tata McGraw-Hill
  2. Statistical and Thermal Physics, S. Lokanathan and R.S. Gambhir. 1991, Prentice Hall
  3. Thermodynamics, Kinetic Theory and Statistical Thermodynamics, Francis W. Sears and Gerhard L. Salinger, 1986, Narosa.
  4. Modern Thermodynamics with Statistical Mechanics, Carl S. Helrich, 2009, Springer
  5. An Introduction to Statistical Mechanics & Thermodynamics, R.H. Swendsen, 2012, Oxford Univ. Press

Session 3

Phase space, Entropy and thermodynamic probability.

https://www.youtube.com/watch?v=YM-uykVfq_E&t=14s

Session 5

Session 6

Thermodynamic functions of an ideal gas

https://www.youtube.com/watch?v=G-L81TfsqKI

Session 7

Practice-1 (2 hours)

Plot Maxwell-Boltzmann Distribution function vs. speed at different temperatures. (Using Scilab or Python)

Maxwell-BoltzmannDistribution_vs_speed.ipynb - Colaboratory

Session 8

Practice-2 (2 hours)

Under Maxwell-Boltzmann Statistics compute Most Probable Speed, Mean Speed and RMS Speed and compare their ratios with theoretical values; (Using Scilab / Python).

Maxwell-BoltzmannDistribution_vs_speed.ipynb - Colaboratory

Session 9

Classical entropy expression

https://www.youtube.com/watch?v=t4zxgJSrnVw

Session 12

Session 13

Applications of equipartition of energy to specific heat and its Limitations .

https://www.youtube.com/watch?v=1u207DoAdzE

Session 14

Flip class-1

Application of law of equipartition energy in specific heat of a gas. Meyer's relation.

Session 15

Practice-3 (2 hours)

Plot specific heat of solids according to Dulong-Petit law for high temperature and low temperature and compare them for these two cases. (Using Scilab / Python)

Session 16

Practice-4 (2 hours)

Plot specific heat of solids according to Einstein distribution function for high temperature and low temperature and compare. (Using Scilab / Python)

SpHeat_DP-Einstein-Debye.ipynb - Colaboratory

Session 17

Practice-5 (2 hours)

Plot specific heat of solids according to Debye distribution function for high temperature and low temperature and compare them for these two cases. (Using Scilab / Python)

SpHeat_DP-Einstein-Debye.ipynb - Colaboratory

Session 18

Properties of thermal radiation

https://www.youtube.com/watch?v=5GoZZKcNZiQ

Session 19

Black body radiation. Pure temperature dependence

https://www.youtube.com/watch?v=_0tkbp8yk-w

Session 20

Group Quiz-1

Properties of Thermal Radiation, Blackbody radiation, Pure Temperature Dependence of Blackbody Radiation

https://www.youtube.com/watch?v=3eOBT6XdBmg

Session 21

Session 22

Stefan-Boltzmann law-Thermodynamic proof.

https://www.youtube.com/watch?v=sUp_WZKZID4

Session 23

Flip class-2

Stefan Boltzmann law practical application .

https://www.youtube.com/watch?v=LidveOARCec

Session 24

Radiation pressure, Effects of solar radiation pressure on satellite orbits

https://www.youtube.com/watch?v=DDJXFSwMCTw

https://www.youtube.com/watch?v=AVO

Session 25

Wien’s displacement law, Wien’s distribution law

https://www.youtube.com/watch?v=4piwVbihZz4

Session 26

Group Quiz-2

Radiation Pressure, Effects of Radiation Pressure, Wien Distribution Law, Wien's Displacement Law

Wien’s displacement law, Wien’s distribution law

https://www.youtube.com/watch?v=4piwVbihZz4

Session 28

Saha Ionization Formula-2

Session 29

Flip class -3

Rayleigh - Jeans Law and Ultraviolet Catastrophe

https://www.youtube.com/watch?v=FXfrncRey-4

https://www.youtube.com/watch?v=7BXvc9W97iU

Session 30

Practice-6 (2 hours)

Plot Wien’s distribution law. (Using Scilab / Python)

Wien_Raylkeigh-Jeans.ipynb - Colaboratory.

Session 31

Practice-7 (2 hours)

Plot Rayleigh-Jeans law. (Using Scilab / Python).

Wien_Raylkeigh-Jeans.ipynb - Colaboratory

Session 32

Planck's Quantum Postulates, Derivation of Planck's Law of Blackbody Radiation

https://www.youtube.com/watch?v=IQvJ4gHs_sg

.https://www.youtube.com/watch?v=ktvsucLC_WY

Session 33

Planck's Blackbody Radiation Law - Experimental Verification, Spectral Distribution of Blackbody Radiation

https://www.youtube.com/watch?v=0_YdVo3BS5w

Session 34

Practice-8

Plot Planck’s law for Black Body radiation and compare it with Raleigh-Jeans Law at high temperature and low temperature. (Using Scilab / Python).

Wien_Vs_PlanckDistribution.ipynb - Colaboratory

Session 35

Deduction of Wien’s distribution law from Planck’s law.

https://www.youtube.com/watch?v=d-CmdePjdKQ

Session 36

Deduction of Rayleigh-Jeans Law from Planck’s law

https://www.youtube.com/watch?v=WL1GrzXhbwA

Session 37

Flip class-4

Deduction of  Stefan-Boltzmann law and Wien’s Displacement law from Planck’s law.

https://www.youtube.com/watch?v=sUp_WZKZID4&t=37s

Session 38

Practice-9 (2 hours)

Implications of Planck's Blackbody Radiation formula: Wien's Displacement Law & Stefan-Boltzmann law: Numerical Simulation;  (Using Scilab / Python).

PlanckDistributionLaw_Implications.ipynb - Colaboratory

Session 39

Bose-Einstein Distribution Law - I (Derivation)

https://www.youtube.com/watch?v=iLU6NvVgZc4

Session 40

Bose-Einstein Distribution Law - II

.

Session 41

Thermodynamic functions of a strongly degenerate Bose gas - I (Derivation)

https://www.youtube.com/watch?v=okZmnB3Hhb4

https://www.youtube.com/watch?v=Vv_ED4F1tAk

Session 42

Thermodynamic functions of a strongly degenerate Bose gas - II (Derivation to be continued)

https://www.youtube.com/watch?v=2Z6UJbwxBZI

https://www.youtube.com/watch?v=QcTj5cc4Ws0

Session 43

Flip class-5 (2 hours)

Bose-Einstein Condensation, Properties of Liquid helium

https://www.youtube.com/watch?v=GPspau1DeRA

ttps://www.youtube.com/watch?v=YSgk78ToKrs

Session 44

Practice-10 (2 hours)

Plot Bose-Einstein distribution and Maxwell-Boltzmann Distribution functions with energy at different temperatures. (Using Scilab / Python)

MB_BE_distributions.ipynb - Colaboratory.

Session 45

Group Quiz -3

Bose- Einstein Statistics, Its Implications and Applications

Session 46

Fermi-Dirac Distribution Law - I (Derivation)

https://www.youtube.com/watch?v=VVeNM4SoRcs

Session 47

Fermi-Dirac Distribution Law - II (Derivation to be continued)

https://www.youtube.com/watch?v=OMRsmNpgZQ0

Session 48

Thermodynamic functions of a completely and strongly Degenerate Fermi Gas - I (Derivation)

https://www.youtube.com/watch?v=juIaEk8AB2w

https://www.youtube.com/watch?v=AdhcrmcZcSA

Session 49

Thermodynamic functions of a completely and strongly Degenerate Fermi Gas - II (Derivation to be continued), Definition of Fermi Energy, Fermi Momentum etc.

Session 50

Flip class-6 (2 hours)

Electron gas in a metal,   Application of Fermi-Dirac distribution to semiconductors, difference between Maxwell Boltzman , Bose Einstein and Fermi Dirac statistics and their uses.

Session 51

Practice-11 (2 hours)

To  Plot  Fermi-Dirac Distribution and compare with Bose-Einstein distribution and Maxwell-Boltzmann Distribution functions with energy at different temperatures. (Using Scilab / Python)

Session 52

Group Quiz-4

Fermi-Dirac Statistics, Its Implications and Applications

Our Main Teachers

  Dr. Nibedita Nayak is working as an Assistant Professor in the Department of Physics, School of Applied Sciences, Bhubaneswar Campus of Centurion University of Technology and Management, Odisha. She completed her Ph.D from Berhampur University in 2018. She is working on computational tools and new material development. She has more than six years of […]

Dr. Subrata Sarangi has a Masters’ Degree in Physics from IIT, Kanpur and a Ph.D in Nuclear Structure Theory from Physical Research Laboratory, Ahmedabad. He has 25 years’ experience in teaching at UG, PG and PhD levels. He has published over 20 peer reviewed research articles in areas of Atomic Nuclei, Nuclear Matter, Materials Sciences […]