Physics 611   Statistical Mechanics    (Spring 2020) 

Course info   | Prerequisites  | Plan of lectures  | Homeworks and Solutions   | Useful Links   | E-mail


Room: ARC-205
           Monday:        3:20-4:40 pm
           Wednesday:   3:20-4:40 pm

Instructor: Sergei Lukyanov
         office: Serin E364
         office phone: (848) 445-9060 
         e-mail: (preferred)

Office hours: Friday 10:00 am -12:00 pm
                        There will be no office hours on Friday, February 14

The main reference text will be: Mehran Kardars,"Statistical Physics of Particles".
Online reference material can be found at
Mehran Kardar's MIT Lectures on Statistical Mechanics.

Additional text: Evergreen L.D. Landau and E.M. Lifshitz "Statistical Physics, Part 1".
You can easily find the PDF file online

Homework: There will be homework assignments. Late homework will not be accepted. Homeworks will be graded and give 30% contribution to your final grade.

Exams: There will be midterm (March 4) and final (exam period May 7-13) exams.

Final grade: Score % = 30% Homework + 20% Midterm + 50% Final

Students with Disabilities:

If you have a disability, it is essential that you speak to the course supervisor early in the semester to make the necessary arrangements to support a successful learning experience. Also, you must arrange for the course supervisor to receive a Letter of Accommodation from the Office of Disability Services.

Download this info in PDF format



I will assume that you are familiar with

(I)  Undergraduate Thermodynamics at the level of 351 or Rutgers placement test program, which includes

  •   Basic: Laws of thermodynamics-definitions, temperature scales, heat transfer by conduction, properties of ideal gas, relation between temperature and kinetic energy, Maxwell distribution, work and PV diagrams, Carnot cycle.  

  •   Intermediate: Thermodynamic variables, macro and micro states, heat engines and refrigerators, thermodynamic potentials, kinetic theory, phase transititions, transport phenomena, Van der Waals gas.

  •   Advanced: Boltzmann distribution, phase transformations in binary mixtures, statistics of ideal quantum systems, black body radiation, Bose-Einstein condensation.

  • (II)  Graduate Classical Mechanics at the level 507 or Rutgers challenge exam program:

  •   Basic: Lagrangian mechanics, invariance under point transformations, generalized coordinates and momenta, curved configuration space, phase space, dynamical systems, orbits in phase space, phase space flows, fixed points, stable and unstable, canonical transformations, Poisson brackets, differential forms, Liouville's theorem, the natural symplectic 2-form and generating functions, Hamilton-Jacobi theory, integrable systems, adiabatic invariants.

    (III) Graduate Quantum Mechanics at the level 501 or Rutgers challenge exam program :

  •   Basic: Vector spaces, eigenvalues and eigenvectors, position and momentum operators, Schroedinger equation, one dimensional potentials, harmonic oscillator, symmetries in quantum mechanics, identical particles, translations and rotations in three dimensions, hydrogen atom, energy levels, degeneracy, spin, Pauli matrices.
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    This is a tentative schedule of what we will cover in the course. It is subject to change, often without notice. These will occur in response to the speed with which we cover material, individual class interests, and possible changes in the topics covered. Use this plan to read ahead from the text books and my lecture notes, so you are better equipped to ask questions in class. I would also highly recommend you to watch Prof. Kardar's lectures online.




    •  Probability: Definitions. Examples: Buffon's needle, lucky tickets, random walk in one dimension. Saddle point method. Diffusion equation. Fick's law. Entropy production in the process of diffusion.

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    • One random variable:  General definitions: the cumulative probability function, the Probability Density Function (PDF), the mean value, the moments, the characteristic function, cumulant generating function. Examples of probability distributions: normal (Gaussian), binomial, Poisson.

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    • Many random variables:   General definitions: the joint PDF, the conditional and unconditional PDF, expectation values. The joint Gaussian distribution. Wick's theorem. Central limit theorem.

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    •   Elements of Classical Mechanics: Virial theorem. Microscopic state. Phase space. Liouville's theorem. Poisson bracket.

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    •   Statistical description of a system at equilibrium:  Mixed state. The equilibrium probability density function. Basic assumptions of statistical mechanics.

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    •   Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy:  Derivation of the BBGKY equations. Collisionless Boltzmann equation. Solution of the collisionless Boltzmann equation by the method of characteristics. Vlasov equations.

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    •   Boltzmann equation:   Length and time scales in the BBGKY hierarchy. Binary collisions. Differential cross section. Mean free path. Dilute gas approximation. Bogoliubov's condition (Boltzmann's hypothesis of molecular chaos). Bogoliubov's form of the collision integral. Boltzmann's collision integral. Heuristic "derivation" of the Boltzmann equation.

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    •   General consequences of the Boltzmann equation:   H - theorem and irreversibility. Equilibrium properties: the equilibrium distribution, the ideal gas entropy. Collision-time approximation for the Boltzmann equation.

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                         FUNDAMENTAL   PRINCIPLES   OF  STAT   MECH

    •   Microcanonical ensemble in classical Statistical Mechanics:  Role of the integrals of motion. Microcanonical distribution. The classical density of states, the number of states, the statistical weight of a macroscopic state. Ergodic hypothesis. Partial equilibrium and macroscopic states. Classical density of states for the monoatomic ideal gas and polar rods. Statistical integral. Laplace transform. Density of states for the polar rods.

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    •   Normal systems in Statistical Thermodynamics (II):   Quasi-static adiabatic processes in statistical mechanics: adiabatic theorem in classical mechanics, adiabatic invariants, particle in an infinite well. Adiabatic theorem in statistical mechanics. The first law of thermodynamics. Pressure. One dimensional gas with nearest neighbor interactions. Gibbs free energy. Enthalpy. Chemical potential. Gibbs-Duhem relation. Clausius-Clapeyron relation.

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      Video-recorded lecture 03.30.2020

      Video-recorded lecture 04.01.2020


                         GIBBS   DISTRIBUTION


                         IDEAL  QUANTUM  GASES

    •   Ideal  quantum gases:  Hilbert space of identical particles. Canonical and grand canonical ensembles.
    •   Ideal Fermi gas:  Equation of state of an ideal Fermi gas. Examples: Pauli paramagnetism, white dwarf stars.

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    Homeworks and Solutions  

    The assignments and solutions are stored in PDF format. The absolute cutoff time for homework is 7pm due date.

    Assigned on
    Due Date
    1. Jan. 22, 2020 pdf   Feb. 3, 2020 pdf  
    2. Jan. 29,2020 pdf   Feb. 10, 2020 pdf  
    3. Feb. 3, 2020 pdf   Feb. 17, 2020 pdf  
    4. Feb. 5, 2020 pdf   Feb. 24, 2020 pdf  
    5. Feb. 19, 2019 pdf   Mar. 2, 2020 pdf  
    6. Feb. 19, 2020 pdf   Mar. 9, 2019 pdf  

    Midterm exam:

    March 4, 2020
               Download program and ground rules

    Midterm solutions  
    7. Mar 6, 2020 pdf   Mar 30, 2020 pdf  
    8. Mar 19, 2020 pdf   Apr 6, 2020 pdf  
    9. Mar 19, 2020 pdf   Apr 13, 2020 pdf  
    10. Apr 6, 2020 pdf   Apr 20, 2020 pdf  
    11. Apr 6, 2020 pdf   Apr 27, 2020 pdf  
    12. Apr 20, 2020 pdf   May 4, 2020 pdf  

    Final exam:

    May 11 (Mon), 2020, 12:00pm:-5:00 pm;
    Room: Home

               Download program and ground rules

    Useful Links  
    1. "Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables",
    M. Abramowitz and I. Stegun
    2. Mehran Kardar's MIT Lectures on Statistical Mechanics.
    3. Buffon's needle. An Analysis and Simulation.
    4. Gamma Function.
    5. Beta Function.
    6. Jacobi Theta functions.
    7. Modified Bessel Function of the First Kind.
    8. Modified Bessel Function of the Second Kind.
    9. (Complementary) Error function.
    10. W. Sutherland,``The viscosity of gases and molecular force'',The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, vol. 36, p.507 (1893)
    11. Lewi Tonks,``The complete equation of state of one, two and three-dimensional gases of hard elastic spheres'', Phys. Rev. 50, p. 955 (1936)

    Course info  |  Prerequisites  |   Plan of lectures  |   Homeworks and Solutions  |   Useful Links  |  E-mail