Physics 611   Statistical Mechanics    (Spring 2018) 


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

COURSE INFO

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: sergei@physics.rutgers.edu (the best way)

Office hours: Friday 10:00 am -12:00 pm



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 (February 28) and final (exam period May 3-9) 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.
http://www.physics.rutgers.edu/ugrad/disabilities.html

Download this info in PDF format

 

PREREQUISITES  


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.
  • Download prerequisites

     
    PLAN OF LECTURES in


    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


    •  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.
    • 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.
    •   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.

      Download lecture notes

     

     
     

                         ELEMENTS OF THE KINETIC THEORY OF GASES


    •   Elements of Classical Mechanics: Virial theorem. Microscopic state. Phase space. Liouville's theorem. Poisson bracket.
    •   Statistical description of a system at equilibrium:  Mixed state. The equilibrium probability density function. Basic assumptions of statistical mechanics.
    •   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.
    •   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.
    •   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.

     

     
     

                         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.
    •   Normal systems in Statistical Thermodynamics (I):  Asymptotic forms of the number of states and state density of a macroscopic system. Entropy of normal systems. Statistical temperature. Helmholtz free energy. Legendre transformation. Mixing entropy and Gibbs paradox. The Tonks gas. Partition of energy between two systems in thermal contact.The zeroth law of thermodynamics.

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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.
    •   Normal systems in Statistical Thermodynamics (III):   Additivity of entropy for systems in equilibrium. Increase of entropy by the establishment of a new equilibrium. The second law of thermodynamics: Clausius and Kelvin formulations. Clausius theorem. Maximum work done by a body in an external medium. Stability conditions.

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    •   Microcanonical ensemble in quantum Statistical Mechanics:  Equipartition theorem. The splendors and miseries of classical Statistical Mechanics. The third law of thermodynamics. Quantum micro and macro states. The density matrix and its properties. The two-level system. Quantum microcanonical distribution. Entropy in the quantum microcanonical ensemble. Thermal density matrix and von Neumann entropy. Thermodynamics of the two level system.

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                         GIBBS   DISTRIBUTION


    •   Canonical ensembles:  Physical interpretation of the thermal density matrix. Energy PDF in the canonical ensemble. Gibbs canonical ensemble. Fluctuations of the fundamental thermodynamic quantities. Grand canonical ensemble. Landau free energy. The particle number fluctuations.
    •   Canonical examples:   Dilute polyatomic gases. Vibrations of a solid: phonons, Einstein and Debye models. Black-body radiation: Stefan-Boltzmann and Wien's displacement laws.

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                         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.
    •   Ideal Bose gas:   The degenerate Bose gas. Bose-Einstein condensation. Superfluids.

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    Download syllabus

     

    Homeworks and Solutions  

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




    Assigned on
    Assignment
    Due Date
    Solutions
     
    1. Jan. 17, 2018 pdf   Jan. 31, 2018 pdf  
    2. Jan. 31, 2018 pdf   Feb. 7, 2018 pdf  
    3. Feb 5, 2018 pdf   Feb. 19, 2018 pdf  
    4. Feb. 12, 2018 pdf   Feb. 26, 2018 pdf  

    Midterm exam:


    February 28 (Wednesday), 2018
               Download program and ground rules



    pdf  
    5. Feb. 26, 2018 pdf   Mar 26, 2018 pdf  
    6. Mar 4, 2018 pdf   Apr 4, 2018 pdf  
    7. Mar 19, 2018 pdf   Apr 16, 2018 pdf  
    8. Apr 4, 2018 pdf   Apr 25, 2018 pdf  
    9. Apr 23, 2018 pdf   pdf  

    Final exam:


    May 7 (Monday), 2018, 10:00 am-1:00 pm;
    Room SEC 204


               Download program and ground rules


    Final grades  
    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