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, 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. 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, the expectation values.The joint Gaussian distribution. Wick's theorem. Central limit theorem.
     
     

                         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 characteristic. Vlasov equation.
    •   Boltzmann equation:   Length and time scales in the BBGKY hierarchy. Binary collisions. Differential cross section. Mean free path. Dilute gas approximation. Bogolyubov's condition (Boltzmann's hypothesis of molecular chaos). Bogolyubov'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.
    •   Chapman-Enskog method:   Conservation laws. Zero and first order hydrodynamics.
     
     

                         FUNDAMENTAL   PRINCIPLES   OF  STAT MECH


    •   Microcanonical ensemble in classical Statistical Mechanics:  Role of the integral of motions. Microcanonical distribution. The classical density of states, the number of states, the statistical weight of a macroscopic state. Example: the monoatomic ideal gas. Ergodic hypothesis. Partial equilibrium and macroscopic states.
    •   Normal systems in Statistical Thermodynamics:  Asymptotic forms of the number of states and state density of a macroscopic system. Entropy of normal systems. Partition of energy between two systems in thermal contact. Statistical temperature. The zeroth law of thermodynamics. Quasi-static adiabatic process 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. Chemical potential. Mixing entropy and Gibbs paradox. Gibbs-Duhem relation. Additivity of entropy for systems in equilibrium. Increase of entropy by the establishment a of new equilibrium. The second law of thermodynamics: Clausius and Kelvin formulations. Clausius theorem. Stability conditions.
    •   Microcanonical ensemble in quantum Statistical Mechanics:   The splendors and miseries of classical Statistical Mechanics. The third law of thermodynamics. Quantum micro and macro states. The density matrix and its properties. Quantum microcanonical distribution. Entropy in the quantum microcanonical ensemble. Two-level systems.
     
     

                         GIBBS   DISTRIBUTIONS


    •   Canonical ensemble:  Thermal interaction with a reservoir. Canonical distribution. Density matrix for the canonical ensemble. Thermodynamics of the canonical and Gibbs canonical ensembles. Energy fluctuations in the canonical ensemble. Examples: two-level systems,localized harmonic oscillators and rotators, dilute polyatomic gases, phonons, black body radiation.
    •   Grand Canonical Ensemble (GCE):  Density fluctuations in the GCE. Examples.
     
     

                         IDEAL  QUANTUM  GASES


    •   Ideal  quantum gases:  Hilbert space of identical particles. Microcanonical and grand canonical ensembles.
    •   Ideal Fermi gas:  Equation of state of an ideal Fermi gas. Examples: white dwarf stars, Landau diamagnetism, De Haas-Van Alphen effect, quantized Hall Effect, Pauli paramagnetism.
    •   Ideal Bose gas:   The degenerate Bose gas. Bose-Einstein condensation. Superfluids.

    Download syllabus

     

    Homeworks and Solutions  

    The assignments and solutions are stored in PDF format. The absolute cutoff time for homework is 4pm due date.


    Assigned on
    Assignment
    Due Date
    Solution
     
    1. Jan. 17, 2018 pdf   Jan. 31, 2018 pdf  
    2. Jan. 31, 2018 pdf   Feb. 14(?), 2018 pdf  
    3. Feb 14(?), 2018 pdf   Feb. 28(?), 2018 pdf  
    4. Feb. 28(?), 2018 pdf   Mar. 19(?), 2018 pdf  
    5. Mar. 19(?), 2018 pdf   Apr 4(?), 2018 pdf  
    6. Apr 4(?), 2018 pdf   Apr 23(?), 2018 pdf  
    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.
    2. Buffon's needle. An Analysis and Simulation.

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