PhD courses offered in English (SP-BP-PQS)

Statistical Physics, Biological Physics and Physics of Quantum Systems

Statistical physics of polymers and membranes

  • Polymers:ideal chains (lattice chain, Gaussian chain, freely jointed chain, bead-rod chain, worm like chain); Flory model of excluded volume, interaction with solvent; Langevin dynamics, Rouse model, Zimm model; scaling properties (semi-dilute solutions, confinement, weak adsorption); electrophoresis.
  • Membranes:physical properties, elastic models; Monge representation, thermal fluctuations, surface tension; membrane tubes; membrane adhesion; interaction with the cytoskeleton.


Pattern formation in complex systems

  • Introduction, definitions: spatial and/or temporal patterns in systems far from equilibrium; homogeneous, ordered (periodic) and chaotic states.
  • Theoretical description: methods, dissipative dynamics, stability and bifurcations, linear stability analysis and nonlinear basic states. model equations.
  • Nonlinear behavior in classical mechanics.
  • Nonlinear behavior in chemistry (Bjelousov-Zhabotinsky and Turing instabilities)
  • Shear induced (flow) instabilities: Taylor-Couette, Rayleigh, Rayleigh-Taylor, Kelvin-Helmholtz, Benard-Marangoni instabilities.
  • Thermally driven convection: Rayleigh-Benard instability. Role of anisotropy; thermal convection in liquid crystals.
  • Electroconvection.
  • Interfacial patterns: viscous fingering, linear stability analysis.
  • Nonequilibrium solidification: dynamics of a solid-liquid interface.
  • Nonlinear optics.
  • Computer simulation methods: DLA, phase-field model.
  • Experimental techniques: shadow graph, image processing.


Disordered systems

This course is an introduction to the statistical mechanics of disordered systems, focused on spin glasses and related subjects.

  • The required background: statistical mechanics, and, possibly, phase transitions.
  • Topics covered:
    • Basic phenomena: history dependence of susceptibilities, nonlinear susceptibility, ageing.
    • Edwards-Anderson model and Sherrington-Kirkpatrick model, complex combinatorial optimization problems.
    • Annealed and quenched averages, frustration. Mattis model.
    • Energy barriers, the problem of the lower critical dimension, characteristic times.
    • The order parameter. The replica trick. Replica symmetric solution of the Sherrington-Kirkpatrick model. The entropy paradox. Almeida-Thouless instability. Replica symmetry breaking.
    • The Parisi solution. Physical interpretation. Ultrametricity, lack of self-averaging. The stability of the Parisi solution.
    • A hint at the rigorous solution of the Sherrington-Kirkpatrick model.
    • Other spin-glass-like systems: complex combinatorial optimization, Hopfield model and elements of the theory of neural networks. Algorithmic phase transitions.
  • Literature: M. Mezard, G. Parisi. M.A. Virasoro: Spin glasses and beyond (World Scientific, Singapore, 1987) ISBN: 9971-50-115-5


Disordered Networks I-II (Fall and Spring semesters)

Special seminar for IV-V year and PhD students

  • These lectures treat the by now classic subjects of long range interaction spin glasses and neural networks from the viewpoint of equilibrium statistical mechanics. It is meant for students with a strong theoretical interest. I. As an exercise we describe the thermodynamical properties in the solvable case of p-spin interaction spin systems. Then we turn to random systems, discuss Derrida's Random Energy Model, the "simplest spin glass", give a picture of pure and mixed thermodynamical states, and finally treat the random exchange p-spin system by the replica method. We analyze why the replica symmetric solution is untenable for low T, show how to break replica symmetry and present Parisi's solution. Some feeling is given of experimental properties of spin glasses. II. After a sketchy introduction to biological neural systems, we present basics of artificial neural networks like the Hopfield model, the simple perceptron and multilayer feedforward networks. The storage problem of the perceptron is discussed by combinatorics, and adaptive methods like the perceptron and backpropagation algorithms are presented. The last part of the semester deals with the statistical mechanical approach of Gardner, where the replica method borrowed from spin glass theory is used to describe storage and prediction abilities of the perceptron.


Classical radiation theory (Fall semester)

Special seminar for III year students

  • The first half of the semester is an introduction into classical field theory, then electrodynamics is treated from the field theoretical viewpoint, and in the end some salient problems of radiation theory are discussed. The title derives from a course running for decades, but since experience showed that students were mostly ignorant about elements of classical field theory, emphasis was shifted towards the latter. As an introduction the connection of the Maxwell equations with XIXth century experimental results is discussed. Then the general framework of classical field theory, variational approach, equations of motion, Lagrange and Hamiltonian formalism, the energy-momentum tensor together with conservation laws follows. A short reminder to the basics of the special theory of relativity is included. Classical field theory is exemplified on ideal fluid dynamics, both non-relativistic and relativistic, in Euclidean and curved coordinate systems, ending up with the general relativistic hydrodynamical equations of motion. Then the electromagnetic field interacting with a continuum of charges is treated by a variational approach. Subsequently discussed are the energy, momentum, and angular momentum of a wave field, the radiative Green function, the Lienard-Wiechert potentials, radiated power both for small velocities and relativistically, elementary estimates about radiative loss in accelerators, natural spectral line broadening, the Abraham-Lorentz electron, Dirac's relativistic classical equation of motion for a charged particle, the simplest picture of Cherenkov radiation, interaction of charged particles and the calculation of the first relativistic correction to the Coulomb potential.


Elements of classical field theory (Fall semester)

Student seminar for III year students

  • This goes in parallel to and is meant for the participants of the previous course. Here lectures are given by students on introduction to Riemann geometry to be able to handle curvilinear coordinates, on problems from continuum mechanics like elasticity by variational principles, on the variational approach to Einstein's equations for general relativity and Hilbert's energy-momentum tensor, surface effects in field theory, and other problems possibly by the speakers' choice.


Extreme value statistics and its physical applications (Spring semester)

Special seminar for III-V year and PhD students

  • The problem of extremal value fluctuations is introduced through empirical statistical problems like water level or appliance failure records. Then the classical basic extremal "universal" limit classes, Gumbel, Frechet, and Weibull are introduced. They are thoroughly, albeit non-rigorously, analyzed with regard to finite size effects, namely, the dependence of the empirical average, the variance, and the correction to the universal asymptote on the number of variables N. Treated are also the distribution of the k-th maximum, and the joint statistics of the first k maxima. The emergence of extreme value statistics in surface fluctuations, like the width (roughness) statistics and extremal excursions of Brownian and more general random walks are discussed, in the light of most recent results.


Student seminars in statistical physics (Spring semester)

For IV-V year physics students

  • Fundamental issues and methods in statistical physics, like the treatment of stochastic processes by path integrals, random walks, Langevin and Fokker-Planck equations, master equations, birth-and-death processes, and other problems possibly by the students' choice, are worked out and presented by the students. The aim is also to have students to gain experience with creating Power Point or Latex slide presentations.


Physics of environmental fluid flows

In this introductory course we summarize the basic physics of the largest scale flow phenomena in the atmosphere and oceans.

  • Necessary background: classical hydrodynamics.
  • Subjects: Rotating homogeneous fluids, Coriolis force, Rossby number, Navier-Stokes equation in rotating reference frame, dimensionless representation, Froude number, dynamic pressure, geostrophic equilibrium, Taylor-Proudman theorem linearized equations, inertial oscillation, inertial waves, dispersion, phase and group velocities, shallow water equations, potential vorticity conservation, free surface Rossby waves, Kelvin waves at closed boundaries, surface curvature: f-plane and beta-plane approximations, planetary Rossby waves, finite viscosity: Ekman number, Ekman spiral, Ekman transport, stratification, Brunt-Vaisala frequency, adiabatic atmosphere, potential temperature, potential density, Boussinesq approximation, internal waves at continuous stratification, equations for two layer fluids, geostrophic solutions, Margules formula, thermal wind, baroclinic instability.
  • Literature: 1) B. Cushman-Roisin: Introduction to Geophysical Fluid Dynamics (Prentice-Hall, London, 1994) ISBN: 0-13-353301-8 2) J. Pedlosky: Geophysical Fluid Dynamics (Springer, New-York, 1987) ISBN: 0-387-96387-1


Application of chaos theory: Nonlinear time series analysis

We provide a concise summary of nonlinear methods developed from theories of dynamical systems. Emphasize on computer exercises.

  • Necessary background: introductory courses on chaos and nonlinear dynamics.
  • Subject: - Linear tools, testing for stationarity, linear filters, linear predictions, phase space methods, delay reconstruction, false neighborhood, Poincare surface of section, recurrence plots, simple nonlinear prediction, nonlinear noise reduction, measuring Lyapunov exponent, attractor geometry and fractals, correlation dimension, interpretation and pitfalls, temporal correlations, scaling laws, detrended fluctuation analysis, testing nonlinearity with surrogate data, nonlinear statistics, transients, intermittency, quasi-periodicity.
  • Literature: H. Kantz, T. Schreiber: Nonlinear Time Series Analysis (Cambridge University Press, Cambridge, 2004) ISBN 0-521-52902-6


Theory of quantum phenomena

  • The density matrix and its representations
  • Complementarity and environment-induced decoherence
  • The master equation and its solutions
  • The measurement problem and the quantum-classical border
  • Quantum Zeno effect
  • Quantum jumps
  • Quantum non-demolition
  • Berry phases
  • A quick introduction to quantum information


Recent experiments in quantum mechanics

  • Neutron interferometry
  • Ion and atom traps, laser cooling
  • Bose-Einstein condensation
  • Micromasers
  • Atom optics
  • Schrödinger cats
  • Photon detectors, two-photon interference
  • Entanglement, EPR correlations, Bell inequalities
  • Vibrating mirrors


Experimental methods in Biophysics

  • X-ray crystallography
    • structure factors
    • convolution
    • Patterson function
    • Phase refinement
  • NMR spectroscopy
    • Semiclassical theory
    • Quantummechanical foundations
    • Bloch equations
    • chemical shift
    • spin-spin coupling
    • 2D (pulse) NMR
    • Solomon equations and Overhauser relaxation
  • Mass spectroscopy
    • analysators: Magnetic sector, quadrupole, ion trap and time of flight (TOF)
    • ion sources: ESI and MALDI
  • Optical spectroscopy
    • absorption
    • fluorescence
    • dichroism
    • IR and Raman
  • Chromatography
    • gel filtration
    • ion exchange
    • affinity
    • HPLC
  • ?
  • Electrophoresis
    • Debye-Huckel theory
    • electroosmosis
    • zeta potential and pH
    • SDS-PAGE, blotting, 2D electrophoresis and differential protein display
  • Sedimentation and Centrifugation
    • diffusion
    • Svedberg equation
    • zonal and density-gradient ultracentrifugation
  • Optical waveguides
    • electromagnetic theory
    • applications
  • Tomography
    • Radon transformation
    • CT
    • PET
    • MRI


Fullerenes and Carbon Nanotubes

  • Discovery of C60, historical survay, isolated cagelike molecules
  • Properties of C60 in gas- liquid- and solid phases
  • Doped fullerenes, fullerites; fullerene polymers
  • ?
  • Carbon nanotubes: geometrical, vibrational and electronic properties
  • Applications


Theoretical Studies on Biopolymers

    • structure calculations of polysaccharides
    • studies on mechanical properties of DNA
    • molecular mechanics and molecular dynamics studies on proteins
    • statistical approach to study protein structure elements
    • studies on transmembrane proteins
    • studies on intrinsically unstructured proteins
  • Topics:


Physics of Semiconductors

In this course fundamentals of semiconductor materials and device physics are introduced - differences between general solid state physics and specific semiconductor properties are emphasized. Most recent applications are treated in some more details.

  • Background: Elements of solid state physics and basic knowledge of quantum mechanics.
    • Crystal physics - structure, structure of thin layers, interfaces, epitaxy, organic and amorphous semiconductors.
    • Electron states - band structure, role of bonding, k.p perturbation, hole states.
    • Motion of electrons - effective Hamiltonian, effective mass tensor, valence band, impurity states, shallow and deep impurities, scattering of electrons and holes.
    • Statistics of semiconductors - intrinsic, extrinsic semiconductors, degeneracy, compensation, semi-insulators.
    • Basic transport - phenomenology, Onsager relation, microscopic kinetic model, Boltzmann equation, conductivity, Hall effect, magnetoresistivity.
    • Advanced electron transport - high field, instability, Gunn effect.
    • a.c. transport - microwave, propagation of light below band-gap, cyclotron resonance.
    • Thermal transport - thermal conductivity of electrons, bipolar (heat) conductivity, Peltier effect, thermoelectric heating and cooling.
    • Quantum transport - reduced dimensionality, space-charge layers, quantum hall effect (integer, fractional).
    • Inhomogeneous semiconductors - diffusion, lifetime, injection, p-n junction, MOS structure.
    • Fundamental experimental techniques for characterization of semiconductor materials and device structures.
    • Basic devices - p-n junction diode, transistor, tunnel diode, MOS device, Zener diode.
    • Light and semiconductors: optoelectronics - LED, laser diode, light detectors, MOS imaging, CCD physics.
    • Light and semiconductors: solar cells - basics, theoretical limits, practical limits, future.
    • Highlights of Hungarian pure and applied semiconductor research.
  • Subjects:
Specific literature is given at the end of each Section.


Káosz kialakulása mechanikai rendszerekben

This is an introductory course to the emergence and properties of chaotic motion in simple dissipative and conservative mechanical systems.

    • Basic concepts and simple motion in dynamical systems: phase space, trajectory, fixed point, limit cycle, Poincaré map, Poincaré-Bendixson theorem
    • Chaotic behavior in general: main features of chaos, Lyapunov exponents, examples: periodically kicked rotator, Lorenz model
    • Dissipative systems:
      • one-dimensional maps: logistic map, period doubling bifurcations, chaotic bands, universality, Feigenbaum constants, fully developed chaos, tent map, Bernoulli shift, Lyapunov exponents, probability distributions, symbolic dynamics
      • two dimensional maps: Hénon model, baker map, strange attractor, fractal dimension period doubling bifurcations and classification of strange attractors in systems of differential equations, intermittency, quasiperiodic transition to chaos, transient chaos, review of experiments
    • Conservative systems: structure of the phase space in integrable and nonintegrable systems with examples, Kolmogorov-Arnold-Moser theorem, Poincaré-Birkhoff theorem, standard map, breakup of the last torus, ergodicity, mixing, Kolmogorov-Sinai entropy, Arnold diffusion, billiards (e.g. Sinai, Bunimovich), chaotic scattering
  • Subject:
Literature: A káosz (szerk. Szépfalusy P., Tél T., Akadémiai Kiadó, 1982) H.-G. Schuster, Deterministic Chaos: An Introduction (VCH, Weinheim, 1995) Tél Tamás-Gruiz Márton, Kaotikus dinamika (Nemzeti Tankönyvkiadó Rt., 2002)


Mesosopic Systems I.

In this introductory course we summarize the basic physics of mesoscopic systems.

    • Two-dimensional electron gas - nanoscale wires and quantum dots
    • Electronic transport - Landauer approach
    • Scattering matrix and transfer matrix method
    • Green's function method (Fisher-Lee relation)
    • Resonant tunneling
    • Aharonov-Bohm effect
    • Weak localizations
    • Universal conductance fluctuations
  • Subjects:
Literature: S. Datta: Electronic Transport in Mesoscopic Systems, (Cambridge University Press, Cambridge, 1995) C. W. J. Beenakker and H. van Houten in Quantum Transport in Semiconductor Nanostructures, Solid State Physics, Vol. 44, pp. 1-228, edited by H. Ehrenreich and D. Turnbull, (Academic Press, Inc., Boston, 1991) Y. Imry: Introduction to Mesoscopic Physics, (Oxford University Press, Oxford, England, 1997) D. K. Ferry and S. M. Goodnick: Transport in Nanostructures, (Cambridge University Press, Cambridge, 1997) T. Heinzel: Mesoscopic Electronics in Solid State Nanostructures,(Wiley-VCH GmbH & Co. KGaA, Weinheim, 2003) H-J. Stöckmann: Quantum Chaos, An Introduction,(Cambridge University Press, Cambridge, 2000) ?


Mesosopic Systems II.

In this course we give a review about the most relevant experiments and theories related to mesoscopic systems. The course is based on the course of Mesosopic Systems I.

    • Integer quantum Hall effect
    • Fractional quantum Hall effect
    • Quantum point contact
    • Coulomb blockade
    • Persistent current
    • Quantum dots, artificial atoms
    • Antidot lattice, classical and quantum chaos
    • Lateral magnetic super lattice (magnetically modulated mesoscopic systems)
    • Photonic crystals
    • Shot noise
    • Spintronics
  • Subjects:
    • Supriyo Datta: Electronic Transport in Mesoscopic Systems (Cambridge Studies in Semiconductor Physics and Microelectronic Engineering) (Paperback)
    • C. W. J. Beenakker and H. van Houten in Quantum Transport in Semiconductor Nanostructures, Solid State Physics, Vol. 44, pp. 1-228, edited by H. Ehrenreich and D. Turnbull, (Academic Press, Inc., Boston, 1991)
    • Y. Imry: Introduction to Mesoscopic Physics, (Oxford University Press, Oxford, England, 1997)
    • D. K. Ferry and S. M. Goodnick: Transport in Nanostructures, (Cambridge University Press, Cambridge, 1997)
    • T. Heinzel: Mesoscopic Electronics in Solid State Nanostructures,(Wiley-VCH GmbH & Co. KGaA, Weinheim, 2003)
    • H-J. Stöckmann: Quantum Chaos, An Introduction,(Cambridge University Press, Cambridge, 2000)
    • C. W. J. Beenakker and C. Schönenberg: Quantum Shot Noise, Pysics Today May 2003, and References therein
    • D. Weiss, G. Lütjering, and K. Richter: Chaotic Electron motion in Macroscopic and Mesoscopic Antidot Lattices, Chaos, Soliton, & Fractals, Vol. 8, pp. 1337-1357
    • R. B. Laughlin: Noble Lecture: Fractional quantization, Review of Modern Physics, Vol. 71, 863 (1999) and References therein; H. L. Stromer: Noble Lecture: The fractional quantum Hall effect, Review of Modern Physics, Vol. 71, 875 (1999) and References therein
    • Spintronics and Quantum Computation, edited by D. D. Awschalom, D. Loss, and N. Samarth (Springer, Berlin, 2002) and References therein ; S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnár, M. L. Roukes, A. Y. Chtchelkanova and D. M. Treger, Science Vol. 294, 1488 (2001) and References therein
  • Literature:


Mesosopic Superconductivity

In this course we give a review of the present status of the mesoscopic superconductivity.

    • Bogoliubov - de Gennes equation
    • Andreev reflection, proximity effect
    • Currents in superconductors
    • Scattering at normal-superconducting interface
    • Conductance of normal-superconducting hybrid systems
    • Superconducting-normal-superconducting systems: mesoscopic Josephson junctions
    • Gauge transformation of the Bogoliubov - de Gennes equation
    • Excitation spectrum for Andreev billiards
  • Subjects:
    • P. G. de Gennes: Superconductivity of Metals and Alloys (Benjamin, New York, 1996)
    • F. Andreev, Zh. Eksp. Teor. Fiz. 46, 1823 (1964), [Sov. Phys. JETP, 19, 1228 (1964)]
    • A. Abrikosov: Fundamental of the Theory of Metals, (Elsevier Science Publishers, Amsterdam, The Netherlands, 1988)
    • M. Tinkham: Introduction to Superconductivity, (McGraw-Hill, Inc., New York, 1996)
    • J. B. Ketterson and S. N. Song: Superconductivity, (Cambridge University Press, Cambridge, 1999)
    • C. J. Lambert and R. Raimondi, J. Phys. Condens. Matter 10, 901 (1998)
    • C. W. J. Beenakker, in Mesoscopic Physics, Les Houches Summer School, edited by E. Akkermans, G. Montambaux, J. L. Pichard, and J. Zinn-Justin (Elsevier Science B. V., Amsterdam, 1995); C. W. J. Beenakker, Review of Modern Physics 69, 731 (1997)
    • M. Brack and R. K. Bhaduri: Semiclassical Physics, (Addison-Wesley Pub. Co., Inc., Amsterdam, 1997)
  • Literature:


Trapped atomic gases II.

In this special course we continue the topics started in Trapped Atomic Gases I. The main interest is put on the current experiments performed in ultracold trapped gases. During the course theoretical issues of a few selected experiments are reviewed.

Necessary backgrounds: Quantum Mechanics and Statistical Physics. Background on many body physics is beneficial.

    • Field theoretical descriptions of weakly interacting particles in external trapping potential.
    • A simple finite temperature approximation for the collective excitations: The Popov approximation for bosons.
    • Experiments with rotating condensates with vortices.
    • Thomas-Fermi approximation for a single vortex. Excitations of a stable single vortex ground state using the hydrodynamical approach. Sum-rule approach.
    • Experiments with trapped fermions, observations of different phenomena in the BEC-BCS transition.
    • The mean-field description of the BEC-BCS transition: the Leggett model.
  • Subjects:
    • Bose-Einstein Condensation in Atomic Gases, Editors: M. Inguscio, S. Stringari and C. E. Wieman, (IOS Press, Amsterdam, 1999) ISBN: 0-9673355-5-8
    • F. Dalfovo, S. Giorgini, L. Pitaevskii and S. Stringari: Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys. 71, 463-512 (1999).
    • Qijin Chen, Jelena Stajic, Shina Tan, Kathryn Levin: BCS-BEC Crossover: From High Temperature Superconductors to Ultracold Superfluids, Physics Reports 412, 1-88 (2005).
  • Literature:


Magnetism I-II

(two-terms special course. Usually, Magnetism I is in the autumn term, Magnetism II in the spring term)

    • Magnetic phenomena are considered as electron correlation effects. The Hubbard model is introduced and used to interpret the most basic correlation effect: the Mott metal-insulator transition. The first argument is qualitative, and later a more sophisticated variational theory is given which also allows the understanding of heavy fermion behavior.
    • The antiferromagnetic Heisenberg model is introduced as the effective hamiltonian of the large-U Hubbard model at half filling. Other kinetic exchange processes, including ring exchange with application to the magnetism of solid He3, are discussed. A detailed treatment of the two-site Coulomb processes allows the introduction of direct exchange.
    • The survey of various mean field theories of magnetic order begins with the Stoner theory. Weak itinerant ferromagnets like ZrZn2 and MnSi are discussed in some detail.
    • Students attending the course are expected to solve 4-6 homework problems before taking oral exam.
  • Magnetism I
      This is the direct continuation of Part I. The basic concepts and results from Magnetism I are assumed to be familiar.
    • The variety of magnetic ordering phenomena is surveyed, the conditions of ordering, and the nature of the excited states over ordered ground states, are discussed in various theoretical frameworks.
    • The discussion of weak itinerant ferromagnets is continued with an introduction of the quantum critical point. The same concept is used for rare earth systems with non-fermi-liquid behavior.
    • The last example of itinerant mean field states is the spin density wave state, illustrated on chromium.
    • Localized-spin order is considered next. Spin wave theory is described both for ferromagnets and antiferromagnets. A detailed discussion of quantum fluctuations in the ground state is given, including recent results on the possibility of spin liquid ground states.
    • A central theme of Part II is the generalization of the previous considerations to systems with spin and orbital degrees of freedom. The discussion of the Mott transition in the orbitally degenerate Hubbard model is followed by the derivation of the spin-orbital kinetic exchange interaction, and surveying the experimental evidence for orbital ordering in d- and f-electron systems.
    • A particular kind of magnetic cooperative behavior gives rise to the integer and the fractional Quantum Hall Effect.
    • Students attending the course are expected to solve 4-6 homework problems before taking oral exam.
    • Much of the material covered in Magnetism I-II is described in P.F.: "Lecture Notes on Electron Correlation and Magnetism" (World Scientific, Singapore, 1999). However, illustrative examples are usually taken from more recent literature, and the arrangement and emphasis is different from that found in the book.
  • Magnetism II


Experimental Methods in Solid State Physics I

    • Amplitude and intensity of the diffracted beam; reciprocal lattice, Bragg's law and the Ewald sphere; atomic scattering factor and the structure factor; systematic extinction, Debye-Waller temperature factor.
    • X-ray sources; detectors; absorption spectroscopy; radiography; single crystal diffraction; powder diffraction; phase identification; quantitative phase analysis; indexing and lattice parameter determination; X-ray topography.
    • The particularities of electron diffraction; deviation parameter; transmission electron microscopy; image formation in TEM and diffraction; Kikuchi lines; the basic equations of the two-beam kinematic and dynamic theory; rocking curve; images of dislocations and other crystal defects; high-resolution electron microscopy. Scanning electron microscopy: priciples of SEM and imaging modes; analytical electron microscopy; electron energy loss spectroscopy.
    • Neutron sources and detectors; absorption; nuclear scattering; magnetic scattering; basic properties of neutron diffraction; time-of-flight method; applications: structure analysis; Rietveld refinement; hydrogen atom location; residual stresses; neutron inelastic scattering and its application.
    • Positron sources; positron annihilation; positronium; positron in solid matter; measuring methods: angular correlation, time-life, Doppler broadening; meauring apparatuses; applications: fermiology; positron trapping at crystal defects; positron emission tomography.
    • Low temperature calorimetry: low temperature measurement of specific heat; elemental excitations; critical parameters; phase transformations; low temperature measuring methods; instrumentation. High temperature calorimetry: classical DTA; Boersma DTA; differential scanning calorimeter (DSC); phase transformations; kinematical investigations.
  • Elements of kinematic diffraction theoryX-ray methodsElectron methodsNeutron methods Positron annihilation spectroscopyCalorimetry


Experimental Methods in Solid State Physics II

    • Scanning tunneling microscopy (STM): theoretical background; one-dimensional elastic tunneling; imaging and spectroscopy; STM design and instrumentation; basics of the work; applications: metals and semiconductors, layered materials. Scanning force microscopy (SFM): design and instrumentation; imaging in contact mode; non-contact force microscopy; applications.
    • Physical background of the resonance-absorption; recoil energy loss; Doppler-broadening; recoil-free emission; Mössbauer-Lamb factor; experimental techniques; sources and detectors; Doppler velocity drive; measuring possibilities; hyperfine interactions: isomer shift; quadrupole splitting; magnetic hyperfine structure; relativistic effects; application in solid state physics.
    • The common characteristics of the high energy ion beam methods; Rutherford backscattering (RBS): kinematic factor and mass resolution; elastic scattering cross sections; energy loss in solids; applications: composition and stoichiometry of the sample; thickness measurement; depth profiling; heavy ion backscattering; non-Rutherford backscattering. Channeling: channeling equipment; crystal alignment; defect analysis methods; surface relaxation. Elastic recoil detection (ERD): principles of ERD; experimental setup; applications. Proton induced X-ray emission (PIXE). Charged particle activation analysis.
    • Classical and quantum properties of the angular momentum; transformation in rotating coordinate system; Bloch equation; spin-spin and spin lattice relaxation; experimental methods: continuous-wave method; Fourier method; instrumentation; different method in pulsed NMR for measuring relaxation times; applications in solid state physics and chemistry: Knight shift; Korringa relation; chemical shift; NMR tomography.
    • Infrared spectra; Raman spectra; atomic spectroscopy; laser spectroscopy.
  • Scanning probe microscopy and spectroscopyMössbauer spectroscopy High-energy ion beam spectroscopy Nuclear magnetic resonance spectroscopyElectronic and vibrational spectroscopy


Introduction to Micro- and Nanotechnology

    • Size ranges and technologies; specific properties of nanosize particles; ancient nanotechnology; instinctive nanotechnology; natural nanotechnology; beginning of conscious nanotechnology; nanoscience and nanotechnologies; ways to the nanoworld.
    • Microscopy: transmission electron microscopy; high resolution TEM; scanning electron microscopy; scanning probe microscopy (AFM, STM).
    • Diffraction: X-ray diffraction; electron diffraction;
    • Spectroscopy: photoelectron spectroscopy; optical spectroscopy; infrared spectroscopy; Raman spectroscopy; nuclear magnetic spectroscopy; positron annihilation spectroscopy.
    • Characterization of nanoparticles; synthesis of nanoparticles; structure; fluctuations; reactivity; conduction electrons and dimensionality; partial and total confinement.
    • Disordered nanostructured materials: glasses containing metallic nanoclasters, porous silicon. Ordered nanomaterials: multilayers; giant magnetic resistance; nanostructured crystals; crystals of metal nanoparticles; arrays of nanoparticles in zeolites; photonic crystals.
    • Nature of carbon bond; graphite, diamond; carbon nanoparticles; fullerenes; fullerene crystals; carbon nanotubes; geometrical structure; dispersion relation; vibrational properties; mechanical properties; fabrication of nanotubes; applications.
    • Intermolecular interactions; molecular recognition; colloid systems, properties of amphypatic molecules; self-assemby in 3D: micelles; membranes; nanoporous systems; self-assembled polymers; dendrimers, applications.
    • Self-assembled monolayers (SAMs), chemical composition, preparation and structure; application in surface chemistry, biosensors, nanolitography.
    • Formation and properties of the Lagmuir-film; Langmuir-Blodgett technics; surface patterning, applications.
    • Quantum wells, wires and dots; conduction electrons and dimensionality; electron transport in nanosytems; ballistic and quasi-ballistic transport; quantum coherency; weak localization; conductivity fluctuation; Aharonov-Bohm effect; quantum Hall effect; Landauer formula; quantum resistance; single electron tunneling; Coulomb blockade; single electron transistor;
    • Short history of microtechnology, tendencies and limitations of the development; photolithography; nanolithography: X-ray lithography; electron lithography; ion beam lithography; the state of the art of nanolithography.
    • Micro-elecromechanical systems (MEMS); fabrication technologies, integration of micromachining with microelectronics; micromechanical sensors; applications; tendencies in development; nano-electromechanical systems (NEMS).
    • Application of new materials and devices today and in the near future; survey of the main fields: informatics; computer techniques; medicine; energy and environment; military use; space research; short-time and long-time risks of the new technology; ethics issues.
  • Basics of nanotechnologyMethods of measuring nanopropertiesProperties of individual nanoparticles Bulk nanostructured materials Carbon nanostructuresSelf-assemblyOrganic nanolayersElectron transport through nanoobjectsMicro- and nanolithographyIntegrative systemsApplications, risks and ethics issues


Electronic states in solids

    • Basic formalism of the density functional theory
    • Thomas-Fermi model
    • Hartree model
    • The Hohenberg-Kohn theorem
    • Ground state energy functional
    • Exchange-correlation energy functionals. Local density approximation. Generalized gradient approximation
    • Space groups
    • Irreducible representation of space groups
    • Group of the wave vector. Stars of wave vectors. Compatibility relations
    • Properties of the density of states. Van Hove singularities
    • Wannier functions
    • Tight binding method
    • Wigner-Seitz method
    • Pseudopotential method
    • Muffin-tin orbitals. Linearized muffin-tin orbitals method
    • Nearly free electron model
    • Friedel model
    • Cohesion in metals
    • Stability of metal structures
    • Spin paramagnetism
    • Stoner model
    • Spin polarized density functional theory
    • Stability of magnetic structures
  • Description of the electronic statesSymmetry of the electronic states in crystalsSolutions of Schrödinger's equation in periodic potentialThe metallic bond. Total energy of metalsMagnetism in metals


Non-equlibrium Thermodynamics II

  • Rational thermodynamics
  • Extended irreversible thermodynamics
  • Microscopic foundation - Onsager, Grad
  • Non-local equilibrium equation of state
  • Non-local equilibrium entropy
  • Axiomatic formulation of EIT
  • Hyperbolic Heat conduction


Introductory course to quantum chaos with emphasis on random matrix theory for quantum eigenstates and transport, together with related experiments in mesoscopic systems

    • introduction and related experiments
    • the role of symmetries
    • universality classes
    • Gaussian ensembles (orthogonal, unitary and symplectic)
    • joint probability distribution of eigenenergies
    • determination of the density of states by the Coulomb gas method and by
    • use of the Green function
    • level spacing statistics
    • numerical studies and theory
    • experiments
    • scattering and transfer matrix method
    • weak localization and RMT
    • universal conductance fluctuations
    • H-J. Stöckmann, Quantum Chaos: an introduction (Cambridge, 1999)
    • C.W.J. Beenakker, Random-matrix theory of quantum transport Rev. Mod. Phys. 69, 731-808 (1997), (
    • F. Haake, Quantum Signatures of Chaos (Springer, 1991)
    • M.L. Mehta, Random Matrices (Academic Press San Diego, 1991 and earlier)
  • Random Matrix TheoryScarsQuantum transportLiterature:


Many body problem I.

Necessary background: Quantum mechanics, Statistical physics.

The course develops the quantum theory of many particle systems. The lectures of the first semester are devoted to normal systems using the temperature Green's function technique. The first part is devoted to the general formalism, while the second part concentrates on the electron gas.

    • Occupation number representation: creation and annihilation operators.
    • Definition of the temperature Green's function in grand canonical ensemble.
    • Expression of equilibrium physical quantities in terms of them.
    • Perturbation theory, Wick's theorem, Feynman diagrams.
    • Self-energy and the Dyson equation.
    • Hartree-Fock approximation.
    • Electron gas in homogeneous positive background.
    • Calculation of the correlation energy.
    • Spectral function, retarded Green's function, analytic properties, elementary excitations.
    • Density propagator and its spectral function: perturbation theory and analytic properties.
    • Collective excitations in the electron gas (the plasmon)
    • The problem of stability of normal systems.
  • Subjects:
    • E.M. Lifsic and L.P. Pitaevskii: Statistical Physics, Part 2 : Volume 9 (Pergamon, 1980)
    • A.L. Fetter and J.D. Walecka: Quantum Theory of Many-Particle Systems (McGraw-Hill, 1971)
    • P. Szépfalusy and G. Szirmai: Véges hõmérsékleti soktestprobléma (lecture notes available in Hungarian under
  • Literature:


Many body problem II.

Necessary background: MANY BODY PROBLEM I.

The second semester is devoted to superfluid Bose and Fermi systems. The applied techniques include canonical transformation, equation of motion method, perturbation theory.

    • Bose-Einstein condensation in interacting systems.
    • Bogoliubov theory of the elementary excitations in the Bose gas. Discussion of the Landau spectrum of elementary excitations in superfluid He II.
    • Developing the perturbation theory of Bose systems in the presence of a Bose-Einstein condensate. Beliaev equations.
    • Hugenholtz-Pines theorem, phonon spectrum.
    • Bogoliubov-Hartree approximation at finite temperature.
    • Treating two-body collisions by summing up ladder diagrams.
    • Elementary excitations in terms of the exact s-wave scattering length.
    • Application to Bose gas in traps.
    • Cooper instability in the attracting Fermi gas. Pair correlations, elements of the Bardeen-Cooper-Schrieffer theory.
    • Determination of normal and anomalous Green's functions by applying the method of equation of motion.
    • Energy gap as order parameter: temperature dependence.
    • Crossover from BCS to Bose condensation.
  • Subjects:
    • E.M. Lifsic and L.P. Pitaevskii: Statistical Physics, Part 2 : Volume 9 (Pergamon, 1980)
    • A.L. Fetter and J.D. Walecka: Quantum Theory of Many-Particle Systems (McGraw-Hill, 1971)
    • L.P. Pitaevskii and S. Stringari: Bose-Einstein Condensation (Clarendon Press, 2003)
  • Literature:



  • Thermodynamical properties of superconductors.
  • Order parameter, Ginzburg Landau equations.
  • Second kind superconductors, vortex state.
  • Description of the condensed state, BCS theory.
  • The self-consistent field method, the Bogoljubov equations.
  • Microscopic analysis of the Ginzburg-Landau equations, the Bogoljubov-deGennes equations.
  • Andreev scattering.
  • Josephson tunneling.

Proposed literature: de Gennes: Superconductivity of metals and alloys, Tinkham: Introduction to superconductivity.


Traffic Modeling in Communication Networks

Goals: to study the statistical properties of traffic in modern communication networks

One semester, 2 hours per week

    • Types of communication networks and their basic properties from the point of view of traffic
    • Poisson modeling of classical telephone systems. Markovian models.
    • Queuing models and Erlang's formula
    • Traffic modeling in the Internet: heavy tailed distributions, Long Range Dependence, scaling and Hurst exponents
    • Fractal traffic models, On-Off processes, file length distributions
    • Measuring traffic in real and simulated networks
    • Round Trip Time, Packet Delay and Packet Loss statistics, 1/f noise, congested-non congested phase transition
    • Protocols and traffic
    • TCP as a dynamical system, modeling small networks with stochastic maps
    • Traffic modeling with network simulator
    • Routing and traffic
    • Ad-hoc networks and load on random graphs
  • Lectures:
    • Kleinrock, L., Queueing Systems, Volume I: Theory, Wiley Interscience, New York, 1975
    • Kocarev L., Vattay G., Complex Dynamics in Communication Networks, Springer Verlag, 2005
  • Recommended literature

Few publications of the Lecturer:

  • A. Fekete, G. Vattay and A. Veres, Improving the 1/sqrt(p) law for single and parallel TCP flows In Teletraffic Engineering in the Internet Era, ed.: J.M. de Souza,N.L.S. da Fonseca, E.A.S. Silva, North-Holland Amsterdam (2001)
  • A. Veres, Zs. Kenesi, S. Molnar, and G. Vattay, On the Propagation of Long-Range Dependence in the Internet ACM SIGCOMM 2000 and Computer Communication Review 30, No 4, 243-254 (2000)
  • A. Fekete and G. Vattay, Self-Similarity in Bottleneck Buffers Proceedings of Globecom, (2001)