Physics MSc final examination informations

The final examination consists of two parts: the thesis defense and a comprehensive exam. The duration of the examination is about half an hour. The topics of the comprehensive exam comprise the topics of the field of the given program/specialization/module (as listed below). With regard to the epidemiological emergency the final examination will be carried out electronically (detailed later).

Research Physicist Specialization

 Module  Date and Time  Location
Astrophysics 2020. jun. 24. (We) 9:00 online
Atomic and Molecular Physics


Nuclear and Heavy Ion Physics -  
Biological Physics 2020. jun. 19. (Fr) 13:00 online
Condensed Matter Physics 2020. jun. 22. (Mo) 9:00 online
Medical Biophysics -  
Particle Physics

2020. jun. 25. (Th) 9:00

Statistical Physics and Complex Systems

2020. jun. 19. (Fr) 10:00

Computational Physics

2020. jun. 23. (Tu) 9:00


Other Specializations

 Specialization  Date and Time  Location
Biophysics 2020. jun. 19. (Fr) 13:00 online
Environmental Physics -  
Scientific Data Analytics and Modeling 2020. jun. 23. (Tu) 9:00 online

Committees and topics for each module and specialization

▼ Astrophysics


chair Frei Zsolt egy. tanár Atomfizikai Tsz.
member (vice-chair) Csótó Attila egy. tanár Atomfizikai Tsz.
member Bagoly Zsolt egy. doc. Komplex Rendszerek Fizikája Tsz.
member Petrovay Kristóf egy. tanár Csillagászati Tsz.
member Kiss László tud. tanácsadó MTA CSFK
member Ábrahám Péter tud. tanácsadó MTA CSFK 


  1. Astronomical observations in the visible band. Photometry and photometric systems in astrophysics. Types of telescopes, their construction, and mounts.
  2. Spectroscopy in astrophysics. Observations in the non-visible spectral bands. Space observatories and their most important results.
  3. Stellar physics: the Lane-Emden equation and the polytrope model. The energy production in stars (the p-p chain and the CNO cycle). The HR diagram. The stellar evolution on the HR diagram.
  4. The endpoints of stellar evolution: nuclear burning in high mass stars, stellar nucleosynthesis, supernovae.
  5. Types and classification of galaxies. The structure of the various types of galaxies and their components. The spiral structure of galaxies.
  6. Galaxy clusters and superclusters in the Universe. The Abell and the Zwicky catalogues.
  7. The large scale structure of the Universe. Observations and surveys aimed at exploring the large scale structure (CfA slice, pencil beam observations, SDSS) and their results: the statistical description of the large scale structure.
  8. The hierarchical model of galaxy formation and evolution. The physics of quasars, observational results.
  9. Fundamentals of cosmology: Friedmann equations, the expanding universe, Big Bang, and the cosmological parameters.
  10. Observational pillars supporting the standard Big Bang theory, questions left open by the model, and cosmic inflation as a possible solution.
  11. Precision cosmological measurements and their results: the power spectrum of CMB fluctuations, the large-scale structure of the Universe, and SNIa experiments.
  12. Active galactic nuclei (AGN) and the unification model: the properties of the different AGN classes (Seyfert galaxies, quasars, blazars, radiogalaxies), the evidence for supermassive black holes, the unification model.
  13. Gamma-ray bursts (GRBs): the detection of various kinds of GRBs, the fireball model, GRB afterglows.
  14. The astrophysics of the hot intra-cluster medium (ICM): the properties of the ICM, cooling flows and AGN feedback, the observational techniques of the ICM (X-ray spectroscopy and the Sunyaev-Zeldovich effect)
  15. The mathematics and physics of general relativity.
  16. Correspondence principle, Einstein equations, their solutions, the conservation laws from the point of view of general relativity.
  17. Observations and experiments that provide the foundations and support for general relativity.
  18. The theory of gravitational waves: general relativistic background, the production of GWs, GW interaction with an interferometer.
  19. The detection of gravitational waves: principles of interferometric detection, noise sources, detections with multiple detectors.
  20. Gravitational waves in astrophysics: types of GWs, fundamentals of signal searches, detections achieved, multi-messenger astronomy.

▼ Atomic and Molecular Physics


chair Csordás András egy. tanár Komplex Rendszerek Fizikája Tsz.
member (vice-chair) Kürti Jenő egy. tanár Biológiai Fizika Tsz.
member Koltai János adj. Biológiai Fizika Tsz.
member Geszti Tamás prof. emer. Komplex Rendszerek Fizikája Tsz.
member László István egy. doc. BME Fizikai Int.
member Udvardi László tud. fmts. BME Fizikai Int.

Topics A:

  1. The principles of the quantum mechanical many-body problem (approximate methods: variational principles and methods, perturbation theory; identical particles, Pauli principle, Slater determinants; spin wave functions, Coulomb- and exchange integrals)
  2. Hartree-Fock approximation and electron correlation (mean-field, closed and open shell systems, Koopmans-theorem: ionization an electron affinity, Roothan method, electron correlation, configuration interaction, electron pair methods)
  3. Atoms with several electrons (Helium, screening, periodic system, spin-orbit interaction, Hund’s rules, multiplets and configurations, coarse, fine and hyperfine level structures, Grotrian diagram)
  4. Theoretical descriptions of molecules (Born-Oppenheimer approximation, potential energy surface (PES): bindings, oscillations, dissociation, predissociation, virial theorem, Hellmann-Feynman theorem, hydrogen molecule ion, hydrogen molecule, VB method, MO method)
  5. Diatomic and polyatomic molecules (diatomic molecules: homonuclear and heteronuclear diatomics, symmetries, configurations, number of bonds, para-magnetism, oxygen molecule, correlations diagram, rule of avoided level crossings; polyatomic molecules: symmetries, water molecule, carbon dioxide, localized and delocalized molecular orbits, hybridization, single pair, planar molecules, s- and p-bonds, benzene molecule)
  6. Interactions of atoms, interactions of molecules, interactions with external electric and magnetic fields (Lennard-Jones interaction, van der Waals interactions, dipol interaction, hydrogen bond, the structure of the water molecule, Stark-effect, normal and anomalous Zeeman-effects, electric polarization, magnetic susceptibility, diamagnetic shielding, g-factor, magnetic resonance)
  7. Principles of atomic and molecular spectroscopy (electron transitions in atoms and molecules, symmetries, selection rules, UV-VIS spectroscopy, charge transfer, chromophore, ORD-CD, fluorescence, phosphorescence, Franck-Condon principle, photoelectron spectroscopy)
  8. Experimental methods for atomic and molecular structures (UV, VIS, IR and microwave spectroscopies, Raman-, resonant Raman-spectroscopy, ORD-CD, NMR, ESR, Mössbauer-spectroscopy, mass-spectroscopy, photoelectron spectroscopy, X-ray, electron microscopy, STM, AFM)

Topics B:

  1. Calculations of electron structures for atoms and molecules (traditional quantum chemistry, density functinal theory, ab initio and semi-empirical methods, occupation number representation, one and two particle operators)
  2. Macromolecules (structures and statistical properties of flexible polymers with chains, polymers with conjugated carbon chains, linear chain: one dimensional instabilities, effects of doping on the electric, magnetic and optical properties, biological macromolecules: non-informational (e.g., polysaccharide) and informational (e.g., protein) macromolecules, secondary and tertiary structure, DNS, genetic code)
  3. Complex molecules (theory of atomic spectra, spin-orbit interaction, hyperfine interaction, crystal field theory, ligand field theory, vibronic interactions, Jahn-Teller instability)
  4. Carbon nanostructures (the discovery of C60, historical overview, isolated cage like molecules, fullerenes in liquid and solid phases, doped fullerenes, fullerites, transport properties, optical properties, polymers, endohedral compounds, carbon nanotubes)
  5. Semiclassical dynamics (WKB approximation, Bohr-Sommerfeld quantization, Maslov index, Berry-Tabor conjecture, Gutzwiller’s trace formula, Poisson and Wigner level spacing statistics)
  6. Principles of mesoscopic systems (2D electrongas, open and closed channels, the role of magnetic field, Landauer-Büttiker formula, relationship between S-matrix and Green’s function, Aharonov-Bohm effect, weak localization, universal conductivity fluctuation)
  7. Many-body theory at zero temperature (Green’s functions, physical quantities that can be expressed by Green’s function, Feynman diagrams, elementary excitations, applications, the method of canonical transformations for Bose-gas and superconductors)
  8. Many-body theory at finite temperature (Green’s functions, the thermodynamic potential expressed by the Green’s function, Matsubara frequencies, Feynman-diagrams, applications)
  9. New experiments in quantum mechanics (neutron interference, ion- and atom traps, micromaser, atomoptics, two-photon interference)
  10. Quantum phenomena (decoherence, quantum jumps, quantum Zenon-effect, quantum non-demolition, Berry-phases)
  11. Statistical quantum optics (quantum theory of electro-magnetic fields, quasi-probability distribution functions, non-classical states of light, photon statistics, optical parametric processes, quantum theory of damping: Langevin-equation, atom-field interaction: Jaynes-Cummings model, methods of quantum information transfer: discrete and continuous teleportation.)
  12. Trapped atomic systems (non-interacting bosons in harmonic potentials, finite size effects in non-interacting systems, Gross-Pitaevskii equation, collapse for bosons with negative scattering length, Thomas-Fermi approximation for bosons, elementary excitations, Bogoliubov-equations, hydrodynamical approach, fermions in traps, BCS-BEC crossover)

▼ Nuclear and Heavy Ion Physics


chair Csótó Attila egy. tanár Atomfizikai Tsz.
member (vice-chair) Papp Gábor egy. tanár Elméleti Fizikai Tsz.
member Csanád Máté egy. doc. Atomfizikai Tsz.
member Varga Dezső tud. tanácsadó MTA Wigner FK
member Wolf György tud. tanácsadó MTA Wigner FK


  1. Ground-state properties, and electromagnetic and weak interactions of nuclei
  2. Nuclear forces (nucleon-nucleon interaction, phase shift analysis, three-body forces)
  3. Nuclear models (liquid drop model, Fermi gas model, shell model, unified model)
  4. Nuclear scattering theory (two- and three-body scattering, multichannel scattering, polology, scattering-theoretic description of nuclear reactions)
  5. Bound and resonant states in few-nucleon systems, effective interaction
  6. Nuclear reactions (medium-energy neutron capture reactions, astrophysically important reactions)
  7. Production and reactions of radioactive nuclear beams
  8. The quark-quark interaction potential in perturbation theory and beyond (asymptotic freedom, bag model of quark confinement)
  9. Field-theoretic descriptions of nuclear matter and its equation of state (Johnson-Teller model, Walecka model of symmetric- and neutron matter)
  10. Effective theories of chiral symmetry and its spontaneous breaking (Nambu--Jona-Lasinio, Gell-Mann--Levy and the nonlinear sigma model)
  11. Phases of nuclear matter, phase transitions, and the signatures of the quark-gluon plasma
  12. Modelling methods of medium-energy heavy-ion reactions (VUU, BUU, (r)QMD, hydro, initial states, fragmentation)
  13. Mechanisms of particle production in heavy-ion reactions (resonances, mesons, subthreshold production, direct reactions)
  14. The basic principles of particle detection (interaction of particles and matter, tracking of charged particles, calorimetry)
  15. Complex detector systems (types of particle detectors, event reconstruction, particle identification, the structure of a modern high-energy detector system, neutrino experiments)
  16. Particle accelerators and beams (fixed target, colliding beams, generation of electron-, proton-, antiproton-, pion-, muon-, and neutrino beams, the LHC accelerator complex)

▼ Biological Physics Module, Biophysics Specialization


chair Derényi Imre egy. tanár Biológiai Fizika Tsz.
member (vice-chair) Meszéna Géza egy. tanár Biológiai Fizika Tsz.
member Lőw Péter egy. doc. Anatómiai, Sejt- és Fejl.biol. Tsz.
member Vicsek Tamás prof. emer. Biológiai Fizika Tsz.
member Simon István tud. tanácsadó MTA TTK

Information for the defense: 15 minutes are available for the defense of the dissertation, as well as an additional 5 minutes for answering the questions of the committee and the audience. Using a projectable presentation (PowerPoint, PDF) is recommended, which can be uploaded to the notebook provided for the presentation in advance.


  1. The cell as a basic unit of living system (IB)
  2. Fundamentals and biological aspects of thermodynamics and statistical physics (BP, SPB)
  3. Structure and function of proteins (BP, BET, MM, IB)
  4. Structure and function of DNA (RNA), the genetic code, transcription, translation (BP, IB)
  5. Gene technology (BP, IB)
  6. Classical and molecular genetics (IB)
  7. Biological membranes (BP, MM, IB)
  8. Bioenergetics: anaerobic and aerobic processes, chemiosmosis (BP, IB)
  9. Bioenergetics: photosynthesis (BP)
  10. Membrane potential, ion transport (BP, IB)
  11. Neurons and synapses, nerve signalling, Hodgkin-Huxley theory (BP, IB)
  12. Spectroscopic methods in biology (BET, IB)
  13. Biophysical experimental methods (BP, BET, IB)
  14. Biophysics of perception (BP, IB)
  15. Cell communication, signal transduction, cell movement (QM, IB)
  16. Collective behavior (synchronization, networks, motion) (BP, SPB)

The topics are based on the following subjects*:

BP:  Biophysics I. and II.
BET: Biophysical Experimental Techniques
QM:  Quantitative Models in Cell and Developmental Biology
SFB: Statistical Physics of Biological Systems
MM:  Macromolecules
IB:  Introduction to Biology 1., 2., 3.

* Those who have not taken any of these subjects, please inform the committee before the exam.

▼ Condensed Matter Physics


chair Groma István egy. tanár Anyagfizikai Tsz.
member (vice-chair) Cserti József egy. tanár Komplex Rendszerek Fizikája Tsz.
member Nguyen Quang Chinh egy. tanár Anyagfizikai Tsz.
member Kriza György tud. tanácsadó MTA Wigner FK
member Lendvai János prof. emer Anyagfizikai Tsz.
member Tichy Géza prof. emer. Anyagfizikai Tsz.
member Tüttő István tud. tanácsadó MTA Wigner FK


  1. Bonding in solid states (van der Walls, ionic, covalent, and  metallic bonding)
  2. Crystalline structures, experimental determination of crystalline structures (basic structures, symmetry properties, direct and reciprocal lattice, X-ray-, electron-, and neutron-diffraction)
  3. Disordered systems ( alloys, amorphous materials, liquid crystals, quasicrystals)
  4. Lattice vibrations (classic and quantum description, acoustic and optic  banches, expeeimental determination of phonon spectra)
  5. Thermal properties of crystalline materials (heat capacity of phonon gas, anharmonicity, thermal expansion, heat conductivity by phonons), Superfluidity.
  6. Electron states in periodic systems (Bloch and Wannier functions, band structure and its relation to the symmetry of the crystal, experimental determination of band structure, metals, semiconductors, insulators)
  7. Pure and doped semiconductors,  (Fermi energy, donor and acceptor levels,  excitons). Schottky barrier, p-n transition, transistor.
  8. Semiclassical description of the dynamic of electrons. Exp erimental determination of Fermi surface (motion of electron in electric and magnetic fields, cyclotron resonance, size effects, magneto-acoustic effect)
  9. Electron is strong magnetic field (Landau levels, de Haas-van Alphen effect, quantum Hall effect)
  10. Electron-phonon interaction, (limits of adiabatic decoupling, obtaining the electron- phonon interaction, polaron, Kohn anomaly).
  11. Theory of conduction (Boltzmann equation, relaxation time approximation and its limitations, transport coefficients in metals and semiconductor, scattering of impurities,  phonons and electrons, scattering of electron on magnetic impurities, Kondo effect).
  12. Interaction of solid materials with photons and neutrons (optical properties, Brillouin and Raman scattering, neutron scattering). Experimental methods for studying the properties of solid materials (Mössbauer effect, NMR, ESR, positron annihilation). 
  13. The phenomenological theory of superconductivity (thermodynamics of superconductors, I. II. type superconductors, Ginzburg-Landau theory, superconducting vortex)
  14. BCS theory of superconductivity, tunnel effect, Josephson effect
  15. Many electron systems, electron-electron interaction (Hartree-Fock approximation, correlations, dielectric constant, screening, plasmons, density function theory).      
  16. Basic properties of magnetism (Hund’s rules, splitting of atomic levels in magnetic field,  dia and  paramagnetism, paramagnetic resonance, Pauli paramagnetism, exchange interaction,  direct exchange, super exchange, RKKY exchange)
  17. Ordered magnetic materials (ferro and antiferromagnets, mean field theory, spin waves, magnetic anisotropy, magnetic domains) disordered magnets (spin glasses)
  18. Low-dimensional systems (magnetic models in 1 and 2D, super-lattice,   magnetic coupling through and non-magnetic layer, 1D electron system, Luttinger liquid)  
  19. Defects in crystals, and their role in material properties, Point defects (equilibrium concentration, mixing and formation entropy and enthalpy, evolution of point defects), dislocations (topological properties, continuum description, force acting on a dislocation, partial dislocations), internal surfaces (grain boundary, phase boundary, energy of the boundaries, nanostructural materials).
  20. Diffusion in solid states (macroscopic and microscopic descriptions, concentration dependence of diffusion constant, Kirkendall-effect, Darken equations)
  21. Mechanical properties (plasticity, work hardening, hardening by solid solution and precipitates)
  22. Solid solutions (thermodynamic of multicomponent systems, quasi chemical theory,  ideal and ordered solid solutions, factors determining the solubility limit)
  23. Phase diagram (equilibrium and non equilibrium phase diagrams, calculating the phase diagram from free energy), solidification (homogeneous and heterogeneous   nucleation, crystal growing, cleaning by zone melting, non equilibrium solidification)
  24. Phase transformations in solid phase (precipitation, spinodal decomposition, phase transformations without diffusion, martensitic transformation)
  25. Ceramics and composite materials (chemical structure of ceramics, properties, properties of composites, how to select materials for composites)

▼ Medical Biophysics


elnök Derényi Imre egy. tanár Biológiai Fizika Tsz.
tag (pótelnök) Horváth Ákos egy. doc. Atomfizikai Tsz.
tag Czirók András egy. doc. Biológiai Fizika Tsz.
tag Fröhlich Georgina sugárfizikus Országos Onkológiai Int.
tag Kotek Gyula egy. doc.

Kaposvári Egyetem

Információ a védéshez: A diplomamunka védésére 15 perc áll rendelkezésre, valamint további 5 perc a bizottság és a hallgatóság kérdéseinek megválaszolására. A védéshez javasoljuk kivetíthető prezentáció (PowerPoint, PDF) használatát, melyet érdemes a védések megkezdése előtt feltölteni a vetítésre biztosított notebookra.


A kötelezően választható tárgyak közül a hallgató csak azokból kap kérdést, amelyeket a tanulmányai során elvégzett.


  1. Az emberi szív szerkezete és fejlődési rendellenességei
  2. Az emberi légzőszervrendszer morfológiai felépítése
  3. Az ember vázizomrendszerének általános jellemzői, a törzs (mellkas, a has és a hát) izmai és működésük


  1. Testfolyadékok és homeosztázis
  2. A szív és a keringés működése
  3. Idegi működések


  1. A sugárvédelem hármas alapelve
  2. Dózisfogalmak, a lakossági- és foglalkozási korlátok
  3. A három legnagyobb, ember által okozott nukleáris szennyezés

MR-Fizika I.

  1. Mágneses rezonancia képalkotás (mágneses momentumok mágneses térben, kvantummechanikai és klasszikus leírás)
  2. Rezonancia, gerjesztés, MR jel
  3. Jelakvizíciós eljárások (FID, echo, spektroszkópia, képalkotó eljárások, Fourier, back-projection)

MR-Fizika II.

  1. Szekvenciák (Spin echo, Fast spin echo, Gradiens echo 2D, -3D, EPI)
  2. RF gerjesztések, akvizíciós technikák (Phased Array, Parallel Imaging)
  3. Kontraszt-mechanizmusok és alkalmazásaik, fiziológiai folyamatok detektálása (áramlás, diffúzió, perfúzió)

Ionizáló sugárzások a gyógyításban

  1. Teleterápia (folyamata, gyorsítók, kobaltágyú, RTG-terápia, dózis-profilok, mélydózis-görbék, kollimáció, besugárzási mezők, ékek, verifikáció)
  2. Sugárterápiás besugárzás-tervezés (ICRU ajánlások, 2D/3D-s tervezés, céltérfogat, védendő szervek, DVH, minőségi indexek, konformális besugárzási technika)
  3. Brachyterápia (folyamata, típusai, izotópok, besugárzó készülékek, dózis-teljesítmény, dózis-előírási technikák, optimalizálási módszerek, különböző lokalizációk besugárzása, IGABT)

Sugárterápiás fizika

  1. Speciális teleterápiás eljárások és technikák, sugárvédelem és biztonság
  2. Brachyterápiás besugárzástervezés és képvezérlés (dozimetriai rendszerek, 2-, 3- és 4D-s BT, TG-43 dózisszámítási formalizmus és modell-alapú algoritmusok, optimalizálás és kiértékelés)
  3. Sugárbiológia, prosztata, nőgyógyászati, emlő és egyéb daganatok brachyterápiája

Kvantitatív modellek a sejt- és fejlődésbiológiában

  1. Sztochasztikus reakciókinetika (Gillespie-algoritmus)
  2. Transzkripciós regulációt leíró egyenletek: transzkripció, transzláció, visszacsatolások
  3. Többsejtű szerveződés biofizikai modelljei (felületi feszültség analógia, sejtmozgás, mint pozitívan visszacsatolt sztochasztikus rendszer)

Sejtszignalizációs hálózatok kvantitatív analízise

  1. Jelátviteli hálózatok matematikai modellezése. Kinázok, foszfatázok, transzkripciós faktorok, mRNS-ek és promóterállapotok leírása reakcióegyenlet rendszerekkel. Fixpont analízis, Gillespie-féle stochasztikus szimuláció
  2. Robosztusság és visszacsatolások. Baktériumok kemotaxis rendszere és molekuláris kapcsolók. PID szabályzók

Fejlődésbiológiai mechanizmusok kvantitatív modelljei

  1. Többsejtű rendszerek matematikai modelljei. Sokrészecske, Potts és folytonos modellek.
  2. Diffúzív morfogén faktorok vezérelte mintázatképződés: Turing mechanizmus, gerjeszthető közegek

Preklinikai modellek a daganatkutatásban

  1. A főbb ellenanyag alapú vizsgálati módszerek (immunoblot, IP, IF, IHC, ELISA, FACS) egy-egy konkrét daganatkutatási példával
  2. Egy-egy konkrét in vivo modellt, amely alkalmas (I) a klasszikus kemoterápia, (II) a célzott terápia és (III) az immunterápia kísérletes vizsgálatára
  3. A xenograft tumormodellekben használható képalkotási módszerek összehasonlítása

▼ Particle Physics


chair Katz Sándor egy. tanár Elméleti Fizikai Tsz.
member (vice-chair) Veres Gábor egy. doc. Atomfizikai Tsz.
member Nógrádi Dániel adj. Atomfizikai Tsz.
member Forgács Péter tud. tanácsadó MTA Wigner FK
member Takács Gábor tud. tanácsadó BME Fizikai Int.
member Balog János tud. tanácsadó MTA Wigner FK


  1. Basic principles of particle detection, tracking, calorimetry.
  2. Accelerators and beams (fix target, colliding-beam, e, p beams).
  3. Complex detector systems, hardware, software, presentation of a modern detector system. 
  4. Description of some important experiments (P, CP, J/Ψ).
  5. Geometrical symmetry groups; rotation group, Poincaré group, reflection symmetries.
  6. The quantum field theory of free fields, symmetries.
  7. S matrix in quantum field theory, functional integrals, Feynman graphs.
  8. Gauge theories.
  9. The renormalization of QED and QCD.
  10. Electron-photon interaction (processes)
  11. The interaction of electromagnetic and ionizing radiation with condensed matter, penetrating power and particle showers.
  12. The basics of the strong interaction: conserved quantities, properties of particles and resonances.
  13. Internal symmetry groups: SU(2), SU(3) and the particle multiplets, the basics of the quark model.
  14. The dynamics of the strong interactions, the basics of QCD.
  15. The dynamics of strong interactions at low energies, chiral symmetry breaking and the effective Lagrangian based description.
  16. Basics of high energy physics, renormalization group equations and their application, running coupling constant, deep inelastic scattering, jet physics.
  17. Classification of the weak processes, conserved quantum numbers and selection rules, the theory of β decay, V-A coupling.
  18. Elements of current algebra: conserved vector current, PCAC, Cabbibo theory, GIM mechanism.
  19. Fundamentals of the electroweak theory: spontaneous symmetry breaking, Goldstone bosons, Higgs mechanism, masses and couplings of the W ans Z bosons, lepton and quark multiplets.

General relativity branch

The above first 4 topics are replaced by:

  1. Topological and metric properties of spacetime, Einstein equations and their action principle derivation.
  2. Exact solutions (Minkowski, De Sitter, Robertson-Walker), cosmological models (Schwarzschild, Reisner-Nordström, Kerr, Gödel, Taub-Nut), Penrose diagrams.
  3. Causal structure, space and time orientability, causal curves, achronal boundaries, Cauchy surfaces, causal boundary of spacetime.
  4. Gravitational collapse, black holes, Schwarzschild black hole, remnant black holes, Hawking radiation, thermodynamics, final state, experimental data.

▼ Statistical Physics and Complex Systems


chair Vattay Gábor egy. tanár Komplex Rendszerek Fizikája Tsz.
member (vice-chair) Csordás András egy. tanár Komplex Rendszerek Fizikája Tsz.
member Kaufmann Zoltán egy. doc. Komplex Rendszerek Fizikája Tsz.
member Palla Gergely tud. tanácsadó MTA Statisztikus és Biológiai Fizika Kcs.
member Pollner Péter tud. fmts. MTA Statisztikus és Biológiai Fizika Kcs.


  1. Foundation of statistical physics and thermodynamics.
  2. Ideal Fermi and Bose gases, occupation number representation.
  3. Manybody problem, perturbation calculus, Feynman diagrams.
  4. Interacting fermion and boson systems.
  5. Elementary excitations. Superfluidity.
  6. Transport processes.
  7. First-order and continuous phase transitions.
  8. Critical phenomena, scale hypothesis and renormalization.
  9. Response and correlation functions, fluctuation-dissipation theorem.
  10. Fundamentals of statistical physical simulations.
  11. Non-equilibrium and stochastic processes.
  12. Fractal geometry, fractal measures and fractal growth.
  13. Transport in nanosystems.
  14. Structure and dynamics of complex networks.

▼ Computational Physics


chair Csabai István egy. tanár Komplex Rendszerek Fizikája Tsz.
member (vice-chair) Papp Gábor egy. tanár Elméleti Fizikai Tsz.


Oroszlány László adj. Komplex Rendszerek Fizikája Tsz.
member Palla Gergely tud. tanácsadó MTA Statisztikus és Biológiai Fizika Kcs.
member Pollner Péter tud. fmts. MTA Statisztikus és Biológiai Fizika Kcs.
member Barnaföldi Gergely tud. fmts. MTA Wigner FK


  1. Many-body systems (molecular dynamics, Hartree-Fock, collision integral Vlasov / Boltzmann equation, Vlasov (Boltzmann) -Uhling-Uhlenberg equation)
  2. Transport processes (Boltzmann equation, diffusion, thermal conductivity)
  3. Generation of random graphs, properties (small-world, clustering, robustness)
  4. First order and continuous phase transitions (eg. Ising model)
  5. Response and correlation functions, fluctuation-dissipation theorem
  6. Stochastic processes (Kauffman network, spin glasses, Markov chain)
  7. Fundamentals of statistical physical simulations and the Monte Carlo method
  8. Dynamic systems, chaotic behavior (based on Complex and Adaptive Dynamical Systems).
  9. Data analysis: linear and nonlinear regression (demonstrated on a model)
  10. Data analysis: bootstrap models
  11. Operation of TCP networks
  12. Data analysis: ARCH, GARCH processes
  13. Numerical methods ((adaptive) Runge-Kutta, relaxation, etc.)
  14. Visualization methods

▼ Environmental Physics Specialization


elnök Jánosi Imre egy. tanár Komplex Rendszerek Fizikája Tsz.
tag (pótelnök) Tél Tamás egy. tanár Elméleti Fizika Tsz.
tag Horváth Ákos egy. doc. Atomfizikai Tsz.
tag Török Szabina tud. tanácsadó MTA EK
tag Kocsonya András tud. mts. MTA EK


  1. Az akusztika alapjai (a hullámegyenlet általánosságban, eredete különböző mechanikai rendszerekben, megoldásai 1 és 3 dimenzióban, transzverzális és longitudinális hullámok, vízfelszíni hullámok sekély és mély folyadékokban, a hang fizikája)
  2. Elektromágneses hullámok környezetünkben (fizikai alapok, elektromágneses spektrum, a Napból érkező sugárzás, elektroszmog)
  3. A Coriolis-erő és környezeti jelentősége (definíció, megjelenése a lokális koordinátarendszerben, béta-hatás, Eötvös-effektus, Rossby-szám)
  4. A geosztrofikus áramlás tulajdonságai (dinamikai egyensúly, homogén folyadék belsejében, sekély folyadék felszínén, rétegzett folyadék belsejében)
  5. Laboratóriumi jelenségek rétegzett folyadékokban (lineáris hullámok kétrétegű folyadék belsejében, Kelvin–Helmholtz-instabilitás, holt víz effektus, baroklin instabilitás)
  6. Radioaktív izotópok a környezetünkben (uránsor, tóriumsor, cézium szennyezések, radioaktív egyensúly, gamma spektroszkópia)
  7. Radon mozgása a környezetünkben (a radon forrása, radonexhaláció, radon eljutása az emberhez, egészségügyi hatásai, radon detektorok)
  8. Ismertesse a megújuló energiák felhasználásának alapjait! Elemezze a) energia előállítás gazdaságossága b) környezeti hatás szempontjából!
  9. Ismertesse a nukleáris fűtőanyagciklust és térjen ki azoknak az aggályoknak a fizikai hátterére, amelyeket a társadalom aggályosnak tarthat!
  10. Populációdinamika korcsoportstruktúrával (Leslie mátrix karakterisztikus egyenlete). A domináns sajátérték közelítő megoldása (ha a növekedési ráta közel 1, a stabil koreloszlás alakja, reproduktív érték, demográfiai paraméterek és jelenlétük)
  11. Kétfajos préda-predátor típusú ökológiai modellek (a klasszikus Lotka–Volterra-modell dinamikai viselkedése, globális és lokális analízis), a Holling II. és a Holling III funkcionális válasz, préda-predátor modell dinamikája Holling II funkcionális válasszal. Hiszterézis jelenség a predációs nyomás változásával (túllengetés és fázisátmenet)
  12. Környezeti fényszennyezés (alapfogalmak, poláros fényszennyezés, detektálásának alapfogalmai, kísérleti bizonyítékok, ökológiai következmények, a táplálkozási láncban bekövetkező hatások, fényszennyezés megelőzése)

▼ Scientific Data Analytics and Modeling Specialization


chair Csabai István egy. tanár Komplex Rendszerek Fizikája Tsz.
member (vice-chair) Papp Gábor egy. tanár Elméleti Fizikai Tsz.


Oroszlány László adj. Komplex Rendszerek Fizikája Tsz.
member Palla Gergely tud. tanácsadó MTA Statisztikus és Biológiai Fizika Kcs.
member Pollner Péter tud. fmts. MTA Statisztikus és Biológiai Fizika Kcs.
member Barnaföldi Gergely tud. fmts. MTA Wigner FK


  1. Measurement data and measurement error. Types and sources of errors and noses. The statistical and systematic error. Stochastic modeling of errors. Data modeling, function fitting. Kernel estimation.
  2. Bootstrap methods. Maximum likelihood method. Hypothesis testing. Extreme value statistics. Post hoc analysis. Regression. Independence analysis. Exact tests.
  3. Random number generation, numerical integration. Newton type formulas, Gauss type formulas. Monte-Carlo method, Markov chain Monte-Carlo, hierarchical Bayesian networks.
  4. Simulation of thermodynamic systems. Ising model, Metropolis algorithm.
  5. Fractal dimension, self similar mathematical fractals, fractals in nature, cellular automata.
  6. Numerical solution of differential equations, Euler's method, Runge-Kutta method, stability of solution methods. Partial differential equations, dynamical systems and chaotic behavior (based on "Complex and Adaptive Dynamical Systems").
  7. Molecular dynamics, Verlet and velocity-Verlet algorithms, determination of thermodynamical properties, relaxation.
  8. Signal processing, time series analysis, Fourier methods, FFT, spectrum and spectrogram. Transfer and window functions, Wiener filter. Correlation functions, Wiener-Khinchin theorem and the power spectrum. Convolution and deconvolution. Analog and digital realization of filters, RLC circuits, FIR and IIR filters.
  9. Image processing - Digital representation of image data, color models. Interpolation, convolution and deconvolution methods, homomorphic filtering. Edge, corner and spot detection. Morphological analysis, feature detection. Perspective correction, noise filtering methods. 
  10. AD and DA conversion. AD conversion with successive approximation. Quantization noise and  Nyquist-Shannon sampling theorem. Compression of digital data: delta modulation and delta-sigma modulation. Digital representation of numbers and arithmetic operations. 
  11. Machine learning - Prediction and classification methods, Supervised and unsupervised learning. Training set, validation and over fitting. K-means, Support Vector Machine, Random Forest, k-NN method. 
  12. Neural networks - fully connected networks, convolutional networks, back propagation, optimizers (SGD, Adam), batch normalization, autoencoders, word2vec
  13. Methods for dimension reduction - Statistical properties of higher dimensional data. Principal component analysis and its applications, t-SNE.
  14. Computer networks - The physical realization of computer networks, ethernet, optical fibers. Wireless networks. IP networks, endpoints, switches, routers, firewalls, IP address, TCP, UDP, DNS, NAT. Bandwidth, queuing, congestion. Network topologies. Cryptography.
  15. Relational databases - relational data model, logical and physical operators (seek, scan and join and variants, aggregation), data storage models (row store, column store), indexing (clustered index, non-clustered index), B-tree, keys and constraints, transactions. Query Optimization. The process of data loading.
  16. Multidimensional data - Representation of geographic and spatial data, point clouds. Basic search problems: interval search, spatial search, nearest neighbors. Spatial indices, space-filling curves (Z-index, Peano-Hilbert index), kD-tree, R-tree, the role of bit coding. Spatial partitioning methods: Delaunay triangulation, Voronoi cell. Sphere indexing, Quad-tree, HEALPix, HTM.
  17. Visualization - Techniques for two- and three-dimensional representation of scientific data, representation of scalar, vector and tensor data. Colors. Spatial data display, volumetric and level representation. Three-dimensional rendering techniques, ray casting, ray tracing. Stereoscopic imaging technologies. Computer geometry.