Master study programme **Mathematical and Computational Modelling in Physics** is designed for students of physics. (The programme for students of mathematics is called **Mathematical Modelling in Physics and Technology**, details concerning this study programme can be found here.)

## Contacts

If you have any **questions concerning the study of mathematical modelling**, please contact Martin Čížek (cizek@mbox.troja.mff.cuni.cz) or Vít Průša (prusv@karlin.mff.cuni.cz).

## Admission

Concerning the general overview of the admission procedure, please see the webpages of the Faculty of Mathematics and Physics. **If you are not yet familiar with advanced subjects such as functional analysis but you are still strongly motivated to study mathematical modelling, please contact us, and we will try to recommend you textbooks and other material for self-study.**

Note also, that if you have a sufficiently strong academic record, you can apply for the exemption from the entrance examination.

It is expected that you have sufficient knowledge of the topics listed below. Note that most of the topics are at a sufficient level also covered by courses Linear Algebra I, Linear Algebra II, Mathematical Analysis I, Mathematical Analysis II, Mathematics for Physicists I, Mathematics for Physicists II and Mathematics for Physicists III, which are a part of the standard curriculum in the bachelor study programme Physics at the Faculty of Mathematics and Physics.

**Differential and integral calculus**(single variable and multivariable). Limits, derivatives, line, surface and volume integrals. Foundations of calculus of variations.**Foundations of measure theory, Lebesgue integral**. These topics are taught, for example, in the undergraduate course Measure and Integration Theory I, see the course syllabus for what is in this case meant by “foundations”.**Foundations of linear algebra**. Vector spaces, matrices, determinants, Jordan canonical form, orthogonalization, eigenfunctions and eigenvalues, multilinear algebra, quadratic forms.**Foundations of numerical solution of systems of linear algebraic equations**. Schur theorem, QR decomposition, LU decomposition, singular value decomposition, least squares method, conjugate gradient method, GMRES, backward error, sensitivity and numerical stability. These topics are taught, for example, in the undergraduate course Analysis of Matrix Calculations I, see the course syllabus for what is in this case meant by “foundations”.**Foundations of complex analysis**. Cauchy theorem, residue theorem, conformal mapping, Laplace transform. These topics are taught, for example, in the undergraduate course Introduction to Complex Analysis, see the course syllabus for what is in this case meant by “foundations”.**Foundations of functional analysis and theory of metric spaces**. Banach and Hilbert spaces, operators and functionals, Hahn-Banach theorem, dual spaces, bounded operators, compact operators, theory of distributions. These topics are taught, for example, in the undergraduate course Introduction to Functional Analysis, see the course syllabus for what is in this case meant by “foundations”. (If you have studied the standard bachelor study programme Physics, then Introduction to Functional Analysis will be anyway your compulsory subject in the first year of your master study. The topics listed above are at a sufficient level covered by the undergraduate course Mathematics for Physicists III.)**Foundation of theory of ordinary differential equations**. Existence of solution, maximal solution, systems of linear ordinary differential equations, elements of stability theory. These topics are taught, for example, in the undergraduate course Ordinary Differential Equations, see the course syllabus for what is in this case meant by “foundations”.**Foundations of the classical theory of partial differential equations**. Quasilinear equations of the first order, Laplace equation and heat equation – fundamental solutions and maximum principle, wave equation – fundamental solution, finite differences method. These topics are taught, for example, in the undergraduate course Introduction to Partial Differential Equations, see the course syllabus for what is in this case meant by “foundations”.**Foundations of classical mechanics**. Newton laws of motion, Lagrange equation, Hamilton equations, variational principles, rigid body dynamics. These topics are taught, for example, in the undergraduate course Theoretical Mechanics, see the course syllabus for what is in this case meant by “foundations”.

## Curriculum

The expected length of the study is **two years**. You are expected to choose the topic of your thesis during the first year of your study. There are three sets of courses: **compulsory**, **compulsory elective** and **optional**.

Concerning the **compulsory** courses you **must complete all of them**. The strange term **compulsory elective** means that you **must complete some of these courses**. (The term “some” means that you need certain number of credits from these subjects, see faculty webpages for details.) The choice is yours, but it should reflect the topic of your thesis. Talk to your thesis supervisor if you want to check which courses are suitable for you! The choice of the **optional** courses is completely up to you. Ask your thesis supervisor, classmates or the people responsible for the study programme, see contacts, for a hint.

The (new) study programme Computational and Mathematical Methods in Physics is expected to start in 2015/2016. Detailed curriculum should appear on this page soon, in the mean time please see the faculty webpages.

### Compulsory courses

You **must complete all these courses**. The next to the last column indicates the recommended year of study. Besides the courses listed below you must also complete three “virtual” one semester courses called “Diploma Thesis Seminar”. You will get the credit for this course provided that your thesis supervisor confirms that you work on the thesis.

Course | Annotation | Year | Semester |
---|---|---|---|

Introduction to Functional Analysis | normed linear spaces, Banach and Hilbert spaces, linear functionals and operators, Hahn-Banach theorem, uniform boundedness principle, open mapping theorem, duality and reflexive spaces, Riesz-Schauder theory of compact operators, Fredholm theorems, convolutions and basic properties of L_p, distributions and tempered distributions | 1st | winter |

Partial Differential Equations 1 | notion of the weak solution, Lebesgue spaces, Sobolev spaces, linear elliptic equations | 1st | winter |

Analysis of Matrix Calculations 1 | fundamentals of numerical linear algebra, solving of linear algebraic equations, least squares method, eigenvalue problems | 1st | winter |

Finite Element Method 1 | foundations of the theory of finite element method, linear elliptic equations | 1st | winter |

Numerical Solution of Ordinary Differential Equations | one-step and multi-step methods (algorithms, analysis, convergence), discrete and continuous dynamical systems | 1st | winter |

Computer Simulations in Many-particle Physics | principles of the Monte Carlo and molecular dynamics methods | 1st | winter |

Partial Differential Equations 2 | Bochner spaces, parabolic and linear hyperbolic equations of the second order | 1st | summer |

The lecture was discontinued in 2020, and it was replaced by the redesigned lecture Analysis of Matrix Iterative Methods – Principles and Interconnections, see below. |
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Analysis of Matrix Iterative Methods – Principles and Interconnections | iterative methods, Krylov subspace methods | 2nd | summer |

### Compulsory elective courses

The courses are grouped into five categories — continuum mechanics, molecular dynamics, quantum systems, general relativity and particle physics. Once you have chosen the particular branch of physics you want to specialize in, you should focus on the courses in the corresponding category. You **must complete some of these courses**. (The term “some” means that you need certain number of credits from the given category, see faculty webpages for details.) The choice is yours, but it should reflect the topic of your thesis. Talk to your thesis supervisor if you want to check which courses are suitable for you!

#### Continuum mechanics

Course | Annotation | Year | Semester |
---|---|---|---|

Continuum Mechanics | introduction to continuum mechanics, kinematics, balance equations, linearized elasticity, Navier–Stokes fluid | 1st/2nd | winter |

Mixture Theory | mechanics and thermodynamics of mixtures | 2nd | winter |

Computer Solutions of Problems in Continuum Mechanics | basic commercial/open source software for numerical computation (Matlab, Femlab, FEniCS) and its application to solution of partial differential equations, basic numerical libraries (Blas, Lapack, Petsc), finite element libraries (Feat, Featflow), libraries for parallel computation (MPI, OpenMP) | 1st | summer |

Thermodynamics and Mechanics of non-Newtonian Fluids | non-Newtonian phenomena, constitutive relations for non-Newtonian fluids, thermodynamics of non-Newtonian fluids | 1st/2nd | summer |

Thermodynamics and Mechanics of Solids | mechanics and thermodynamics of nonlinear solids, constitutive relations for non-linear solids | 1st/2nd | summer |

In 2020 the lecture has been renamed to Nonlinear Differential Equations, see below. |
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Nonlinear Differential Equations | nonlinear operators in Banach and Hilbert spaces, monotone, pseudomonotone and potential operators | 2nd | summer |

Numerical Solution of Evolutionary Equations | theoretical and practical aspects of the numerical solution of evolutionary differential equations | 2nd | summer |

Solution of Nonlinear Algebraic Equations The lecture is in the academic year 2020/2021 taught for the last time. Starting from 2021/2022 it will be replaced by the lecture Numerical optimization methods 1, see below. |
theoretical and practical aspects of the numerical solution of nonlinear equations and their systems | 2nd | winter |

Numerical optimization methods 1 | theoretical and practical questions regarding the numerical solution of non-linear equations and minimization of functionals | 2nd | winter |

#### Molecular dynamics

Course | Annotation | Year | Semester |
---|---|---|---|

Methods of Mathematical Statistics | principles of estimation and hypothesis testing | 2nd | summer |

Advanced Simulations in Many-particle Physics | advanced methods of Monte Carlo and molecular dynamics and their applications to various problems (critical phenomena, complex molecules, non-equilibrium phenomena, transport coefficients, kinetic MC, growth processes, optimalization problems, quantum MC, ab initio simulations) | 2nd | summer |

Introduction to Plasma and Computational Physics I | basics of computational physics, introduction to plasma physics | 2nd | winter |

Introduction to Plasma and Computational Physics II | elementary processes in plasma, transport effects in plasma, advanced techniques of particle modelling, particle and fluid modelling in plasma physics and in plasma chemistry | 2nd | summer |

Computer Modelling of Biomolecules | drug design, searching and visualization of biomolecular structures | 2nd | summer |

Thermodynamics and Statistical Physics II | thermodynamic limit, Gibbs paradox, identical particles, quantum statistical ensembles, the classical limit, fluctuation theory, equivalence of statistical ensembles, ideal Bose and Fermi gas, interacting systems (virial expansion, critical phenomena, mean field approximation, the scaling hypothesis), transport phenomena, Boltzmann kinetic equation | 2nd | summer |

#### Quantum systems

Course | Annotation | Year | Semester |
---|---|---|---|

Quantum Theory of Molecules | Born-Oppenheimer and adiabatic approximation, Hückel method, Hartree, Hartree-Fock and Roothaan equations, semiempirical and ab initio methods, correlation energy, symmetry of molecules, intermolecular interactions, polarizability, vibrations of molecules, chemical reactivity | 2nd | summer |

Quantum Scattering Theory | theory of atomic processes with applications in non-relativistic astrophysics and chemical physics, formal collision theory and methods for solving scattering problems, single-particle approach in atomic physics | 2nd | winter |

Theory of Collisions of Atoms and Molecules | advanced theory of atomic processes with applications in non-relativistic astrophysics and chemical physics, introduction to many-particle atomic and molecular theory, computational methods for collision processes and applications to collisions of electrons with atoms and molecules | 2nd | summer |

Group Theory and its Applications in Physics | group theory and representation theory for both finite and continuous (Lie) groups | 2nd | winter |

The lecture was in 2021/2022 replaced by the lecture Quantum Theory II, see below. These lectures are equivalent from the perspective of the study plans. |
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Quantum Theory II | advanced quantum theory; multi-particle systems, scattering | 2nd | summer |

Quantum theory – selected applications Lectures Quantum theory — selected applications a Quantum theory — selected topics are taught alternately. It is possible to attend both lectures. |
first principles calculations for multiparticle systems, basic principles of model construction | 2nd | summer |

Quantum theory — selected topics Lectures Quantum theory — selected applications a Quantum theory — selected topics are taught alternately. It is possible to attend both lectures. |
quantum nonlocality and information, quantum-classical correspondence, quantum statistical physics, parametrically dependent systems | 2nd | summer |

#### Relativistic physics

Course | Annotation | Year | Semester |
---|---|---|---|

General Theory of Relativity | general relativity and its applications in astrophysics and cosmology | 2nd | summer |

Geometrical Methods of Theoretical Physics I | elements of topology, differentiable manifolds, tangent bundles, vector and tensor fields, affine connection, parallel transfer and geodesic curves, torsion and curvature, Riemann and pseudo-Riemann manifolds, Riemann connection, Gauss theory of surfaces, Gauss formula, Lie derivative, Killing vectors, exterior calculus, integration on manifolds, integrable densities | 2nd | winter |

Relativistic Physics I | tensor analysis, curvature and Einstein gravitational law, Schwarzschild solution of Einstein equations, black holes and gravitational collapse, astrophysics of black holes, general relativity in other branches of physics, linearised theory of gravitation, gravitational waves | 2nd | winter |

Foundations of Numerical Study of Spacetimes | basic numerical methods for solving Einstein equations of general relativity | 2nd | winter |

Geometrical Methods of Theoretical Physics II | Riemann geometry in terms of forms, Hodge theory, Lie groups and algebras, fibre bundles, geometry of gauge fields, two-component spinors | 2nd | summer |

Introduction to Analysis on Manifolds | foundations of differential geometry | 2nd | winter |

#### Particle physics

Course | Annotation | Year | Semester |
---|---|---|---|

Elementary Particle Physics | basic properties of elementary particles, models (SU(3), the eightfold way, the quark model, interactions (strong, electromagnetic, weak) and their unification | 2nd | winter |

Fundamentals of Electroweak Theory | phenomenological V-A theory of the weak interactions, idea of unification of weak and electromagnetic interactions, nonabelian gauge fields and the Higgs mechanism, Glashow-Weinberg-Salam standard model of electroweak interactions | 2nd | summer |

Quarks, Partons and Quantum Chromodynamics | quark model of hadrons, parton model and the deep inelastic scattering of leptons off nucleons, synthesis of mentioned models in the framework of the field theory | 2nd | summer |

Particles and Fields I | gauge theories | 2nd | winter |

Selected Topics on Quantum Field Theory I | path integral in quantum mechanics, functional methods and Green functions, Wick rotation and the partition sum, Berezin integral | 2nd | winter |

Software and data processing in particle physics I | overview of software used in particle physics | 1st | summer |

Software and data processing in particle physics II | simulations of particle collisions and thier passage through detector, statistics in evaluation of data acquired in modern detectors | 2nd | winter |

Neural Nets in Particle Physics | neural nets, training, error backpropagation, the Hessian matrix, regularization in neural network, Bayesian neural networks dual use of neural networks – approximations and decision |
2nd | winter |

#### Other

Course | Annotation | Year | Semester |
---|---|---|---|

GENERIC — Non-equilibrium Thermodynamics | thermodynamics in GENERIC (General Equation for Non-equilibrium Reversible-Irreversible Coupling) formalism | 2nd | winter |

Non-equilibrium Thermodynamics of Electrochemistry | equations describing electrochemical processes | 2nd | summer |

Functional Analysis 1 | topological spaces, weak topology, vector integration, spectral theory | 2nd | winter |

Ordinary Differential Equations 2 | dynamical systems, Poincaré-Bendixson theory, Carathéodory theory, optimal control, Pontryagin maximum principle, bifurcations, stable, unstable and central manifolds | 2nd | winter |

Partial Differential Equations 3 | linear and nonlinear evolution equations, semigroup theory, asymptotic behaviour of the solutions to differential equations, pptimal control of evolution equations | 2nd | winter |

Theory of Nanoscale Systems I | independent fermion and boson models, Hartree-Fock theory for fermions and bosons, Brueckner-Hartree-Fock theory, density functional theory, quantum dots in a magnetic field, Monte Carlo methods | 2nd | winter |

Modern Computational Physics I | genetic algorithms, wavelet transform | 2nd | winter |

Modern Computational Physics II | neural networks, advanced algoritms for molecular dynamics | 2nd | summer |

Statistical Methods in High Energy Physics | basic statistical methods frequently used in analysis of experimental data in high energy physics | 2nd | summer |

Theory of Nanoscale Systems I | independent fermion and boson models, Hartree-Fock theory for fermions and bosons, Brueckner-Hartree-Fock theory, density functional theory, quantum dots in a magnetic field, Monte Carlo methods | 2nd | winter |

## State exam

After two years the study is completed by the state exam that has two parts — thesis defense and oral part.

### Thesis defense

The student presents, usually in twenty minutes, the results of his/her thesis. After that he/she must discuss the issues pointed out in the referee reports. (The referee reports are available in the Study Information System beforehand. It is absolutely crucial to be ready to discuss referee’s comments in a highly qualified manner.) At the end the student answers question by the committee members and the audience.

### Oral part

The oral part of state exam consists of answering several question concerning the topics specific for the given study programme. In principle the student will be answering questions regarding the topics studied in the compulsory courses listed in the curriculum above.

The oral part of the state exam proceeds as follows. The student answers, after a preparation, in total six questions regarding

- theory of partial differential equations (one question),
- functional analysis (one question),
- finite element method (one question),
- theory of solution of systems of algebraic equations (one question),

the *remaining two questions are chosen according to student’s specialisation — continuum mechanics, molecular dynamics, quantum systems, general relativity and particle physics*.

Answering one question is expected to take ten minutes. The students is expected to demonstrate deeper insight into the discussed topics and the ability to see the subject matter in a broad context. (The student is not expected to give *extreme* details of each proof. He/she should be able to describe the basic ideas behind the proofs and explain why the definitions/notions/theorems are stated as they are.) Detailed list of the topics for the state exam is listed below. **You can also download the list of topics as a PDF file**.

#### Partial differential equations

The topics are covered in courses Partial differential equations 1 a Partial differential equations 2. It is expected that the student is also familiar with the classical theory of partial differential equations at the level of the course Introduction to partial differential equations. (See Admission.)

##### Sobolev spaces

Weak derivative, definition and basic properties of Sobolev spaces W^{k,p} — reflexivity, separability, density of smooth function, extension operator for W^{1,p} functions and domain with Lipschitz boundary. Theorems concerning the continuous and compact embedding of Sobolev spaces into Lebesgue and Hölder spaces. Definition of the trace operator for functions in Sobolev spaces, trace theorem, inverse trace theorem.

##### Weak solution of linear elliptic partial differential equations in bounded domain

Definition of the weak solution to linear elliptic partial differential equation with various boundary conditions. Existence of a solution via Riesz representation theorem (symmetric operator), via Lax–Milgram lemma and via Galerkin method. Compactness of the solution operator, eigenvalues and eigenvectors. Fredholm alternative and its applications. Maximum principle for the weak solution. W^{2,2} regularity via finite differences technique. Selfadjoint operator, equivalence with the minimization problem for a quadratic functional.

##### Weak solution of nonlinear elliptic partial differential equations in bounded domain

Fundamental of calculus of variations, fundamental theorem of calculus of variations, dual formulation, relation to convexity. Existence and uniqueness of the solution to nonlinear problems via fixed point theorems (nonlinear Lax–Milgram). Existence via Galerkin method and Minty trick — monotone operator and semilinear term.

##### Second order linear parabolic partial differential equations

Bochner spaces and their basic properties, Gelfand triple, Aubin–Lions lemma. Definition of the weak solution. Initial conditions. Existence of a solution via Galerkin method, uniqueness and regularity of the solution (spatial and temporal), smoothing property, maximum principle.

##### Second order linear hyperbolic partial differential equations

Definition of the weak solution. Initial conditions. Existence of a solution via Galerkin method, uniqueness, regularity (spatial and temporal), finite propagation speed.

#### Numerical mathematics

The topics are covered in courses Finite element method and Matrix iterative methods 1. It is expected that the student is also familiar with fundamentals of numerical mathematics at the level of the course Analysis of matrix calculations 1 and Fundamentals of numerical mathematics. (See Admission.)

##### Finite element method for solution of linear elliptic partial differential equations

Galerkin and Ritz methods for solution of abstract linear elliptic equations. Estimate on the discretization error, Céa lemma. Defintion of the abstract finite element, simple examples of finite elements of Lagrange and Hermite type. Approximation theory in Sobolev spaces, approximation properties of polynomial preserving operators. Application of these results to Lagrange and Hermite type finite elements. Rate of convergence of approximate solutions to linear elliptic partial differential equations. Estimate of the rate of convergence in L^{2} norm, Nitsche lemma.

Fundamentals of numerical integration in finite element method.

##### Solution of systems of algebraic equations and eigenvalues computation

Methods for solution of systems of linear algebraic equations and eigenvalues computation. Spectral decomposition of operators and matrices. Invariant subspaces and spectral information, normality. Comparison of direct and iterative methods for solution of systems of linear algebraic equations. Projection process. Description of the convergence of the iterative methods. Relation between iterative methods for linear equations and methods for eigenvalues computation. Comparison of methods for solution of linear and nonlinear systems of algebraic equations. Numerical stability and algebraic error.

#### Functional analysis

The topics are partially covered in course Functional analysis I. Theory of function spaces is also partially discussed in courses Partial differential equations 1 and Partial differential equations 2. It is expected that the student is also familiar with fundamental of functional analysis at the level of the course Inroduction to functional analysis. (See Admission.)

##### Hilbert and Banach spaces

Definition, norm, scalar product, examples. Linear functionals. Hahn-Banach theorem. Dual space, representations of some dual spaces (Hilbert spaces, Lebesgue spaces). Riesz representation theorem. Weak and weak-* topology. Banach-Alaoglu theorem. Weak compactness. Reflexivity.

##### Continuous linear operators

Definition, basic properties, norm, space of linear operators, adjoint operator. Spectrum and its basic properties, spectral radius, Gelfand-Mazur theorem. Compact operators, symmetric operator, selfadjoint operator, closure, closed operator, definition and properties of adjoint operator. Eigenvalues and eigenvectors of symmetric elliptic operators.

##### Fixed point theorems

Banach theorem, Brouwer theorem, Schauder theorem, Schaefer theorem.

##### Integral transformations and fundamentals of the theory of distributions

Definition of Fourier transform in L^{1} and its basic properties, Fourier inversion theorem, Fourier transform of convolution and derivative. Space of test functions, characterization of a distribution, order of a distribution, operations with distributions (derivative, multiplication), Schwarz space and tempered distributions, Fourier transformation for functions in Schwarz space and tempered distributions, its basic properties. Fourier transform in L^{2}.

#### Continuum mechanics

The topics are covered in courses Continuum mechanics, Thermodynamics and mechanics of nonnewtonian fluids and Thermodynamics and mechanics of solids.

##### Kinematics

Description of the motion of continuous media. Deformation of line, surface and volume elements, deformation, deformation gradient, polar decomposition of deformation gradient and its interpretation, right and left Cauchy–Green tensor, Green–Saint-Venant tensor. Rate of deformation of line, surface and volume elements. Velocity, velocity gradient, symmetric velocity gradient, material time derivative. Isochoric deformation. Streamlines and pathlines. Kinematic condition for material surface. Lagrange and Euler description. Compatibility conditions for linearized strain tensor. Isotropic tensor functions, representation theorem for isotropic tensor functions.

##### Dynamics

Balance equations (mass, momentum, angular momentum, total energy) in Euler and Lagrange description. Integral form for the balance equations, localization principle. Cauchy stress tensor, first Piola–Kirchhoff stress tensor, Piola transformation. Balance equations in non-inertial reference frame.

##### Simple constitutive relations

Compressible and incompressible Navier–Stokes–Fourier model (viscous heat conducting fluid), equation of state for ideal gas. Geometric linearization, linearized elasticity. Boundary conditions, displacement and traction boundary conditions.

##### Nonnewtonian fluids

Balance equations in the case of non-newtonian fluids, identification of entropy production. Clausius–Duhem inequality. Assumption on the maximization of the entropy production and its application in the design of mathematical models for fluids, concept of natural configuration. Overview of non-newtonian phenomena — shear dependent viscosity, normal stress differences, activation/deactivation criteria, stress relaxation, non-linear creep. Principle of frame indifference and its consequences, frame indifferent quantities in fluid mechanics, frame indifferent rates. Application of representation theorem for isotropic tensorial functions. Overview of standard models for non-newtonian fluids. Power-law fluids, fluids with pressure dependent viscosity, Bingham type fluids. Viscoelastic fluids and simplified spring-dashpot models. Korteweg fluids.

##### Solids

Principle of frame indifference and its consequences, frame indifferent quantities in solid mechanics. Elastic materials in finite elasticity theory, linearized elasticity. Incompressible materials in finite elasticity and in linearized elasticity. Elastic material as a material that does not produce entropy, relation between the stress tensor and free energy. Hyperelastic materials, examples of hyperelastic materials, behaviour with respect to the determinant of deformation gradient. Variational formulation of the static problem for deformation of hyperelastic solids. Viscoelastic solids — Kelvin–Voigt model — and simplified spring-dashpot models.

#### Molecular dynamics

The topics are covered mainly in courses Simulations in many-particle physics, Advanced Simulations in many-particle physics} and Computer modelling in plasma physics I.It is expected that the student is also familiar with basic concepts of statistical physics at the level of course Thermodynamics and statistical physics II.

##### Basics of statistical physics

Statistical description of thermodynamics. Liouville equation. Basic statistical ensembles, micro canonical, canonical and grand canonical ensemble. Quantum statistical mechanics. Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein distributions. Basic formulas of statistical mechanics for simulations (calculation of thermodynamic variables, ergodic theorem) and role of numerical simulations.

##### Basics of Monte Carlo simulations of physical systems

Basics of the Monte Carlo method (integration using MC method, Markov chains, error estimate of MC integration). Implementation of Monte Carlo step (natural and preferential sampling, Metropolis method). Methods of random number generation. MC simulation of discrete models (Ising model, percolation threshold). MC simulation of simple models of liquids.

##### Basics of molecular dynamics

Principles of the method of molecular dynamics. Equations of motions of the classical many-body system. Interaction potentials and boundary conditions. Simulations in different statistical ensembles — NVE, NVT, NPT and grand canonical.

##### Determination of thermodynamics and structural properties from simulations

Calculation of specific heat and susceptibility using fluctuations. Radial distribution function.

##### Advanced methods of many-particle simulations

Phase transitions and critical phenomena. Phase and reaction equilibria. Simulation of growth processes. Quantum simulations. Calculation of kinetic coefficients. Kinetic Monte Carlo. Determination of energy barriers using molecular statics. Optimization methods.

##### Basics of modelling in plasma physics

Characterization of plasma and types of plasma. Quasineutrality and Debye screening length. Theoretical description of plasma, kinetic description, Boltzmann equation, conservation laws, magnetohydrodynamic description.

#### Quantum systems

The topics are covered mainly in courses Quantum scattering theory, Theory of collisions of atoms and molecules. It is expected that the student is also familiar with quantum mechanics which is covered by any of the courses Quantum mechanics II (NTMF067) or Quantum mechanics II (NJSF095) or Quantum theory II and their first parts.

##### Basics of quantum mechanics

Description of states and observable quantities. Operators, commutation relations. Time dependence in quantum mechanics. Description of measurement. Stationary states and integrals of motion.

##### Solvable systems

Particle in potential well, linear harmonic oscillator, Coulomb field.

##### Angular momentum and spin

Definition of angular momentum, spectrum and eigenfunctions. Addition of angular momenta, Clebsch-Gordan coefficients. Vector and tensor operators, irreducible components and Wigner-Eckart theorem.

##### Basic approximation methods

Variational method and perturbation theory. Many-particle systems: symmetrization postulate, bosons, fermions, Slater determinant, role of spin.

##### Scattering theory

M{\o}ller operators and S-matrix. Scattering cross section. Time independent formulation of scattering, Lippmann-Schwinger equation. S-matrix poles and eigenphases. Basics of multichannel scattering theory.

##### Basic methods of many-particle quantum physics

Mean field approximation, correlation energy and methods for its calculation. Second quantization. Basics of theory of atoms and molecules: electronic structure, vibrational and rotational states of molecules, application of group theory, optical transitions.

##### Methods for scattering calculations

Partial wave expansion. Born series. Variational principles. R-matrix theory.

#### General theory of relativity

The topics are covered in courses General theory of relativity, Relativistic physics I and Foundations of numerical study of spacetimes.

##### The underlying principles of special and general theory of relativity

Spacetime, four-vector formalism, coordinate transformations. Metric, affine connection, covariant derivative. Parallel transport and geodesic equation. Gravitational redshift. Lie derivative and Killing vectors, tensor densities. Integration on manifolds (densities, integral theorems). Spacetime curvature.

##### Einstein law of gravitation and its consequences

Energy-momentum tensor, conservation laws and equations of motion. Einstein gravity field equations. Schwarzschild and Reissner-Nordström metric. Kerr and Kerr-Newman metric.

##### Relativistic astrophysics and cosmology

Relativistic models of stars. Gravitational collapse and black holes. Critical behavior of gravitational collapse. Relativistic cosmology FLRW models.

##### Properties of Einstein equations

Linearized theory of gravitation and gravitational plane-waves. Lagrangian formalism in general relativity, conservation laws. Hamilton formalism in general relativity, initial value problem. Conformal decomposition of constraints, initial-value data. Einstein equations as hyperbolic system of partial differential equations.

#### Particle physics

The basic theoretical information is partially covered by course Elementary particle physics, and practical topics in Software and data processing in particle physics II, Neural nets in particle physics.

##### Basic concepts and methods of quantum field theory

Equations of relativistic quantum mechanics. Quantization of free fields. Interaction of fields, Feynman diagrams.

##### Classification and properties of elementary particles

Leptons, hadrons and force carriers. Spin, parity, charge parity, strangeness, isospin. Conservation laws.

##### Structure of hadrons

Quark model, color charge, partons, distribution functions.

##### Basics of standard model of elementary particles

Electroweak interaction. Higgs mechanism. Quantum chromodynamics.

##### Interaction of particles with environment and particle measurement methods in experiments

Measurement of energy, momentum and time of flight of particles. Identification of particles. Monte Carlo simulation of particle motion in detector.

##### Methods of data analysis in particle physics experiments

Software tools. Selection rules and multivariate analysis. Neural networks.