Courses: BSc, MSc

Description:

Langton's Ant is a particular cellular automaton involing a walker (called an ant in the original proposal) on a square lattice, making a mark at every step. The deterministic rule which is used to advance the walker at each step can result in both ballistic propagation and random-walk like features. The quantum walk is a generalizations of the random walk. The employed dynamics is deterministic, and yields a ballistic spreading of the quantum walker. The random walk is recovered as the result of some decoherence effect, e.g. external noise. The goal of the project is to review the quantum mechanical generalization of the Langton's Ant problem based on the notion of quantum walks and an additional memory, then next relate it to other quantum walks with memories, and numerically study certain particular case.

Literature:A. Gajardo, A. Moreira, E. Goles: Complexity of Langton's ant. Discr. Appl. Math. 117, 41–50 (2002)

J. Kempe: Quantum random walks - an introductory overview, Contemp. Phys. 44, 307 (2003)

A. Gábris, B. Kollár, J. Novotý, I. Jex, A. Delgado, unpublished

D. Li, M. Mc Gettrick, F. Gao, J. Xu, Q.-Y. Wen: Generic Quantum Walks with Memory. arXiv:1508.07674

Name of the supervisor: Aurél Gábris, PhD / prof. Ing. Igor Jex, DrSc.

Supervisor's office: Břehová 13b

E-mail of the supervisor: gabris.aurel@fjfi.cvut.cz

Language: English/Czech

Date: 13/09/2015

Mathematical formalism , Newtonian mechanics , Lagrange function, constrains, generalised coordinates , Symmetries of the Lagrange function and conservation laws , Virial, Two body problem, Elastic scattering, Oscillations of coupled systems , Dynamics of rigid bodies , Classification of physical principles, the static equilibrium, Differential principles (d´Alembert, Jourdain, Gauss, Hertz) , Integral principles (Hamilton, Jacobi, Maupertius).

Key references:

[1] I.Štoll, J. Tolar, Theoretical Physics, ČVUT 2002 (in Czech)

Recommended references:

[2] V. Trkal, Mechanics of Mass Points and Solid Bodies, ČSAV Praha 1956 (in Czech)

[3] L.D. Landau, E.M. Lifšic, Teoreticeskaja fizika I, FIZMATGIZ Moskva, 2002 (in Russian)

Lessons for the third year of M.Sc. studies, field of study Mathematical Physics. The course concentrates on some advanced topics of statistical mechanics not discussed in the basic course on thermodynamics and statistical physics. Question concerning density matrices, the behaviours of nonideal gases and its macroscopic description, microscopic description of phase transitions, the role of fluctuations are addressed in detail.

Elements of thermodynamics, Bacis of statistical physics, The density matrix, Nonideal gases, Fermi gas, Fluctuations, Microscopic models and phase transitions, Ising model, Elements of kinetic theory, Transport phenomena.

Key references:

[1] R. Balian, From microphysics to macrophysics, Springer, New York, 1991

Recommended references:

[2] J. Kvasnica, Termodynamics, SNTL Praha, 1965 (in czech)

[3] L J. Kvasnica, Statistical physics, Academia Praha, 2003 (in czech)

Elective lessons for M.Sc. students, field of study Mathematical Physics . The analysis of information processes from the point of view of quantum theory lead to the formation of a new field -- quantum information processes. The cources discusses some of the basic concepts of quantum information processing like quantum algorithms, error correction, quantum cryptography.

Elements of quantum theory, Entanglement and density matrices, Models of computation, Complexity, Quantum gates and circuits, Quantum Fourier tranformation, Search algorithm, Realization of quantum computation, Error correction, Quantum operations, Cryptography, Quantum cryptography.

Key references:

[1] M. A. Nielsen, I. L. Chuang, Quantum computation and quantum informaction, Cambridge Univ. Press, 2002.

Recommended references:

[2] M. Dušek, Conceptual question of quantum theory, Olomouc, 2002(in czech)

[3] G. Alber, Quantum Information, Springer, Berlin 2002

Elective lessons for M.Sc. students, field of study Mathematical Physics. The study of instabilities in physics enables a unified description and understanding of many processes and effects in physical, chemical and living systems. The course focuses on the mathematical desription methods and on processes of selforganization in systems like the laser, morphogenesis (chemical reactions), changes in societies.

Mathematical description of instabilities, Laser theory, Selforganization in chemistry, Morphogenesis, Dynamics of sociological systems, Stochastic processes, Instabilities in economy , Selforganization in cosmology, Chaotic dynamics.

Key references:

[1] H. Haken, Synergetics, Springer, Berlin, 1970

Recommended references:

[2] W. Ebeling, R. Feistel, Physics of Selforganization,
Akademie Verlag, Berlin 1986 (in german)

[3] L. J. Krempaský, Synergetics, Vydavatelstvo SAV, Bratislava 1988

[4] G. Nicolis, C. Nicolis, Foundations of complex systems, World Scientific, 2007

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