Matematics for Particle Systems

Garant předmětu:

Milan Krbálek

Lecturer:

Milan Krbálek

Tutor:

Milan Krbálek

Supervisor:

Department of Mathematics

Synopsis:

The aim of the course is to study general mathematical properties of one-dimensional stochastic particle systems whose elements are interacting. Especially, systems fulfilling a so-called balance property are analyzed. For such systems, statistical distributions of headways and multi-headways, interval frequencies, and associate statistical rigidity are examined.

Requirements:

Syllabus of lectures:

1. Selected asymptotic methods and special functions – rough leading, approximation of Laplacian integrals, saddle point approximation, Bessel function, Bessel equations, and Macdonald’s functions.

2. Special classes of densities and convolution - general definition of density and its description, balanced densities, moments and their properties, moment code, convolution and properties.

3. Laplace transform for the class of balanced densities - general introduction, properties, Laplace's decalogue, Lerch's theorem, theorem on the analyticallity of the balanced density image, theorem on inversion,

4. Stochastic one-dimensional particle systems and their description - headways, multi-headways, interval frequencies and their probabilistic description, balance particle systems, Poisson and quasi-Poisson systems, theory of statistical rigidity.

5. Dyson gases and their linkage to balance particle systems.

Syllabus of tutorials:

Study Objective:

Study materials:

Key references:

[1] Kollert, O., Krbálek, M., Hobza, T., Krbálková, M.: Statistical rigidity of vehicular streams - theory versus reality, J.Phys.Commun. 3, 035020, 2019

[2] Krbálek, M., Šleis, J.: Vehicular headways on signalized intersections: theory, models, and reality, J. Phys. A: Math. Theor. 48, 015101, 2015

[3] Dingle, R. B.: Asymptotic Expansions: Their Derivation and Interpretation, Academic Press, 1973.

Recommended references:

[4] Schwartz, L.: Mathematics for the physical sciences, Hermann, Addison-Wesley Pub. Co., Paris 1966

[5] Abramowitz, M., Stegun. I. A.: Handbook of mathematical functions, National Bureau of Standards, Applied Mathematics Series, 55, 1964

Media and tools: MATLAB. A set of lecture notes are available at http://www.krbalek.cz/For_students/mcs/mcs.html

Note:

Time-table for winter semester 2023/2024:

Time-table is not available yet

Time-table for summer semester 2023/2024:

Time-table is not available yet

The course is a part of the following study plans:

• Aplikované matematicko-stochastické metody (compulsory course in the program)