Matematics for Particle Systems
Code  Completion  Credits  Range  Language 

01CAS  Z,ZK  3  2P+1C  Czech 
 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 onedimensional stochastic particle systems whose elements are interacting. Especially, systems fulfilling a socalled balance property are analyzed. For such systems, statistical distributions of headways and multiheadways, 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 onedimensional particle systems and their description  headways, multiheadways, interval frequencies and their probabilistic description, balance particle systems, Poisson and quasiPoisson 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, AddisonWesley 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:
 Timetable for winter semester 2023/2024:
 Timetable is not available yet
 Timetable for summer semester 2023/2024:
 Timetable is not available yet
 The course is a part of the following study plans:

 Aplikované matematickostochastické metody (compulsory course in the program)