Vacuum Simulations in High Energy Accelerators and Distribution Properties of Continuous and Discrete Particle Motions

The underlying thesis on mathematical simulation methods in application and theory is structured into three parts. The first part sets up a mathematical model capable of predicting the performance and operation of an accelerator’s vacuum system based on analytical methods. A coupled species-balance equation system describes the distribution of the gas dynamics in an ultra-high vacuum system considering impacts of conductance limitations, beam induced effects (ion-, electron-, and photon-induced de- sorption), thermal outgassing and sticking probabilities of the chamber materials. A new solving algorithm based on sparse matrix representations, is introduced and presents a closed form solution of the equation system. The model is implemented in a Python environment, named PyVasco, and is supported by a graphical user interface to make it easy available for everyone. A sensitivity analysis, a cross-check with the Test-Particle Monte Carlo simulation program Molflow+ and a comparison of the simulation results to readings of the Large Hadron Colliders (LHC) pressure gauges validate the code. The computation of density profiles considering several effects (as men- tioned above) is performed within a short computation time for indefinitely long vacuum systems. This is in particular interesting for the design of a stable vacuum system for new large accelerat- ors like the Future Circular Colliders (FCC) with 100 km in circumference. A simulation of the FCC is shown at the end of this part. Additionally, PyVasco was presented twice at international conferences in Rome and Berlin and has been submitted in July with the title “Analytical vacuum simulations in high energy accelerators for future machines based on the LHC performance” to the Journal “Physical Review Accelerator and Beams”. The second and third part of the thesis study properties of quasi-Monte Carlo (QMC) methods in the scope of the special research project “Quasi-Monte Carlo methods: Theory and Applications”. Instead of solving a complex integral analytically, its value is approximated by function evaluation at specific points. The choice of a good point set is critical for a good result. It turned out that continuous curves provide a good tool to define these point sets. So called “bounded remainder sets” (BRS) define a measure for the quality of the uniform distribution of a curve in the unit- square. The trajectory of a billiard path with an irrational slope is especially well distributed. Certain criteria to the BRS are defined and analysed in regard to the distribution error. The idea of the proofs is based on Diophantine approximations of irrational numbers and on the unfolding technique of the billiard path to a straight line in the plane. New results of the BRS for the billiard path are reported to the “Journal of Uniform Distribution”. The third part analyses the distribution of the energy levels of quantum systems. It was stated that the eigenvalues of the energy spectra for almost all integrable quantum systems are uncor- related and Poisson distributed. The harmonic oscillator presents already one counter example to this assertion. The particle in a box on the other hand obtains these properties. This thesis formulates a general statement that describes under which conditions the eigenvalues do not follow the poissonian property. The concept of the proofs is based on the analysis of the pair correlations of sequences. The former particle physicist Ian Sloan also exposed this topic and he became spe- cialized as a skilled mathematician in this field. To honour his achievements a Festschrift for his 80th birthday is written and the results of the work of this thesis are published there. The book will appear in 2018.