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The relevant dynamics underlying COVID-19 waves is described from an amplitude space perspective. To this end, the amplitude dynamics of infected populations is considered in different stages of epidemic waves. Eigenvectors and their corresponding amplitudes are derived analytically for low-dimensional models and by means of computational methods for high-dimensional models. It is shown that the amplitudes of all eigenvectors as functions of time can be tracked through the diverse stages of COVID-19 waves featuring jumps at the stage boundaries. In particular, it is shown that under certain circumstances the initial, outbreak stage and the final, subsiding stage of an epidemic wave are primarily determined by the unstable eigenvector of the initial stage and its corresponding remnant vector of the final stage. The corresponding amplitude captures most of the dynamics of the emerging and subsiding epidemics such that the problem at hand effectively becomes one dimensional leading to a dramatic reduction of the complexity of the problem at hand. Explicitly demonstrated for the first-wave COVID-19 epidemics of the year 2020 in the state of New York and Pakistan are given.

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A semi-classical model describing the charge transfer collisions of C60 fullerene with different slow ions has been developed to explain available experimental data. This data reveals multiple Breit-Wigner like peaks in the cross sections, with subsequent peaks of reactive cross sections decreasing in magnitude. Calculations of charge transfer probabilities, quasi-resonant cross sections and cross sections for reactive collisions have been performed using semi-empirical interaction potentials between fullerenes and ion projectiles. All computations have been carried out with realistic wave functions for C60's valence electrons derived from the simplified jellium model. The quality of these electron wave functions have been successfully verified by comparing theoretical calculations and experimental data on the small angle cross sections of resonant C60+C60+ collisions. Using the semi-empirical potentials to describe resonant scattering phenomena in C60 collisions with ions and Landau-Zener charge transfer theory, we calculated theoretical cross sections for various C60 charge transfer and fragmentation reactions which agree with experiments.

In this study, molecular dynamics simulations were conducted to investigate the relaxation of the internal energy in nano-sized particles and its impact on the nucleation of atomic clusters. Quantummechanical potentials were utilized to analyze the growth and collision relaxation of the internal energy of ArnH + clusters in a metastable Ar gas. The results revealed that small nano-clusters are formed in highly excited rotational-vibrational states, and the relaxation of internal energy and growth of these nascent clusters are concurrent processes with a strong mutual influence. Under nonequilibrium growth conditions, the relaxation of internal energy can delay the cluster growth process. The rates of cluster growth and internal energy relaxation were found to be influenced by energytransfer collisions between cluster particles and free Ar atoms of the bath gas. Furthermore, the non-equilibrium growth and internal energy relaxation of small nano-clusters were found to depend on the structure of the cluster's atomic shells. An ensemble of molecular dynamics simulations were conducted to investigate the growth, time-evolution of kinetic and total energies of ArnH + clusters with specified n ≤ 11, and the results were explained by collisional relaxation processes described by the Boltzmann equation. Finally, the general relationship between the rates of internal energy relaxation and non-equilibrium growth of nano-particles is discussed.