This book is based on my research work I did between 1986 and 1992 at the Institut
fur Theoretische Physik und Synergetik, Universitiit Stuttgart. It might be of interest
to all students and scientists who are interested in modern mathematical physics.
It deals with one of the essential topics of modern physical research, namely t.he
problem of universal aspects in statistical physics, and in doing so multi-component
systems, like thermodynamic systems, laser systems and high-dimensional quantum
systems, are considered. Synergetic aspects of multi-component systems will
be discussed. Remarks to biological systems will be made, in particular remarks to
the problem of EEG analysis. Non-linear aspects will be discussed. The problem
of a macroscopic access to multi-component systems was an essential part of my
research work. In this context it has to be emphasized that in this book the concept
of hyper-surface equations will be introduced, which is a totally analytical strategy
to determine distribution function parameters in an exact way by using only given
measurement quantities. This concept can be used to determine both distribution
functions of macroscopic systems and path integrals of quantum systems. In this
context the property self-similarity will playa crucial role. A special extreme principle,
the maximum information entropy principle, can be taken as a basis to introduce
this concept. This principle will be discussed. In order to determine the dynamic
behavior of multi-component systems, special evolution equations are needed. Such
evolution equations will be discussed. In particular, equations of the Langevin type,
the Fokker-Planck type and the Schrodinger type will be considered, and it will be
shown that an elementary access to both equations of the Fokker-Planck and the
Schrodinger type is possible. A special chapter will consider quantum system theory,
in particular the concept of Feynman path integrals. In this context a general concept
to calculate Feynman kernels will be introduced. An additional possibility to
describe the behavior of multi-component systems is given by information theory.
The problem of description of such systems with the tools of information theory
will also playa crucial role in this book. It will be discussed that the concept of
information theory allows a universal access to arbitrary multi-component systems.
Additionally, basics of a relativistic system theory will be considered. In this context
a connection of quantum systems and cosmological systems will be worked out.
Then it will be shown that a universal access to all physical systems discussed in
this book is possible by introducing the notion Riemann universe.