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This book presents Markov and quantum processes as two sides of a
coin called generated stochastic processes. It deals with quantum
processes as reversible stochastic processes generated by one-step
unitary operators, while Markov processes are irreversible
stochastic processes generated by one-step stochastic operators.
The characteristic feature of quantum processes are oscillations,
interference, lots of stationary states in bounded systems and
possible asymptotic stationary scattering states in open systems,
while the characteristic feature of Markov processes are
relaxations to a single stationary state. Quantum processes apply
to systems where all variables, that control reversibility, are
taken as relevant variables, while Markov processes emerge when
some of those variables cannot be followed and are thus irrelevant
for the dynamic description. Their absence renders the dynamic
irreversible. A further aim is to demonstrate that almost any
subdiscipline of theoretical physics can conceptually be put into
the context of generated stochastic processes. Classical mechanics
and classical field theory are deterministic processes which emerge
when fluctuations in relevant variables are negligible. Quantum
mechanics and quantum field theory consider genuine quantum
processes. Equilibrium and non-equilibrium statistics apply to the
regime where relaxing Markov processes emerge from quantum
processes by omission of a large number of uncontrollable
variables. Systems with many variables often self-organize in such
a way that only a few slow variables can serve as relevant
variables. Symmetries and topological classes are essential in
identifying such relevant variables. The third aim of this book is
to provide conceptually general methods of solutions which can
serve as starting points to find relevant variables as to apply
best-practice approximation methods. Such methods are available
through generating functionals. The potential reader is a graduate
student who has heard already a course in quantum theory and
equilibrium statistical physics including the mathematics of
spectral analysis (eigenvalues, eigenvectors, Fourier and Laplace
transformation). The reader should be open for a unifying look on
several topics.
This book presents Markov and quantum processes as two sides of a
coin called generated stochastic processes. It deals with quantum
processes as reversible stochastic processes generated by one-step
unitary operators, while Markov processes are irreversible
stochastic processes generated by one-step stochastic operators.
The characteristic feature of quantum processes are oscillations,
interference, lots of stationary states in bounded systems and
possible asymptotic stationary scattering states in open systems,
while the characteristic feature of Markov processes are
relaxations to a single stationary state. Quantum processes apply
to systems where all variables, that control reversibility, are
taken as relevant variables, while Markov processes emerge when
some of those variables cannot be followed and are thus irrelevant
for the dynamic description. Their absence renders the dynamic
irreversible. A further aim is to demonstrate that almost any
subdiscipline of theoretical physics can conceptually be put into
the context of generated stochastic processes. Classical mechanics
and classical field theory are deterministic processes which emerge
when fluctuations in relevant variables are negligible. Quantum
mechanics and quantum field theory consider genuine quantum
processes. Equilibrium and non-equilibrium statistics apply to the
regime where relaxing Markov processes emerge from quantum
processes by omission of a large number of uncontrollable
variables. Systems with many variables often self-organize in such
a way that only a few slow variables can serve as relevant
variables. Symmetries and topological classes are essential in
identifying such relevant variables. The third aim of this book is
to provide conceptually general methods of solutions which can
serve as starting points to find relevant variables as to apply
best-practice approximation methods. Such methods are available
through generating functionals. The potential reader is a graduate
student who has heard already a course in quantum theory and
equilibrium statistical physics including the mathematics of
spectral analysis (eigenvalues, eigenvectors, Fourier and Laplace
transformation). The reader should be open for a unifying look on
several topics.
Mathematische Modellierung (MM) dient als Planungswerkzeug fur
Entscheidungen, von denen wir alle zunehmend betroffen sind. UEber
das Vorgehen und die Aussagekraft von MM ist daher ein
grundlegendes Verstandnis auch ohne akademische Vorbildung
wunschenswert. MM benoetigt zwei wesentliche Elemente: Das
Auffinden relevanter Einflussgroessen und das Auffinden einer
kleinschrittigen Regel, die den Ablauf von Szenarien schrittweise
erfasst. Im Text werden prototypische Modelle vorgestellt, die
weite Anwendungsbereiche haben. Die Grenzen vom MM ergeben sich aus
unvermeidbaren Beschrankungen in der Genauigkeit und sie fuhren zu
einer Verzweigung von Modellen in vernetzte Modellsysteme.
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