Why writing a book about a specialized task of the large topic
of complex systems? And who will read it? The answer is simple: The
fascination for a didactically valuable point of view, the elegance
of a closed concept and the lack of a comprehensive
disquisition.
The fascinating part is that field equations can have localized
solutions exhibiting the typical characteristics of particles.
Regarding the field equations this book focuses on, the field
phenomenon of localized solutions can be described in the context
of a particle formalism, which leads to a set of ordinary
differential equations covering the time evolution of the position
and the velocity of each particle. Moreover, starting from these
particle dynamics and making the transition to many body systems,
one considers typical phenomena of many body systems as shock waves
and phase transitions, which themselves can be described as field
phenomena. Such transitions between different level of modelling
are well known from conservative systems, where localized solutions
of quantum field theory lead to the mechanisms of elementary
particle interaction and from this to field equations describing
the properties of matter. However, in dissipative systems such
transitions have not been considered yet, which is adjusted by the
presented book. The elegance of a closed concept starts with the
observation of self-organized current filaments in a semiconductor
gas discharge system. These filaments move on random paths and
exhibit certain particle features like scattering or the formation
of bound states. Neither the reasons for the propagation of the
filaments nor the laws of the interaction between the filaments can
be registered by direct observations. Therefore a model is
established, which is phenomenological in the first instance due to
the complexity of the experimental system. This model allows to
understand the existence of localized structures, their mechanisms
of movement, and their interaction, at least, on a qualitative
level. But this model is also the starting point for developing a
data analysis method that enables the detection of movement and
interaction mechanisms of the investigated localized solutions. The
topic is rounded of by applying the data analysis to real
experimental data and comparing the experimental observations to
the predictions of the model.
A comprehensive publication covering the interesting topic of
localized solutions in reaction diffusion systems in its width and
its relation to the well known phenomena of spirals and patterns
does not yet exist, and this is the third reason for writing this
book. Although the book focuses on a specific experimental system
the model equations are as simple as possible so that the discussed
methods should be adaptable to a large class of systems showing
particle-like structures.
Therefore, this book should attract not only the experienced
scientist, who is interested in self-organization phenomena, but
also the student, who would like to understand the investigation of
a complex system on the basis of a continuous description.
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