Verlag des Forschungszentrums Jülich
JUEL-3555
Our treatment consists of three parts. Chapters 2 - 5 are pedagogical in nature.
We introduce some of the basic tools and notions used later and thus provide a
"dictionary" for the correspondence between quantities in probabilistic and
quantum spin language respectively.
The second part is concerned with the theoretical investigation of purely diffusive
systems of hard core particles (Chapters 6 and 7). We use the Bethe ansatz and a
related purely algebraic formulation of the symmetric exclusion process to find a
quasi stationary relaxational behaviour (Chapter 6). Perhaps the most far reaching
result discussed here is the exact derivation of stationary properties of the
asymmetric exclusion process with open boundaries. With the picture of shocks
(domain walls separating regions of low and high density) propagating through
the system we develop an essentially complete understanding of how the interplay
of boundary effects and shock dynamics leads to the boundary induced phase
transitions obtained from the exact solution of the model. These arguments allow
us to predict the phase diagram of quite generic one dimensional driven lattice
gases (Chapter 7).
In the third part we investigate dynamical properties of reaction diffusion
mechanisms (Chapters 8 and 9). We review various approaches for the exact treatment,
particularly the free fermion approach (Chapter 9) which allows for a full
discussion of the dynamics of such systems, even if the initial probability
distributions have a complicated, non translationally invariant structure. The
free fermion property becomes manifest for various models by some suitably chosen
similarity transformation of the stochastic generator and constitutes the common
mathematical ground on which these models stand. Our main message is that all known
equivalences between these models can be generated by two families of similarity
transformations.
Our derivation and the form of these transformations leads us to conjecture
that the models described here are all equivalent single species free fermion
processes with pair interaction between sites. We also discuss some peculiar
non equilibrium phenomena associated with the presence of an external driving
force (Chapter 8).
Schütz, Gunter Markus
Integrable stochastic many-body systems
227 S., 1998
This work investigates classical interacting particle systems for which the
stochastic time evolution is defined by a master equation. We use a quantum
Hamiltonian representation of the master equation to employ the mathematical
tools of manybody quantum mechanics for the derivation of exact results on the
stationary and dynamical properties of these stochastic processes. We consider
mainly integrable lattice reaction diffusion processes of a single species of
particles which hop on a onedimensional lattice and which may undergo simple
nearest neighbour chemical reactions. Integrability in this context refers to
the integrability of the associated manybody quantum Hamiltonian. We treat some
standard models of non equilibrium mechanics, including the exclusion process,
Glauber dynamics and diffusion limited pair annihilation, as well as various
other less well known, but nevertheless rather interesting interacting particle
systems in one dimension. Thus this work attempts to expose the unified mathematical
framework underlying the exact treatment of these systems and to provide insight
in the role of inefficient diffusive mixing for the kinetics of diffusion limited
chemical reactions, in the dynamics of shocks and in other fundamental mechanisms
which determine the behaviour of low dimensional systems far from thermal equilibrium.
Chapter 10 concludes with selected experimental applications of integrable
stochastic processes - gel electrophoresis, kinetics of biopolymerization and
exicton dynamics on polymer chains.
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