164 research outputs found
Relation Between Bulk and Interface Descriptions of Alloy Solidification
From a simple bulk model for the one-dimensional steady-state solidification
of a dilute binary alloy we derive an interface description, allowing arbitrary
values of the growth velocity. Our derivation leads to exact expressions for
the fluxes and forces at the interface and for the set of Onsager coefficients.
We, moreover, discover a continuous symmetry, which appears in the low-velocity
regime, and there deletes the Onsager sign and symmetry properties. An example
is the generation of the sometimes negative friction coefficient in the
crystallization flux-force relation
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Quantum Stochastic Processes and Quantum Many-Body Physics
This dissertation investigates the theory of quantum stochastic processes and its applications in quantum many-body physics.
The main goal is to analyse complexity-theoretic aspects of both static and dynamic properties of physical systems modelled by quantum stochastic processes.
The thesis consists of two parts: the first one addresses the computational complexity of certain quantum and classical divisibility questions, whereas the second one addresses the topic of Hamiltonian complexity theory.
In the divisibility part, we discuss the question whether one can efficiently sub-divide a map describing the evolution of a system in a noisy environment, i.e. a CPTP- or stochastic map for quantum and classical processes, respectively, and we prove that taking the nth root of a CPTP or stochastic map is an NP-complete problem.
Furthermore, we show that answering the question whether one can divide up a random variable into a sum of iid random variables , i.e. , is poly-time computable; relaxing the iid condition renders the problem NP-hard.
In the local Hamiltonian part, we study computation embedded into the ground state of a many-body quantum system, going beyond "history state" constructions with a linear clock.
We first develop a series of mathematical techniques which allow us to study the energy spectrum of the resulting Hamiltonian, and extend classical string rewriting to the quantum setting.
This allows us to construct the most physically-realistic QMAEXP-complete instances for the LOCAL HAMILTONIAN problem (i.e. the question of estimating the ground state energy of a quantum many-body system) known to date, both in one- and three dimensions.
Furthermore, we study weighted versions of linear history state constructions, allowing us to obtain tight lower and upper bounds on the promise gap of the LOCAL HAMILTONIAN problem in various cases.
We finally study a classical embedding of a Busy Beaver Turing Machine into a low-dimensional lattice spin model, which allows us to dictate a transition from a purely classical phase to a Toric Code phase at arbitrarily large and potentially even uncomputable system sizes
Diffusion-Induced Oscillations of Extended Defects
From a simple model for the driven motion of a planar interface under the
influence of a diffusion field we derive a damped nonlinear oscillator equation
for the interface position. Inside an unstable regime, where the damping term
is negative, we find limit-cycle solutions, describing an oscillatory
propagation of the interface. In case of a growing solidification front this
offers a transparent scenario for the formation of solute bands in binary
alloys, and, taking into account the Mullins-Sekerka instability, of banded
structures
Capillary-Wave Description of Rapid Directional Solidification
A recently introduced capillary-wave description of binary-alloy
solidification is generalized to include the procedure of directional
solidification. For a class of model systems a universal dispersion relation of
the unstable eigenmodes of a planar steady-state solidification front is
derived, which readjusts previously known stability considerations. We,
moreover, establish a differential equation for oscillatory motions of a planar
interface that offers a limit-cycle scenario for the formation of solute bands,
and, taking into account the Mullins-Sekerka instability, of banded structures
Capillary-Wave Model for the Solidification of Dilute Binary Alloys
Starting from a phase-field description of the isothermal solidification of a
dilute binary alloy, we establish a model where capillary waves of the
solidification front interact with the diffusive concentration field of the
solute. The model does not rely on the sharp-interface assumption, and includes
non-equilibrium effects, relevant in the rapid-growth regime. In many
applications it can be evaluated analytically, culminating in the appearance of
an instability which, interfering with the Mullins-Sekerka instability, is
similar to that, found by Cahn in grain-boundary motion.Comment: 17 pages, 12 figure
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