Invariance principle for Mott variable range hopping and other walks on point processes

We consider a random walk on a homogeneous Poisson point process with energy marks. The jump rates decay exponentially in the A-power of the jump length and depend on the energy marks via a Boltzmann--like factor. The case A=1 corresponds to the phonon-induced Mott variable range hopping in disordered solids in the regime of strong Anderson localization. We prove that for almost every realization of the marked process, the diffusively rescaled random walk, with arbitrary start point, converges to a Brownian motion whose diffusion matrix is positive definite, and independent of the environment. Finally, we extend the above result to other point processes including diluted lattices.Comment: 47 pages, minor corrections, submitte

Learning From Multi-Frame Data

Multi-frame data-driven methods bear the promise that aggregating multiple observations leads to better estimates of target quantities than a single (still) observation. This thesis examines how data-driven approaches such as deep neural networks should be constructed to improve over single-frame-based counterparts. Besides algorithmic changes, as for example in the design of artificial neural network architectures or the algorithm itself, such an examination is inextricably linked with the consideration of the synthesis of synthetic training data in meaningful size (even if no annotations are available) and quality (if real ground-truth acquisition is not possible), which capture all temporal effects with high fidelity. We start with the introduction of a new algorithm to accelerate a nonparametric learning algorithm by using a GPU adapted implementation to search for the nearest neighbor. While the approaches known so far are clearly surpassed, this empirically reveals that the data generated can be managed within a reasonable time and that several inputs can be processed in parallel even under hardware restrictions. Based on a learning-based solution, we introduce a novel training protocol to bridge the need for carefully curated training data and demonstrate better performance and robustness than a non-parametric search for the nearest neighbor via temporal video alignments. Effective learning in the absence of labels is required when dealing with larger amounts of data that are easy to capture but not feasible or at least costly to label. In addition, we show new ways to generate plausible and realistic synthesized data and their inevitability when it comes to closing the gap to expensive and almost infeasible real-world acquisition. These eventually achieve state-of-the-art results in classical image processing tasks such as reflection removal and video deblurring

Nano-crystalline inclusions as a low-pass filter for thermal transport in a-Si

We use atomistic simulations to study the resonant acoustic modes and compare different calculations of the acoustic mean-free path in amorphous systems with nanometric crystalline spherical inclusions. We show that the resonant acoustic properties are not a simple combination of the vibrations in the inclusions and in the amorphous matrix. The presence of the inclusion affects the transport properties mainly in the frequency range separating simple scattering from multiple scattering processes. However, propagation of acoustic wavepackets is spatially heterogeneous and shows that the amorphous/crystalline interface acts as a low energy pass filter slowing down the high kinetic energy motion whatever the vibration frequency. These heterogeneities cannot be catched by the mean free path, but still they must play an important role in thermal transport, thus raising the question of the correct modeling of thermal transport in composite systems

RKKY Interactions in Graphene: Dependence on Disorder and Gate Voltage

We report the dependence of Ruderman-Kittel-Kasuya-Yoshida\,(RKKY) interaction on nonmagmetic disorder and gate voltage in grapheme. First the semiclassical method is employed to reserve the expression for RKKY interaction in clean graphene. Due to the pseudogap at Dirac point, the RKKY coupling in undoped grapheme is found to be proportional to $1/R^3$. Next, we investigate how the RKKY interaction depends on nonmagnetic disorder strength and gate voltage by studying numerically the Anderson tight-binding model on a honeycomb lattice. We observe that the RKKY interaction along the armchair direction is more robust to nonmagnetic disorder than in other directions. This effect can be explained semiclassically: The presence of multiple shortest paths between two lattice sites in the armchair directions is found to be responsible for the reduceddisorder sensitivity. We also present the distribution of the RKKY interaction for the zigzag and armchair directions. We identify three different shapes of the distributions which are repeated periodically along the zigzag direction, while only one kind, and more narrow distribution, is observed along the armchair direction. Moreover, we find that the distribution of amplitudes of the RKKY interaction crosses over from a non-Gaussian shape with very long tails to a completely log-normal distribution when increasing the nonmagnetic disorder strength. The width of the log-normal distribution is found to linearly increase with the strength of disorder, in agreement with analytical predictions. At finite gate voltage near the Dirac point, Friedel oscillation appears in addition to the oscillation from the interference between two Dirac points. This results in a beating pattern. We study how these beating patterns are effected by the nonmagnetic disorder in doped graphene

Charge transport through bio-molecular wires in a solvent: Bridging molecular dynamics and model Hamiltonian approaches

We present a hybrid method based on a combination of quantum/classical molecular dynamics (MD) simulations and a mod el Hamiltonian approach to describe charge transport through bio-molecular wires with variable lengths in presence o f a solvent. The core of our approach consists in a mapping of the bio-molecular electronic structure, as obtained f rom density-functional based tight-binding calculations of molecular structures along MD trajectories, onto a low di mensional model Hamiltonian including the coupling to a dissipative bosonic environment. The latter encodes fluctuat ion effects arising from the solvent and from the molecular conformational dynamics. We apply this approach to the c ase of pG-pC and pA-pT DNA oligomers as paradigmatic cases and show that the DNA conformational fluctuations are essential in determining and supporting charge transport

Accumulative reservoir construction: Bridging continuously relaxed and periodically refreshed extended reservoirs

The simulation of open many-body quantum systems is challenging, requiring methods to both handle exponentially large Hilbert spaces and represent the influence of (infinite) particle and energy reservoirs. These two requirements are at odds with each other: Larger collections of modes can increase the fidelity of the reservoir representation but come at a substantial computational cost when included in numerical many-body techniques. An increasingly utilized and natural approach to control the growth of the reservoir is to cast a finite set of reservoir modes themselves as an open quantum system. There are, though, many routes to do so. Here, we introduce an accumulative reservoir construction -- an ARC -- that employs a series of partial refreshes of the extended reservoirs. Through this series, the representation accumulates the character of an infinite reservoir. This provides a unified framework for both continuous (Lindblad) relaxation and a recently introduced periodically refresh approach (i.e., discrete resets of the reservoir modes to equilibrium). In the context of quantum transport, we show that the phase space for physical behavior separates into discrete and continuous relaxation regimes with the boundary between them set by natural, physical timescales. Both of these regimes ``turnover'' into regions of over- and under-damped coherence in a way reminiscent of Kramers' crossover. We examine how the range of behavior impacts errors and the computational cost, including within tensor networks. These results provide the first comparison of distinct extended reservoir approaches, showing that they have different scaling of error versus cost (with a bridging ARC regime decaying fastest). Exploiting the enhanced scaling, though, will be challenging, as it comes with a substantial increase in (operator space) entanglement entropy.Comment: 26 pages, 18 figure