arXiv subjects

A. Arnold

Publications and source records attributed to A. Arnold.

12 recordsLinked to original sources

arXiv2025

Long- and short-time behavior of hypocoercive evolution equations via modal decompositions

The long- and short-time behavior of solutions to dissipative evolution equations is studied by applying the concept of hypocoercivity. Aiming at partial differential equations that allow for a modal decomposition, we compute estimates that are uniform with respect to all modes. While the special example of the kinetic Lorentz equation was treated in previous work of the authors, that analysis is generalized here to general evolution equations having a scaled family of generators.

math.DS↗

arXiv2022

The application of encoder-decoder neural networks in high accuracy and efficiency slit-scan emittance measurements

A superconducting radio-frequency (SRF) photo injector is in operation at the electron linac for beams with high brilliance and low emittance (ELBE) radiation center and generates continuous wave (CW) electron beams with high average current and high brightness for user operation since 2018. The speed of emittance measurement at the SRF gun beamline can be increased by improving the slit-scan system, thus the measurement time for one phase space mapping can be shortened from about 15 minutes to 90 seconds. A parallel algorithm and machine learning have been used to reduce the beamlet image noise. In order to estimate the uncertainty in the calculation of normalized emittance, we analyze the main error contributions such as slit position uncertainty, image noise, space charge effects and energy measurement inaccuracy.

physics.acc-ph↗

arXiv2020

The algebra of complete binary trees is affine complete

A function on an algebra is congruence preserving if, for any congruence, it maps pairs of congruent elements onto pairs of congruent elements. We show that on the algebra of complete binary trees whose leaves are labeled by letters of an alphabet containing at least three letters a function is congruence preserving if and only if it is polynomial. This exhibits an example of a non commutative and non associative affine complete algebra. As far as we know, it is the first example of such an algebra.

math.CO↗

arXiv2019

WKB-method for the 1D Schrödinger equation in the semi-classical limit: enhanced phase treatment

This paper is concerned with the efficient numerical computation of solutions to the 1D stationary Schrödinger equation in the semiclassical limit in the highly oscillatory regime. A previous approach to this problem based on explicitly incorporating the leading terms of the WKB approximation is enhanced in two ways: first a refined error analysis for the method is presented for a not explicitly known WKB phase, and secondly the phase and its derivatives will be computed with spectral methods. The efficiency of the approach is illustrated for several examples.

math.NA↗

arXiv2011

2D versus 3D Freezing of a Lennard-Jones Fluid in a Slit Pore

We present a computer simulation study of a (6,12)-Lennard-Jones fluid confined to a slit pore, formed by two uniform planes. These interact via a (3,9)-Lennard-Jones potential with the fluid particles. When the fluid approaches the liquid-to-solid transition we first observe layering parallel to the walls. In order to investigate the nature of the freezing transition, we performed a detailed analysis of the bond-orientational order parameter in the layers. Changing the distance between the slit wall, we studied howthe behaviour changes from a quasi-2d crystallization without hysteresis and an intermediate hexatic phase to a standard 3d nucleation.

cond-mat.soft↗

arXiv2011

How Close to Two Dimensions Does a Lennard-Jones System Need to Be to Produce a Hexatic Phase?

We report on a computer simulation study of a Lennard-Jones liquid confined in a narrow slit pore with tunable attractive walls. In order to investigate how freezing in this system occurs, we perform an analysis using different order parameters. Although some of the parameters indicate that the system goes through a hexatic phase, other parameters do not. This shows that to be certain whether a system has a hexatic phase, one needs to study not only a large system, but also several order parameters to check all necessary properties. We find that the Binder cumulant is the most reliable one to prove the existence of a hexatic phase. We observe an intermediate hexatic phase only in a monolayer of particles confined such that the fluctuations in the positions perpendicular to the walls are less then 0.15 particle diameters, i. e. if the system is practically perfectly 2d.

cond-mat.soft↗

arXiv2010

Elastic properties of graphene flakes: boundary effects and lattice vibrations

We present a calculation of the free energy, the surface free energy and the elastic constants ("Lam'e parameters" i.e, Poisson ratio, Young's modulus) of graphene flakes on the level of the density functional theory employing different standard functionals. We observe that the Lam'e parameters in small flakes can differ from the bulk values by 30% for hydrogenated zig-zag edges. The change results from the edge of the flake that compresses the interior. When including the vibrational zero point motion, we detect a decrease in the bending rigidity by ~26%. This correction is depending on the flake size, N, because the vibrational frequencies flow with growing N due to the release of the edge induced compression. We calculate Grueneisen parameters and find good agreement with previous authors.

cond-mat.mes-hall↗

arXiv2009

2D versus 3D Freezing of a Lennard-Jones Fluid in a Slit Pore: A Molecular Dynamics Study

We present a computer simulation study of a (6,12)-Lennard-Jones fluid confined to a slit pore, formed by two uniform planes. These interact via (3,9)-Lennard-Jones potential with the fluid particles. When the fluid approaches the liquid-to-solid transition we first observe layering parallel to the walls. In order to investigate the nature of the freezing transition we performed a detailed analysis of the bond-orientational order parameter in the layers. We found no signs of hexatic order which would indicate a melting scenario of the Kosterlitz-Thouless type. An analysis of the mean-square displacement shows that the particles can easily move between the layers, making the crystallization a 3d-like process. This is consistent with the fact that we observe a considerable hysteresis in the heating-freezing curves, showing that the crystallization transition proceeds as an activated process.

cond-mat.soft↗

arXiv2007

Harvesting graphics power for MD simulations

We discuss an implementation of molecular dynamics (MD) simulations on a graphic processing unit (GPU) in the NVIDIA CUDA language. We tested our code on a modern GPU, the NVIDIA GeForce 8800 GTX. Results for two MD algorithms suitable for short-ranged and long-ranged interactions, and a congruential shift random number generator are presented. The performance of the GPU's is compared to their main processor counterpart. We achieve speedups of up to 80, 40 and 150 fold, respectively. With newest generation of GPU's one can run standard MD simulations at 10^7 flops/$.

cond-mat.other↗

arXiv2007

Unexpected relaxation dynamics of a self-avoiding polymer in cylindrical confinement

We report extensive simulations of the relaxation dynamics of a self-avoiding polymer confined inside a cylindrical pore. In particular, we concentrate on examining how confinement influences the scaling behavior of the global relaxation time of the chain, t, with the chain length N and pore diameter D. An earlier scaling analysis based on the de Gennes blob picture led to t ~ N^2D^(1/3). Our numerical effort that combines molecular dynamics and Monte Carlo simulations, however, consistently produces different t-results for N up to 2000. We argue that the previous scaling prediction is only asymptotically valid in the limit N >> D^(5/3) >> 1, which is currently inaccessible to computer simulations and, more interestingly, is also difficult to reach in experiments. Our results are thus relevant for the interpretation of recent experiments with DNA in nano- and micro-channels.

cond-mat.soft↗

arXiv2006

Molecular Conductance from Ab Initio Calculations: Self Energies and Absorbing Boundary Conditions

Calculating an exact self energy for ab initio transport calculations relevant to ``Molecular Electronics'' can be troublesome. Errors or insufficient approximations made at this step are often the reason why many molecular transport studies become inconclusive. We propose a simple and efficient approximation scheme, that follows from interpreting the self energy as an absorbing boundary condition of an effective Schroedinger equation. In order to explain the basic idea, a broad introduction into the physics incorporated in these self energies is given. The method is further illustrated using a tight binding wire as a toy model. Finally, also more realistic applications for transport calculations based on the density functional theory are included.

cond-mat.mes-hall↗

arXiv2003

On global classical solutions of the time-dependent von Neumann equation for Hartree-Fock systems

This paper is concerned with the well-posedness analysis of the Hartree-Fock system modeling the time evolution of a quantum system comprised of fermions. We consider quantum states with finite mass and finite kinetic energy, and the self-consistent potential is the unbounded Coulomb interaction. This model is first formulated as a semi-linear evolution problem for the one-particle density matrix operator lying in the space of Hermitian trace class operators. Using semigroup techniques and generalized Lieb-Thierring inequalities we then prove global existence and uniqueness of mild and classical solutions. To this end we prove that the quadratic Hartree-Fock terms are locally Lipschitz in the space of trace class operators with finite kinetic energy. Technically, the main challenge stems from considering the model as an evolution problem for operators. Hence, many standard tools of PDE-analysis (density results, e.g.) are not readily available for the density matrix formalism.

math-ph↗