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We provide a natural simple argument using anistropic flows to prove the existence of weak solutions to Lutwak's $L^p$-Minkowski problem on $S^n$ which were obtained by other methods.

Constructing accurate interatomic potential and overcoming the exponential growth of structural equilibration time are challenges to the atomistic investigations of the composition-dependent structure and dynamics during the vitrification process of deeply supercooled multi-component metallic liquids. In this work, we describe a state-of-the-art strategy to address these challenges simultaneously. In the case of the representative Zr-Cu-Al system, in combination with a general algorithm for generating the neural-network potentials (NNP) of multi-component metallic glasses effectively and accurately, we propose a highly efficient atom-swapping hybrid Monte Carlo (SHMC) algorithm for accelerating the thermodynamic equilibration of deeply supercooled liquids. Extensive calculations demonstrate that the newly developed NNP faithfully reproduces the phase stabilities and structural characteristics obtained from the ab initio calculations and experiments. In the combined NNP-SHMC algorithm, the structure equilibration time in the deeply supercooled temperatures is accelerated by at least five orders of magnitudes, and the quenched glassy samples exhibit comparable stability to those prepared in the laboratory. Our results pave the way for the next-generation studies of the vitrification process and, thereby the composition-dependent glass-forming ability and physical properties of multi-component metallic glasses., Comment: Updated version

The process of crystal nucleation can be accelerated or retarded by ultrasonic vibration, which is particularly attractive for the addictive manufacture and thermoplastic forming of metallic glasses, however, the effect and mechanism of oscillatory loading on the nucleation process are still elusive. By using molecular dynamics simulation, the changes in the time-temperature-transformation (TTT) curve under oscillatory external loading are systematically investigated in two typical binary alloys. A glass forming ability dependent response to the external loading is found, and the shortest incubation time is insensitive to the external loading, while the corresponding temperature can be significantly shifted. Within the framework of classical nucleation theory, a fitting formula is proposed to describe the simulation data quantitatively. In contrast to stationary shear, the elastic stress, rather than the strain rate, is the key parameter to control the evolution of TTT curve under oscillatory loading. Furthermore, the model shows that oscillatory loading can decouple the mobility and nucleation in the deeply supercooled liquid, hence the formation ability can be enhanced while the nucleation is suppressed, which is particularly helpful for the forming and manufacturing of metallic glasses.

An atomistic understanding of annealing embrittlement is a longstanding issue for metallic glasses, which is still lacking due to the insurmountable gap between the thermal history of atomic models and laboratory-made samples. Here, based on a thermal-cycling annealing method that can vary the effective quenching rate over ten orders of magnitude, we perform an atomistic study of the ductile-brittle transition in a ternary model metallic glass, which can be keyed to the annealing embrittlement in bulk metallic glasses. We reveal that thermal annealing can effectively obliterate thermally active-able "defects", which are abundant in the hyper-quenched and ductile glass but gives rise to strain-created shear events in the well-annealed and brittle glass. While the activation of the strain-created events eventually causes single shear banding, other local structural disruptions can be "healed" by the same type of events upon stress reversal, thereby hindering shear band broadening or multiplication, and resulting in annealing embrittlement.

This expository paper presents the current knowledge of particular fully nonlinear curvature flows with local forcing term, so-called locally constrained curvature flows. We focus on the spherical ambient space. The flows are designed to preserve a quermassintegral and to de-/increase the other quermassintegrals. The convergence of this flow to a round sphere would settle the full set of quermassintegral inequalities for convex domains of the sphere, but a full proof is still missing. Here we collect what is known and hope to attract wide attention to this interesting problem., Comment: 19 pages

Cycling of a metallic glass between ambient and cryogenic temperatures can induce higher-energy states characteristic of glass formation on faster cooling. This rejuvenation, unexpected because it occurs at small macroscopic strains and well below the temperatures of thermally induced structural change, is important, for example, in improving plasticity. Molecular-dynamics simulations elucidate the mechanisms by which thermal cycling can induce relaxation (reaching lower energy) as well as rejuvenation. Thermal cycling, over tens of cycles, drives local atomic rearrangements progressively erasing the initial glass structure. This arises mainly from the heating stage in each thermal cycle, linked to the intrinsic structural heterogeneity in metallic glasses. Although, in particular, the timescales in MD simulations are shorter than in physical experiments, the present simulations reproduce many physically observed effects, suggesting that they may be useful in optimizing thermal cycling for tuning the properties of metallic glasses and glasses in general.

We report a metadynamics simulation study of crystallization in a deep undercooled metallic glass-forming liquid by developing appropriate collective variables. Through a combined analysis of free energy surface (FES) and atomic-level behaviors, a picture of an abnormal-endothermic crystallization process is revealed: rejuvenated disorder states with less local fivefold-symmetry and fast dynamics form firstly by changing the local chemical order around Cu atoms, which then act as the precursor for the nucleation of well-ordered crystallites. This process reflects a complex energy landscape with well-separated glassy and crystal basins, giving rise to the direct evidence of intrinsic frustration against crystallization in deep undercooled metallic glass forming liquids. Moreover, the rejuvenated disorder states with distinct physical behaviors offer great opportunities to tailor the performances of metallic glass by controlling the thermal history of a metallic melt., Comment: 20 pages, 12 figures

The activation energy (E_A) spectra of potential energy landscape (PEL) provides a convenient perspective for interpreting complex phenomena in amorphous materials; however, the link between the E_A spectra and other physical properties in metallic glasses is still mysterious. By systematically probing the E_A spectra for numerous metallic glass samples with distinct local geometric ordering, which correspond to broad processing histories, it is found that the shear modulus of the samples are strongly correlated with the arithmetic mean of the E_A spectra rather than with the local geometrical ordering. Furthermore, we studied the correlation of the obtained E_A spectra and various well-established physical parameters. The outcome of our research clearly demonstrates that the soft mode parameter {\Psi} and the E_A spectrum are correlated; therefore, it could be a good indicator of metallic glass properties and sheds important light on the structure-property relationship in metallic glass through the medium of PEL.

In this paper, we study the solvability of a general class of fully nonlinear curvature equations, which can be viewed as generalizations of the equations for Christoffel-Minkowski problem in convex geometry. We will also study the Dirichlet problem of the corresponding degenerate equations as an extension of the equations studied by Krylov., Comment: 33 pages. References added

Mechanical deformation of amorphous solids can be described as consisting of an elastic part in which the stress increases linearly with strain, up to a yield point at which the solid either fractures or starts deforming plastically. It is well established, however, that the apparent linearity of stress with strain is actually a proxy for a much more complex behavior, with a microscopic plasticity that is reflected in diverging nonlinear elastic coefficients. Very generally, the complex structure of the energy landscape is expected to induce a singular response to small perturbations. In the athermal quasistatic regime, this response manifests itself in the form of a scale free plastic activity. The distribution of the corresponding avalanches should reflect, according to theoretical mean field calculations (Franz and Spigler, Phys. Rev. E., 2017, 95, 022139), the geometry of phase space in the vicinity of a typical local minimum. In this work, we characterize this distribution for simple models of glass forming systems, and we find that its scaling is compatible with the mean field predictions for systems above the jamming transition. These systems exhibit marginal stability, and scaling relations that hold in the stationary state are examined and confirmed in the elastic regime. By studying the respective influence of system size and age, we suggest that marginal stability is systematic in the thermodynamic limit.

Structural heterogeneity of amorphous solids present difficult challenges that stymie the prediction of plastic events, which are intimately connected to their mechanical behavior. Based on a perturbation analysis of the potential energy landscape, we derive the atomic nonaffinity as an indicator with intrinsic orientation, which quantifies the contribution of an individual atom to the total nonaffine modulus of the system. We find that the atomic nonaffinity can efficiently characterize the locations of the shear transformation zones, with a predicative capacity comparable to the best indicators. More importantly, the atomic nonaffinity, combining the sign of third order derivative of energy with respect to coordinates, reveals an intrinsic softest shear orientation. By analyzing the angle between this orientation and the shear loading direction, it is possible to predict the protocol-dependent response of plastic events. Employing the new method, the distribution of orientations of shear transformation zones in a model two-dimensional amorphous solids can be measured. The resulting plastic events can be understood from a simple model of independent plastic events occurring at variously oriented shear transformation zones. These results shed light on the characterization and prediction of the mechanical response of amorphous solids., Comment: 3 figures

Understanding the difference between universal low-temperature properties of amorphous and crystalline solids requires an explanation of the stronger damping of long-wavelength phonons in amorphous solids. A longstanding sound attenuation scenario, resulting from a combination of experiments, theories, and simulations, leads to a quartic scaling of sound attenuation with the wavevector, which is commonly attributed to Rayleigh scattering of the sound. Modern computer simulations offer conflicting conclusions regarding the validity of this picture. We simulate glasses with an unprecedentedly broad range of stabilities to perform the first microscopic analysis of sound damping in model glass formers across a range of experimentally relevant preparation protocols. We present a convincing evidence that quartic scaling is recovered for small wavevectors irrespective of the glass's stability. With increasing stability, the wavevector where the quartic scaling begins increases by approximately a factor of three and the sound attenuation decreases by over an order of magnitude. Our results uncover an intimate connection between glass stability and sound damping., Comment: 8 pages, 8 figures

Amorphous materials have a rich relaxation spectrum, which is usually described in terms of a hierarchy of relaxation mechanisms. In this work, we investigate the local dynamic modulus spectra in a model glass just above the glass transition temperature by performing a mechanical spectroscopy analysis with molecular dynamics simulations. We find that the spectra, at the local as well as on the global scale, can be well described by the Cole-Davidson formula in the frequency range explored with simulations. Surprisingly, the Cole-Davidson stretching exponent does not change with the size of the local region that is probed. The local relaxation time displays a broad distribution, as expected based on dynamic heterogeneity concepts, but the stretching is obtained independently of this distribution. We find that the size dependence of the local relaxation time and moduli can be well explained by the elastic shoving model.

Cryogenic rejuvenation in metallic glasses reported in Ketov et al 's experiment (Nature(2015)524,200) has attracted much attention, both in experiments and numerical studies. The atomic mechanism of rejuvenation has been conjectured to be related to the heterogeneity of the glassy state, but the quantitative evidence is still elusive. Here we use molecular dynamics simulations of a model metallic glass to investigate the heterogeneity in the local thermal expansion. We then combine the resulting spatial distribution of thermal expansion with a continuum mechanics calculation to infer the internal stresses caused by a thermal cycle. Comparing the internal stress with the local yield stress, we prove that the heterogeneity in thermomechanical response has the potential to trigger local shear transformations, and therefore to induce rejuvenation during a cryogenic thermal cycling.

We numerically study the evolution of the vibrational density of states $D(\omega)$ of zero-temperature glasses when their kinetic stability is varied over an extremely broad range, ranging from poorly annealed glasses obtained by instantaneous quenches from above the onset temperature, to ultrastable glasses obtained by quenching systems thermalised below the experimental glass temperature. The low-frequency part of the density of states splits between extended and quasi-localized modes. Extended modes exhibit a boson peak crossing over to Debye behaviour ($D(\omega) \sim \omega^2$) at low-frequency, with a strong correlation between the two regimes. Quasi-localized modes instead obey $D(\omega) \sim \omega^4$, irrespective of the glass stability. However, the prefactor of this quartic law becomes smaller in more stable glasses, and the corresponding modes become more localized and sparser. Our work is the first numerical observation of quasi-localized modes in a regime relevant to experiments, and it establishes a direct connection between glass stability and soft vibrational motion in amorphous solids., Comment: 8 pages, 6 figures

We establish interior $C^2$ estimates for convex solutions of scalar curvature equation and $\sigma_2$-Hessian equation. We also prove interior curvature estimate for isometrically immersed hypersurfaces $(M^n,g)\subset \mathbb R^{n+1}$ with positive scalar curvature. These estimates are consequences of an interior estimate for these equations obtained under a weakened condition., Comment: 16 pages

We consider a fully nonlinear partial differential equation associated to the intermediate $L^p$ Christoffel-Minkowski problem in the case $1

Despite the use of glasses for thousands of years, the nature of the glass transition is still mysterious. On approaching the glass transition, the growth of dynamic heterogeneity has long been thought to play a key role in explaining the abrupt slowdown of structural relaxation. However, it still remains elusive whether there is an underlying link between structural relaxation and dynamic heterogeneity. Here we unravel the link by introducing a characteristic time scale hiding behind an identical dynamic heterogeneity for various model glass-forming liquids. We find that the time scale corresponds to the kinetic fragility of liquids. Moreover, it leads to scaling collapse of both the structural relaxation time and dynamic heterogeneity for all liquids studied, together with a characteristic temperature associated with the same dynamic heterogeneity. Our findings imply that studying the glass transition from the viewpoint of dynamic heterogeneity is more informative than expected., Comment: 5 pages, 3 figures