This paper studies dissection propagation subject to internal pressure in a residually-stressed two-layer arterial model. The artery is assumed to be infinitely long, and the resultant plane strain problem is solved using the extended finite element method. The arterial layers are modelled using the anisotropic hyperelastic Holzapfel–Gasser–Ogden model, and the tissue damage due to tear propagation is described using a linear cohesive traction–separation law. Residual stress in the arterial wall is determined by an opening angle  An arterial dissection is a tear within the wall of a large artery, such as the aorta. The dissection can lead to the creation of a false lumen through which blood flows, and propagation of the tear can quickly lead to death as a result of decreased blood supply to other organs, damage to the aortic valve, and rupture of the artery. The loading conditions on the arterial wall, the geometry of the artery and of the tear, and the material properties of the arterial wall determine whether the tear propagates. A prediction of how the critical condition for tear propagation depends on these factors could help to optimize diagnosis and treatment.In the absence of loading, many biological soft tissues are not stress-free, but subject to residual stress. At physiological loading, the residual stress in arteries reduces variation in the stress distribution across the arterial wall and decreases the peak stress  constitutive law  in this study. This, together with the assumption of the plane-strain problem, means that our computed results will be qualitative. However, different values of The actual value of All simulations are based on a residually-stressed configuration Let tituting  and 8)  \documenmentclasspt{minimamentclass2pt{minimTo ensure that the difference between various simulations is only due to the residual stress, we can also determine }Ω0 from , 7), an, an\documentclasspt{minimaWe start from the stress-free configuration The computations are performed using the finite element package ABAQUS 6.13). Four-node plane-strain hybrid elements are used to construct the mesh, as shown in Fig. . Four-nosame unloaded configuration The geometries . For the tear of length The simulated configurations for Our previous study on a 2D strip (Wang et al. Comparison of the critical pressure for different tear lengths Fig.  shows thmentclasspt{minimaHere we vary the fibre orientation. We refer to the fibre orientation listed in Table To ensure that the differences between the simulations are only due to the fibre orientations, we use the same unloaded configuration The critical pressure also changes with different fibre orientation Fig. . Notice We have used both analytical and computational approaches to study the effect of residual stress on the propagation of arterial dissection. Our simulations show that the shortest We now discuss the limitations of this study. Our model is a plane-strain problem and does not include the effect of axial stretch. In our previous work (Wang et al. The other limitation is that we use an isotropic cohesive traction–separation law. The tensile testing of a porcine thoracic aorta performed by MacLean et al.  showed t


The mechanical failure of solid oxide fuel cell (SOFC) components may cause cracks with consequences such as gas leakage, structure instability and reduction of cell lifetime. A comprehensive 3D model of the thermal stresses of an anode-supported planar SOFC is presented in this work. The main objective of this paper is to get an interconnect optimized design by evaluating the thermal stresses of an anode-supported SOFC for different designs, which would be a new criterion for interconnect design. The model incorporates the momentum, mass, heat, ion and electron transport, as well as steady-state mechanics. Heat from methane steam reforming and water–gas shift reaction were considered in our model. The results examine the relationship between the interconnect structures and thermal stresses in SOFC at certain mechanical properties. A wider interconnect of the anode side lowers the stress obviously. The simulation results also indicate that thermal stress of coflow design is smaller than that of counterflow, corresponding to the temperature distribution. This study shows that it is possible to design interconnects for an optimum thermal stress performance of the cell. The solid oxide fuel cell (SOFC) is considered to be one of the most promising new energy technologies for high energy efficiency, environmental friendly and fuel diversity , 2. The Numerical simulation as an economic approach to design materials, cell or stack structure was used to develop the SOFC technology further. The effect of microstructure on effective ionic and electrical electrode conductivities was investigated by numerical optimization, which determines the effective conductivities and degradation of materials . The masThermal stress analysis of a single cell can be used to evaluate the possibility of cracks or flaws that was not able to be detected earlier and easier in experimental works . For a pHowever, the just mentioned studies mainly focused on the thermal stresses distribution, considering the multiphysics process or operating conditions but neglecting the interconnect structural effects. As a component for connecting the electrodes and loads, the interconnect is not only essential to provide paths for electron transport in single cell (or to the neighboring cell in stack) and to protect the electrode material from damages in an ambient environment, but can also maintain the thermal mechanical properties matching to the adjacent electro-active components. Therefore, the selection criteria of interconnect materials and optimization of the structure is more important and stringent than other components of the cell. The ceramic interconnect possesses a good stability and retains a fine compatibility with other components. However, the conductivity of ceramics is not appreciable below 600 °C and the poor sinterability of ceramic interconnects is also a challenge for implementation . While, Normally, the interconnect optimization focused on the electrical performance, degradation processes or temperature distribution, since the structure and the shape of interconnects were related to concentration polarization and electric resistance –30. HoweBesides, altering the width of ribs is easier both in numerical and experimental design. 3D numerical model was built to analyze the effect of rib width on cell performance . The relIn this work, in order to optimize the interconnect structure, based on calculations of the thermal stress field, a 3D comprehensive bi-layer model was developed. Elaborating the reliable strategies for the optimization coupling electrochemical and mechanical properties is of crucial importance for new SOFC designs. Different cell configurations with a geometry design based on the finite element method were applied in this model. The influences from the shape and size of interconnect as well as the components for SOFC involved the thermal stress distribution were investigated in terms of ion/electronic, momentum, mass and heat transport.Conservation and constitution laws were applied to each domain to obtain a comprehensive model containing all the above mentioned phenomena. Stress analysis of an SOFC was performed using the commercial software COMSOL Multiphysics (version 5.0). A half-cell model with bipolar channels operating with humidified hydrogen, carbon monoxide and methane as fuel. The material structure parameters and geometry were selected based on a cell developed and tested at Ningbo Institute of Material Technology & Engineering (NIMTE) . The geoThe reactions considered are shown in Eqs. –5). Car. Car5). E) is reduced due to the internal resistance and polarizations, as shown in Eq.  or bulk fluid within the gas channels (b) and ne is the number of electrons transferred per reaction. In the high current region with sufficiently large ηa whereThe activation polarization/current density relationship is described by the Bulter–Volmer equation:The electronic conductivity in the electrodes  and the internal steam reforming reaction within the anode are included as source term in the model :27\documQh is the heat generation or consumption, keff is the effective thermal conductivity and cp,g is the gas-phase specific heat. The heat source term Qh, due to electrochemical reactions, ion and electron transport and losses through the activation can be defined asSr is the reaction entropy change, rref is the reforming reaction rate and ΔHref is the enthalpy change of the reforming reactions.The temperature is assumed to be locally equal for the solid material and the gases, based on the minor temperature difference between the gas- and the solid-phase. The energy conservation equation can be written asThe materials used in this model are the ones most frequently used for SOFC. Table εxx, εyy, εzz, γyz, γxz, γxy are the longitudinal and shear components for strain, respectively. The thermal strain depends on the temperature, T, the stress-free reference temperature, Tref, and the coefficient of thermal expansion (CTE), α:SOFC components expand with temperature, causing thermal strains to develop in the material when the deformation is constrained. The overall strain results from the summation of elastic and thermal stresses, i.e., the initial contributions are neglected:Tref is the sintering temperature, at which different layers are joined.Determining the stress free temperature is critical as it directly affects the magnitude of the thermal stress induced in the material. For SOFCs it is widely accepted that σ0 is the initial stress, which is treated as the residual stress in model. The elasticity matrix (D) for isotopic material is defined as,E is the Young’s modulus and v the Poisson’s ratio of the material.The stress–strain relationship for the linear material was calculated as5 Pa) is fixed. The fuel inlet fractions are defined as 30 % pre-reformed natural gas , the mole fractions of the fuel are: H2: H2O: CH4:CO: CO2 = 0.263 : 0.493 : 0.171 : 0.0294 : 0.0436) were used as fuel in this model. Note that the ion and electron transport was considered as a testing part in our model, i.e., the H2 mixture and not the 30 % pre-reformed natural mixture was considered in this section, which is sufficient for the study on thermal stress. The air inlet is defined as air, including oxygen and nitrogen. The boundary conditions for the outlets are defined as convective fluxes. The inlet gas temperature is defined by the operating temperature  and the outlet is defined as a convective flux. The boundaries at the top and the bottom walls of the cell are defined as the symmetry conditions, because it is assumed that the cell is surrounded by other ones with identical temperature distribution. Also the boundaries at the side walls are defined as symmetry conditions, because the channel section is assumed to be surrounded by identical channel sections. The potential at the anode current collector is set to zero and the one at the cathode current collector as the cell operating voltage (0.7 V). All other boundaries and interfaces are electrically insulated.The gas inlet velocities are defined as laminar flow profiles, and the boundary condition at the impermeable walls is a nonslip condition for the velocity. At the outlets, the pressure  includes interconnects with the other components of a SOFC, as illustrated in Fig. For optimization of the shape, two types were designed to investigate the support and contact effects of interconnects, as shown in Fig. The temperature distribution for the counterflow case of standard shape is presented in Fig. Coupling the mass and heat transport, reforming reaction, solid mechanical and the first principal stress of SOFC with different designs are shown in Fig. z axis in the model  and the vertical direction represents the x axis . The palliation effects of thermal stress of shape 1 and 2 design was obvious. The interconnect shape design is more effective to reduce the tensile stress in the PEN structure. However, the thermal stress distribution would be extended in the shape 2 design case, because the interconnect has a higher heat conduction compared to the gas, leading to a higher temperature on the surfaces of support layers.The effects of mismatch mechanical properties can be analyzed by comparing the stress distribution at each components’ interface. Since the mismatch between the interconnect and the cathode is larger than for the anode Table , the theThe thermal stress on the interface close to the inlet and outlet for the different designs is shown in Fig. Figure It is observed in Fig. Figure The thermal stresses of the coflow case for an electrolyte with a wider rib shown in Fig. Interconnects are critical to minimize the overall stack resistance and weight as well as to enhance the stability when thermal stresses are considered. A comprehensive model to investigate the relationship between the cell structure, temperature distribution and thermal stresses in SOFCs at certain mechanical properties was built using the finite element method. Proper design of interconnect in conjunction with single-cells were implemented based on the thermal stress optimization performed in this work.The palliation of thermal stress of different designs was obvious. The interconnect shape design was the most efficient to reduce the tensile stress in PEN structure. Besides, thicker interconnect ribs causes a rise at the interconnect close to cathode side while this effect for thermal stress at the anode side is smaller. It is also observed that, wider interconnect of anode side lowers the stress. We show, by numerical calculations, that the thermal stress can be reduced through structural correlations. This optimization can be used as a new strategy for interconnect optimization in SOFCs which is beyond single criterion for electrochemical performance.


Phenomena surrounding the sputtering and associated morphological changes caused by the ion irradiation have been explored. In both the experiments and the simulations, large variations in the sputter yields from individual nanorods were observed. These sputter yields have been shown to correlate with the strength of channelling directions close to the direction in which the ion beam was incident. Craters decorated by ejecta blankets were found to form due to cluster emission thus explaining the high sputter yields.Nanostructures may be exposed to irradiation during their manufacture, their engineering and whilst in-service. The consequences of such bombardment can be vastly different from those seen in the bulk. In this paper, we combine transmission electron microscopy with  Once deployed, nanotechnologies may experience displacing irradiation in a wide variety of situations including nuclear reactors4, extraterrestrial environments5 and medical applications8.With the continued drive to miniaturise technology and to explore the opportunities presented by nanoscience, there comes the need to understand the behaviour of nanomaterials under the various conditions potentially experienced both during manufacture and whilst in-service. Ion irradiation has long been used as a processing technique to modify the properties of materials in the semiconductor industry9, modification of surfaces10 and changes in geometry11. In the proximity of a surface, an atomic collision cascade can cause the mass ejection of atoms via cluster emission13. Such events can result in significantly-enhanced sputter yields in nanostructures with their increased surface-area-to-volume ratios. However, the potential for channelling to increase the depth at which cascades occur can ameliorate such effects.The phenomena of channelling and sputtering are of particular importance to nanostructures exposed to these conditions. Channelling has the potential to vastly dilute the density of energy deposition and to extend the range of ions in a material; as the scale of the target is reduced, this can result in a shift of the nuclear energy deposition peak outside the dimensions of a nanostructure. Sputtering can result in the loss of material14. In high-density targets, the range of a primary knock-on atom is reduced due to the large cross-section for nuclear collisions resulting in denser cascades15. Additionally, in metals with low melting temperatures there arises the possibility of molten flow in the region heated by the thermal spike16. Thus there has been particular interest in such effects during the ion irradiation of bulk and thin-film gold23 and, more recently, of nanostructures25.Cluster emission was first reported in 1958 by Honig for low-energy inert-gas ion irradiation of silver, germanium and germanium-silicon alloy surfaces3 atoms.ion−1 for gold nanorods irradiated with 80 keV Xe ions both observed in situ via transmission electron microscopy (TEM) and simulated using molecular dynamics13. It was demonstrated that the experimental observations could be explained by the explosive emission of clusters and the ways in which the geometry of the nanorods made this significantly-more likely compared to the flat surface of a bulk sample. However, not all the nanorods studied in that work exhibited the same degree of sputtering and this was tentatively attributed to channelling effects dictated by the crystallographic orientations of the individual nanorods relative to the incident ion beam.Recently we reported on enhanced sputter yields exceeding 1013. Therefore, effects due to the emission of clusters of atoms caused by a single ion are expected to be significantly enhanced at this higher energy offering greater possibilities to explore the processes of crater formation, plastic deformation and enhanced sputtering.In the current paper, the aim is to extend that work to explore the effects in the higher-energy regime of 1.7 MeV Au ion irradiation. Theoretically, in a direct head-on collision, a 1.7 MeV Au ion can transfer all of its energy to a gold atom in a nanorod compared to a maximum of 76.8 keV for an 80 keV Xe ion as used in our previous workin situ ion irradiation allows the direct observation of a sample whilst it undergoes dynamic microstructural evolution induced by the ion beam. Furthermore, electron diffraction data can be used to determine the crystallographic orientation of a nanostructure relative to the incident ion beam. Thus it is ideally suited to the explore ion beam sputtering and channelling effects in the individual gold nanorods studied for the work reported here. A schematic of the relative orientations of the electron beam, ion beam and TEM sample grid during the irradiations presented here is shown in Fig. The technique of TEM with θ, and between the ion beam and the [001] direction projected onto the (100) plane, φ. The TEM image series which show the evolution of nanorods AuNR-1 to AuNR-6 over 12 irradiation steps up to 3.6 × 1015 ions.cm−2 are presented in Supplementary Table Morphological changes were observed in all gold nanorods  NanoRod) and a numerical suffix) irradiated with 1.7 MeV Au ions. The magnitude of the sputter yield was found to vary by up to two orders of magnitude with most nanorods exhibiting a sputter yield towards the bottom of this range as summarised in Fig. The sputter rate for the nanorods was found to be relatively constant as a function of fluence; i.e. a nanorod which demonstrated a low sputter yield at low fluences would continue to do so and similarly a nanorod subject to greater sputtering effects in the early irradiations would continue to exhibit the same behaviour in subsequent steps. This is illustrated in Fig. In addition to the gradual loss of material from the nanorods, various structures were observed to form during the irradiation which are consistent with material being quenched whilst undergoing flow processes as shown in Fig. The results of the MDRANGE channelling calculations for 1.7 MeV Au ions incident on a 20 nm thick gold foil are plotted in Fig. 3 atoms in several cases as shown in Fig. Despite the selection of a non-channelling direction, the majority of the molecular dynamics simulations gave a sputter yield of zero. This was because in these cases the ion was transmitted through the nanorod without undergoing any strong nuclear collisions. A systematic analysis of MDRANGE calculations along the direction used for the simulations has demonstrated that this is to be expected (see Supplementary Information). However, a number of simulations also generated significant cascades sufficiently close to a surface to cause cluster emission resulting in sputter yields in excess of 1013 using 80 keV Xe at the MIAMI facility26. Several explanations could account for the large range of sputter yields observed across the nanorods studied as discussed below.Sputtering of the gold nanorods was observed during irradiation with 1.7 MeV Au ions both in the experiments and the simulations. This occurred via both the continuous loss of material due to the sputtering of individual atoms but also in discrete events which resulted in mass ejections leaving behind craters on the nanorods. Molecular dynamics simulations have revealed the latter to be due to an atomic collision cascade near the surface of the nanorod causing the emission of a cluster of atoms. The experimentally-measured sputter yields varied from −2.1 to 30.1 whereas the molecular dynamics simulations gave an average value of 254. The vastly different sputter yields, varying by up to two orders of magnitude, measured in the experiments reported here under 1.7 MeV Au ion irradiation have similarly been observed in previous workFirstly, if undetected inhomogeneities were present in the ion beam then these could potentially cause differing effects across the irradiated area. Figure 27. As can be seen in Fig. Another important consideration is the geometric orientation of the nanorods relative to the incident ion beam. As the projected area presented by a nanorod to the ion beam is taken into account in the calculations of the sputter yields , the importance of geometric orientation for the sputter yields reported here would be in the range of incidence angles made between the nanorod surface and the ion beam as sputtering is a strong function of angle28. The nuclear energy deposition peak for 1.7 MeV Au ions in gold is about 50 nm  and therefore such effects could significantly reduce the number of nuclear collisions occurring within the nanorod volume. Indeed, it should be noted that in the study of the ion-irradiation-induced morphological changes presented in Fig. As no other significant differences were observed between the nanorods, such as the presence of oxide or contamination layers, then the most likely candidate to account for the variations in sputter yield is their crystallographic orientations. Channelling effects are known to be able to significantly extend the range, and thus reduce the energy deposition, of ions29 compared to the annular cones of a few degrees being considered here. It is possible that the combined error from the mechanical accuracy of the TEM x-tilt and from the diffractions conditions (i.e. sg ≠ 0), means that the orientations of the nanorods deviated from those expected sufficiently for the ion beam to have been incident along a direction closer to Experimental investigations into ion channelling often rotate single crystals in the beam path whilst monitoring a quantity such as sputter yield. Channelling manifests itself as a reduction in the measured yield over a small angular range when a channelling direction is traversed. This angular range is normally a fraction of a degreeWhilst the absence of an accessible channelling direction for AuNR-5 likely explains the large disparity between its sputter yield of 30.1 and the next-closest of 2.6 for nanorod AuNR-6, another possible difference could lie in the adhesive forces between the nanorods and the carbon support film. If a nanorod settles into an undulation or an area of contamination on the carbon support film then it is possible that the effective surface area of contact could be increased to such a degree that the ion beam cannot induce a rotation sufficient to access the orientation corresponding to The difference between the experimentally-measured sputter yields and those from the molecular dynamics is expected as the simulations were performed deliberately on a non-channelling crystallographic direction. However, another explanation for the discrepancy may lie in the calculation of the volumes of the nanorods . This procedure involved dividing a nanorod into a series of thin slices which are then treated as short cylinders assuming that the nanorods remain approximately circular in each cross section. Analysis of images from the tilt series taken of irradiated nanorods have shown this to be generally the case in terms of the overall geometry. However, this approach can overestimate the volume, and thus underestimate the sputter yield, in the presence of a large crater especially if the crater does not affect the projected shape of the nanorod in the TEM micrograph. Therefore, at low sputter yields it is possible for a negative sputter yield (i.e. an addition of atoms) to be calculated as can be seen in Fig. 30. The similarity between the arch shown in Fig. Comparison of the experimentally-observed ion-beam-induced features in Fig. 29. Hence island formation must result from diffusion-based agglomeration processes in those situations. For heavier more-energetic ions such as 1.7 MeV Au ions, both sputtering of single atoms and cluster emission become possible12. In the current study, the islands were often observed to appear within a time period of less than one frame of video  concurrently with the formation of a crater as shown in the video frames in Fig. 31 as well as in the simulations reported here, it has been shown that ejected clusters can travel distances comparable to the diameter of a nanorod before disintegrating so it is therefore plausible that these islands did indeed form via cluster deposition in at least some instances.During the ion irradiation experiments, the carbon support films around the nanorods became decorated with islands of gold atoms due to sputter deposition. These can be seen in many of the micrographs shown in Figs in situ ion irradiation at the I3TEM facility at Sandia National Laboratories, large variations in the sputter yield have been observed between different nanorods. Calculations of nuclear energy deposition as a function of crystallographic orientation have revealed a correlation between the experimentally-measured sputter yields and the strength of the channelling directions found within annular cones of ± 3° of the ion irradiation direction. It is hypothesised that the ion beam may induce small rotations to the nanorods and if a channelling direction is reached then the probability of further tilting is vastly reduced and hence the nanorod will tend to remain thus orientated. Further experiments and modelling are required to explore the nanorod rotations which can be induced by ion irradiation and the plausibility of the proposed mechanism for preferential alignment along a channelling direction.Complementary experimental and theoretical studies of 1.7 MeV Au ion irradiation of gold nanorods have produced new insights into the physics of ion-beam-induced sputtering of these nanostructures. Using TEM with −1 (expected to be an underestimate) have been measured experimentally compared to 254 atoms for the non-channelling direction in the molecular dynamics simulations. Such high sputter yields are explained by the observation, in both the experiments and the simulations, of the ejection of clusters of atoms. The emission of such clusters produced craters decorated by ejecta blankets on the surface of the nanorod in both the experiments and the simulations. Moreover, in the case of the experiments, islands of deposited gold on the surrounding carbon support film were observed to appear within a timespan of a video frame. Simulations have shown that the clusters are expected to fragment after emission and therefore it is likely that the observed gold islands are due to both the arrival of single large clusters and the agglomeration of sputtered atoms moving across the surface of the support film.Sputter yields of up to 30 atoms.ion32 and gold nanoparticles33. Our recent work has studied enhanced sputtering of gold nanorods due to cluster emission under 80 keV Xe ion irradiation and identified large differences in sputter yield between individual nanorods13. This has now been extended to the higher-energy regime of 1.7 MeV Au ions identifying channelling as the most likely explanation for the observed differences and capturing the more-pronounced formation of surface structures during cluster emission.From a fundamental perspective, these results give new insights into the atomistic processes at work during the ion irradiation of nanostructures. Previous work has looked at the effects of size and geometry on sputtering in systems such as silicon nanospheres6. Understanding of the radiation response of nanoparticles, especially where it differs from that of bulk material, will help in the design of therapies using optimised combinations of irradiation conditions and nanoparticle composition, size and distribution. This work will be of even greater relevance to the use of such nanoparticles in heavy-ion therapies8 which are more likely to produce the effects reported here. In the area of nuclear energy, nanoparticles have been proposed for use in nanofluids to improve heat transfer efficiency4. These will be bombarded by neutrons in primary coolant loops and detailed understanding of the optional effects this may have will aid the development and deployment of this technology.These results will be important for the development of nanotechnologies in terms of both their manufacture and modification using ion irradiation techniques and also for their deployment in environments in which they are exposed to irradiation. For example, gold nanoparticles are being studied as radiation sensitizers to improve the efficacy of proton beams for cancer treatmentGold nanorods suspended in deionised water were obtained from Nanopartz Inc (product number A12-25-850-25) and deposited on carbon film supports on 400 mesh copper TEM grids. The nanorods were measured to be of diameter 20 nm with typical lengths in the range of 60–100 nm. The majority were found to be <001> orientated along their axes but were otherwise randomly orientated relative to the support grid due to the dispersive nature of the TEM sample preparation technique. Individual isolated nanorods distant from the mesh bars of the TEM grid were selected for the irradiation experiments to avoid shadowing and sputter deposition effects from the mesh bars and from other nanorods.3TEM in situ ion irradiation facility at Sandia National Laboratories  combines a 6 MV tandem accelerator with a JEOL JEM-2100 TEM and is described in detail elsewhere34. The ion beam is orientated horizontally such that the electron beam, ion beam and x-axis of the TEM are all orthogonal to each other as shown in Fig. 3+ ions with a flux of 1.5 × 1013 ions.cm–2.s−1 at room temperature up to a fluence of either 3.6 × 1015 or 4.8 × 1015 ions.cm−2 in steps of 3.0 × 1014 ions.cm−2. All fluences are per unit area in the xz-plane of the TEM. The TEM was operated at 200 kV and the electron beam was not incident on the sample during ion irradiation. Prolonged electron beam exposure resulted in no observable changes to the gold nanorods. However, the carbon film did demonstrate roughening and ultimately decomposition under the electron beam; therefore, it was necessary to minimise electron beam exposure to preserve the integrity of the support.The Iin situ ion irradiation experimental techniques and associated calculations used for the current work are given in the Supplementary Information.Additional details of the TEM with 35. A gold target was generated as a perfect FCC crystal with random thermal displacements of atoms corresponding to 300 K. The simulations were run using the ZBL36 and DMol37 interatomic-pair potentials but both were found to give essentially identical results; the results presented here were produced using the ZBL potential.Nuclear energy deposition as a function of crystallographic direction was calculated for 1.7 MeV Au ion irradiation using molecular dynamics simulations with the recoil interaction approximation in the MDRANGE computer codeθ, and 0–90° for twist, φ, off the [001] direction in increments of 1° and 2°, respectively. For each θ, φ combination, 103 ions were simulated and the average nuclear energy deposition was calculated over a distance of 20 nm in the target. A bilinear interpolation was used to achieve a smooth energy-deposition plot38. The results of these calculations (prior to interpolation) are included in the Supplementary Information.The simulations were carried out in the ranges of 0–89° for tilt, 6 gold atoms were simulated using molecular dynamics. They were then relaxed for 5 ps at 312 K using linear Berendsen temperature control39 and then rotated 13° about the major axis (i.e. θ = 13° and φ = 0° so as to be 13° off a {001}) to reduce the chance of channelling40.Spherocylindrical nanorods measuring 100 × 20 × 20 nm and containing 1.7 × 1041 using the Foiles et al.42 implementation of the embedded atom model (EAM)44. The EAM reliably models surface energy, melting temperature and the liquid state45 which are essential to the modelling of ion impacts and sputtering in metals. This approach has previously demonstrated good agreement with experimental results on surface irradiation effects13; however, Samela et al.46 have highlighted some drawbacks and to help address these the EAM potential was combined with the universal Ziegler-Biersack-Littmark (ZBL) repulsive potential36. The outcome of an atomic collision cascade tends to be determined by the liquid zone41 which is not affected by the choice of repulsive potential and therefore any inaccuracies of the ZBL potential should not be significant here. In addition, ZBL electronic stopping was applied for all atoms in the system which had ≥ 5 eV of kinetic energy35.Molecular dynamic simulations were run with the PARCAS code47.A 1.7 MeV Au atom moving vertically downwards was then generated at a random position in a horizontal plane above the nanorod. Simulations were terminated if the incident atom transferred less than 170 eV to the nanorod for the purposes of computational efficiency by excluding events which would not result in any significant erosion. Although terminated early, such simulations were still included in the calculation of the average sputter yield of 254 atoms reported here. Visualisation and analysis of the results was performed using the Open Visualization Tool (OVITO) softwareThe datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.Supplementary InformationVideo of TEM tilt seriesVideo of molecular dynamics simulationMDRANGE channelling map data file


TBI) by employing the multivariate machine learning approach and introducing a novel task‐related hemodynamic response detection followed by a heuristic search for optimum set of hemodynamic features. To achieve this goal, the hemodynamic response from a group of 31 healthy controls and 30 chronic TBI subjects were recorded as they performed a complexity task.We have explored the potential prefrontal hemodynamic biomarkers to characterize subjects with Traumatic Brain Injury  that contribute in distinguishing between TBI and healthy subjects. As expected, Brodmann areas (BA) 10 within the PFC were isolated as the region that healthy subjects (unlike subjects with TBI), showed major hemodynamic activity in response to the High Complexity task. Overall, our results indicate that identified temporal and spatio‐temporal features from PFC's hemodynamic activity are promising biomarkers in classifying subjects with TBI.The identified optimum feature elements resulted in classification accuracy, sensitivity, and specificity of 85%, 85%, and 84%, respectively. Classification improvement was achieved for  ExecutiAdvancement made in functional neuroimaging provide tools needed for the sensitive assessment of functional abnormalities following TBI in various brain regions, including the PFC. In particular, functional magnetic resonance imaging (fMRI) of the blood‐oxygen‐level‐dependent (BOLD) signal  for comThe objective of this study is to identify the potential prefrontal hemodynamic TBI biomarkers that contribute in characterizing TBI subjects at the individual level through a multivariate feature selection technique. Employing the multivariate analysis instead of the commonly used univariate analysis was motivated by the findings in  from which data from nine subjects were not used in this analysis. Data from these subjects were excluded from the study either due to the problems in data collection or major motion or detector artifacts. Details of the procedure to identify subjects with major artifactual data are explained in the preprocessing section. Final number of subjects available for analysis was 61, 31 healthy controls  and 30 TBI subjects . Table All TBI subjects fulfilled inclusion/exclusion criteria of NCT01797549, “Detection of Hemodynamic Changes in TBI Population With Functional Near Infrared Spectroscopy”, adults between 18 and 55 years of age who suffered a moderate or severe TBI, by DoD criteria, that is a head injury associated with Glasgow Coma Score between 3 and 12, loss of consciousness >30 min, alteration of consciousness or post‐traumatic amnesia >24 hr, or TBI‐related abnormality on neuroimaging (CT or MRI). All except one (examined at 2 months after injury) were tested over 6 months after injury with the median interval from TBI to NIRS testing as listed in Table Thirteen TBI subjects received either inpatient or outpatient TBI rehabilitation after their acute hospitalization for their injury. The type and duration of their therapy reflected the severity of their injury and symptoms during the subacute period after their injury. Only 1 TBI subject was still receiving rehabilitative services at the time of enrollment in the study, a 28‐year‐old who suffered a severe TBI 8.5 years prior to enrollment in NCT01797549.2.2An event‐related paradigm in which subjects are required to evaluate the complexity of certain daily life activities was chosen for this study. The paradigm was originally designed and implemented in an fMRI experiment by Krueger et al.  and has http://www.pstnet.com/eprime.cfm). Participants were first trained with a separate set of stimuli to familiarize them with the experiment. At the beginning of each trial, instructions describing the type of task (Complexity or Font) and the name of a daily‐life activity, for example, “stirring a cup of coffee” was displayed on a computer monitor for 4 s. For the complexity task, participants were asked to make a binary decision as to whether the activity name displayed corresponded to an activity with low complexity, for example, “stirring a cup of coffee” or an activity with high complexity, for example, “planning a wedding”, using a two‐button response pad. For the Font task, participants were asked to decide whether the instructions and the activity name shown represented the same or different fonts. Participants were prompted to respond as quickly and accurately as possible. A trial lasted for 5 s as the stimulus was presented for 4 s and was followed by 1 s after the stimulus disappeared. Trials were separated by a randomly assigned jittered interstimulus interval of varied interval of 5–7 s. 33 Font and 66 Complexity trials were randomly arranged within a 15‐min period of fNIRS data collection. Figure In this paradigm, subjects were exposed to two classes of conditions: An experimental condition or the Complexity task and a control condition or the Font task. Stimulus presentation was controlled by the E‐prime software package . The distance between each source/detector pair was 2.5 cm. The configuration of probes for this device is denoted in Fig. 2.3The raw intensity data measured at two wavelengths was normalized for all channel sites to compute the relative change by dividing each value of the intensity signal by the mean of the baseline signal. The intensity‐normalized data was then used to calculate the change in optical density . Delta‐optical density was computed for each wavelength as the negative logarithm of normalized intensity. For every subject, a differential pathlength factor (DPF) was calculated as the variable of age and the wavelength following the formula obtained in  to investigate the difference in the TBI and healthy populations. We address this issue by extracting 11 time‐ and frequency‐domain features and investigating their potential to be employed for characterizing the TBI and healthy subjects. We identify the optimum feature space for distinguishing TBI and healthy subjects by employing a features selection method in the next section.Typically, the average HbO signal obtained in response to an eliciting stimulus embodies a positive deflection representing the activation in the channel that we refer to as the activity curve. The activity curve is the curve embodied in the HbO signal that is formed by an increase in oxygenation and its return to the same level of oxygenation. Depending on the nature of the features, they are extracted from the entire average HbO or the activity curve as follows:Mean value of the HbO signal (HM),Variance of the HbO signal (HV),Left slope of the activity curve (CSL),Right slope of the activity curve (CSR)Kurtosis value of the HbO signal (HK),Skewness value of the HbO signal (HS),Area under the activity curve (CA),Full width half maximum of the activity curve (CF),Peak amplitude of the activity curve (CP),Activity start time (CAS),Discrete Fourier Transform (DFT) Coefficients of the HbO signal (HDFT)N data points in the time domain into the frequency domain by the following:Figure cn coefficients are a sequence of complex numbers that represents the amplitudes and shifts of a decomposition of the signal into sinusoid functions. HbO(k) is the value of the HbO time series at time k. Keeping a few coefficients and discarding the rest that provides a rough sketch for the original HbO signal is a common time series feature extraction technique : average signal value.n data points as follows:Variance (HV): measure of HbO signal spread. The variance formula for HbO signal with i is the ith data point of the HbO signal.where HM is the average of the HbO signal and HbOSkewness (HS): measure of the asymmetry of signal values around its mean relative to a normal distribution. If the HbO signal is symmetrically distributed, then HS will be 0.Kurtosis (HK): measure of the degree of peakedness of a distribution of signal values relative to a normal distribution.Activity start time (CAS): represents the time instant after stimulus onset when the oxygenation value in the HbO starts to increase toward its peak's amplitude.Full Width at Half Maximum (FWHM): is commonly used to measure the width of a peak on a curve. As is illustrated in Fig. 2.5To determine the optimal feature set  that enables distinguishing TBI subjects from the healthy subjects with the highest accuracy, we employed the wrapper feature selection method  – A subject belongs to the TBI population and is classified correctly as a TBI subject.True Negative (TN) – A subject that belongs to the healthy population and is classified correctly as a healthy subjectFalse Positive (FP) – A subject that belongs to the healthy population and is classified incorrectly as a TBI subject.False Negative (FN) – A subject that belongs to the TBI population and is classified incorrectly as a healthy subject.Since the number of TBI and healthy subjects are comparable , the common metric of accuracy that weights TP and TN equally is appropriate for this classification problem  were discarded from the analysis by applying the trial‐removal preprocessing step. The remaining channels for every subject contained only trials that were the most representative for hemodynamic activation in response to the High complexity task. Figure 3.2Temporal features were extracted from every channel of the subjects. For every subject, these features were averaged across all the channels to obtain the subjects' representative feature set. All the possible combinations of the generated features were considered for distinguishing the TBI from the healthy subjects using the Decision Tree classification algorithm. Table In Fig. It is worth noting that the classification performance for the features of the optimal set: CA, HDFT, CF, CSL, and CSR in Table For the optimal features set of , we attempted to evaluate the performance of other commonly used classifiers and provide a comparison with the Decision Tree performance. In Table 3.3In this section, the efficacy of our proposed preprocessing step of imposing constraints on the selected trials is evaluated. As discussed in the Comparing classification performance in Tables 3.4In addition to the temporal classification, a spatio‐temporal feature extraction and classification procedure was also considered to identify the features that enable distinguishing TBI subjects from healthy. In the spatio‐temporal classification, unlike the temporal classification approach, the extracted features for a subject were not averaged across all the channels. Therefore, for a single feature there were at most 16 different feature values , HbO's, DFT coefficients of the HbO signal (HDFT), activity curve's full width half maximum (CF), activity curve's left slope (CSL), and activity curve's right slope (CSR). As discussed in the For the spatio‐temporal classification, HbO's skewness (HS), activity curve peak value (CP), activity curve's left slope (CSL), activity curve's area under the curve (CA), and HbO signal's variance HV were identified as the optimum feature elements. In Fig. We investigated the parametric effects of the task complexity in distinguishing the TBI and healthy subjects by employing the optimum feature set of the temporal classification for task with different loads. It should be noted that the complexity of the tasks  had been determined in advance and was not dependent on subject's response. As it is shown in Table Overall, we successfully identified a set of hemodynamic biomarkers that enabled identifying and characterizing subjects with TBI from healthy subjects with a significant accuracy . Furthermore, for certain features such as DFT, more number of features with employing 2N or 3N DFT could have been obtained that was not considered in this work. In our future studies, we will consider extracting hemodynamic features from other types of hemodynamic signals as well as more number of features with the purpose of creating larger feature space that can potentially improve classification of the TBI and healthy subjects.It should be emphasized that although we showed that employing our proposed preprocessing step of trial/channel removal improves the classification performance, but it suffers from certain limitations and is susceptible of discarding valuable information. By applying this preprocessing step, channels for which 80% or more of the trials had been discarded were not considered for further analysis. This may result in discarding the majority of the trials from further analysis for certain channels. Furthermore, since certain channels are discarded from the analysis, the spatial distribution of the hemodynamic activity in the temporal classification can vary across subjects.Finally, it is worth mentioning that our proposed approach of identifying TBI functional biomarkers using the fNIRS's hemodynamic signal has the potential to become a common approach in characterization of subjects with neurodegenerative, neurodevelopment disorders to further help clinical investigators to identify the underlying impairments of brain in the patient groups.We acknowledge the funding of the intramural program of the Eunice Kennedy Shriver National Institute of Child Health and Human Development, the National Institute of Neurological Disorder and Stroke, and Department of Defense in Center for Neuroscience and Regenerative Medicine.None declared.


We study the quantum relaxation of the two-dimensional transverse-field Ising model after global quenches with a real-time variational Monte Carlo method and address the question whether this non-integrable, two-dimensional system thermalizes or not. We consider both interaction quenches in the paramagnetic phase and field quenches in the ferromagnetic phase and compare the time-averaged probability distributions of non-conserved quantities like magnetization and correlation functions to the thermal distributions according to the canonical Gibbs ensemble obtained with quantum Monte Carlo simulations at temperatures defined by the excess energy in the system. We find that the occurrence of thermalization crucially depends on the quench parameters: While after the interaction quenches in the paramagnetic phase thermalization can be observed, our results for the field quenches in the ferromagnetic phase show clear deviations from the thermal system. These deviations increase with the quench strength and become especially clear comparing the shape of the thermal and the time-averaged distributions, the latter ones indicating that the system does not completely lose the memory of its initial state even for strong quenches. We discuss our results with respect to a recently formulated theorem on generalized thermalization in quantum systems. In recent years the non-equilibrium dynamics of isolated many-body quantum systems, defined by the unitary time evolution starting from a generic non-eigenstate of the Hamiltonian, has gained tremendous interest123456789101112131415161718192021222324252627286163031323334353637383940β = 1/T is the inverse temperature and is determined under the constraint of maximizing the entropy.The Lagrange multiplier Thermalization is closely linked to the conserved quantities of the system. While in non-integrable systems only the energy is conserved, integrable systems possess additional conserved quantities which avoid thermalization. The canonical Gibbs ensemble thus cannot be applied to integrable systems, but it has been shown that their stationary state can be well described by the generalized Gibbs ensemble (GGE)234789101112142242434445λn are uniquely determined by the initial conditions. However new results show that care has to be taken in the definition of the GGE4748495053The conserved charges are taken into account by the operators In non-integrable systems the energy is the only conserved quantity, but non-integrability is not synonymous to thermalization. Counterexamples are the driven Rabi modelD-dimensional effective O(N)-Hamiltonian close to dynamical critical points based on a renormalization-group method6566Predictions on whether non-integrable systems thermalize or not usually rely on the eigenstate thermalization hypothesis (ETH)2345859606163Here we present for the first time a systematic investigation of the relaxation dynamics of a two-dimensional, non-integrable model going beyond system sizes accessible with exact diagonalization and applicable to large areas of the parameter space. We study the transverse-field Ising model in two dimensions (2D-TFIM) after global interaction quenches in the paramagnetic phase and global field quenches in the ferromagnetic phase. In contrast to the Ising chain in one dimension, the 2D model is non-integrable and cannot be solved analytically. We use a real-time variational Monte Carlo (rt-VMC) method to compute the time evolution of observables like magnetization and correlation functions with high accuracy for long time scales and large system sizes. To answer the question whether the 2D-TFIM thermalizes or not we compare the time-averaged distributions of the observables to their thermal distributions for the system in equilibrium at temperatures determined by the excess energy after the quench. For a system that thermalizes one would expect the asymptotic time-averaged distributions and the thermal distributions to be identical. We discuss our results with respect to the theorem on generalized thermalizationL × L with periodic boundary conditions (PBC). The system is described by the HamiltonianWe study the 2D-TFIM with nearest neighbour interactions on a square lattice of size J and external transverse field h. The total number of spins in the system is N = L2, thus the dimension of the Hilbert space N. As basis of x-basis, in which the operator R, while with coupling strength The model is highly symmetric. Its Hamiltonian is invariant under the global L → ∞) at low temperatures and fields due to spontaneous symmetry breaking of the global spin flip symmetry and is indicated by a non-vanishing ground state magnetization in x-direction h = 0 the equilibrium phase transition occurs at (T/J)crit ≈ 1.135T = 0 at (h/J)crit ≈ 3.0446970T/J and h/J. For this reason we use for the finite systems in our numerical studies the modulus of the magnetization as order parameter. In order to get rid of finite size effects it is renormalized according toFerromagnetic long-range order exists in the thermodynamic limit  temperature of a system in equilibrium for which the excess energy due to the quench is equal to the thermal energy above the ground state energy and compare the thermal distributions and the time averages of the distributions after the quench. The temperature attributed to a quench is called interaction quenches  →  and field quenches  → . These quenches just lower the ground state energy of the system, so that ΔE = 0 and Ef = Ei,0. For the interaction quenches in the initial state all spins are aligned in the direction of the external transverse field:In the following we will focus on J is considered, i.e. The energy of this state is x-basis with the energy of the two fully magnetized basis states of the T/J for different values of h/J for the system with L = 16. The results for Teff for the interaction and the field quenches are shown in Teff/J for h/J ≈ (h/J)crit. A dependency of the effective temperature on the system size can only be observed for interaction quenches ending close to the phase transition. This is due to the increasing correlation length in the vicinity of the phase transition. Field quenches are predicted to drive the system out of the ferromagnetic phase when the external field is quenched to values larger than For the determination of the effective temperature after the quenches the thermal energy see  of the fTeff/J as function of h/J can be understood considering the effects of J, h and T onto the order in the system. In the ground state at T = 0 for h/J < (h/J)crit a parallel orientation of the spins in x-direction is energetically preferred, while for h/J > (h/J)crit the orientation of the spins in z-direction parallel to the external transverse field is favourable. The temperature causes fluctuations of the orientation of the spins and thus disturbs the described order. The strength of this effect depends on h/J. For strong couplings or strong fields higher temperatures are necessary to disturb the order of the spins. This can be seen considerng the energy as a function of T/J for fixed h/J in T/J with T = 0 on the other hand the energy h/J, while for large ratios of h/J it approaches h/J. The closer h/J is to (h/J)crit, the less ordered the spins are and thus can be more easily disturbed by the temperature. As for the interaction quenches the expectation value of the energy of the system in thermal equilibrium at Teff has to be equal to Teff/J close to (h/J)crit. Away from (h/J)crit the effective temperature of the system after the interaction quenches increases. Although for large ratios h/J the energy T = 0 is only slightly smaller than the energy h/J the same argument holds. In this case the order is caused by the coupling between the spins. For the field quenches the effective temperature is determined by the condition that Teff/J = 0 for h/J = 0. In the limit h/J → ∞ on the other hand T = 0 is lowered almost linearly with h in this case.The shape of We prepare the system in the ground state of the initial Hamiltonian i,0 according to with ketΨIn terms of the eigenbasis of The initial state of the system is a pure state and its state remains a pure state under unitary time evolution. A thermal system on the other hand is described by a mixed state. For this reason time averages of the observables after the quench have to be compared to the thermal values as will be shown in the following. Considering the time-evolved expectation value in diagonal ensemble:The stationary state of the system can thus be described by a mixed state in the so-called For the distributions of the (possibly degenerate) eigenvalues As the dimension of the Hilbert space et al. for the Bose-Hubbard model77Rt-VMC was introduced by Carleo Jastrow ansatz7778A common choice for the variational state is the Jastrow factor. For the 2D-TFIM the Jastrow ansatz reads79In the paramagnetic phase we use the Jastrow ansatz for the variational function. This ansatz is constructed from the completely uncorrelated state. Correlations are taken into account by the The operatorsr normalized by their number Nr. The sum over r runs over all independent directions in the lattice, whose number also determines the number of variational parameters αr(t), which is N/8 + 3L/4 for the square lattice with edge length L. For a given r, the average over all dependent directions is taken  is Nm,n. Transitions between subspaces are induced by spinflips. A spinflip increases or decreases m by one and keeps n untouched or changes it by ±2 or ±4. Thus each subspace is linked to up to 10 other subspaces. We denote the total number of transitions between the subspaces  and  with Tm,n;m′,n′. The Tm,n;m′,n′ are symmetric with respect to  ↔ . The determination of the Nm,n and the Tm,n;m′,n′ is a pure combinatorics problem. As for the 2D-TFIM there are no closed-form expressions for their values and due to the high dimensionality of the Hilbert space we determine them with rare event sampling (RES)Nm,n and Tm,n;m′,n′ are independent of the quench parameters, they have to be determined only once for each system size. The possible values of n depend on m in a non-trivial way. In case of the 2D-TFIM not just the maximal number of kinks is a function of m like in one dimension, but also their minimal number. The PBC have to be taken into account, too. In leading order the number of possible values of n grows linearly with m. As the number of possible values of m grows linearly with the system size N, the number of variational parameters thus grows in leading order with N2. Due to the symmetry of the variational parameters with respect to m ↔ N − m and the conservation of this symmetry under time evolution, we can reduce the number of independent variational parameters using αm,n(t) = αN−m,n(t). For the system sizes we studied the numbers of the variational parameters are listed in The dimension of the subspace  by flipping one single spin. The equations of motion of the variational parameters are thus a system of coupled linear differential equations of first order with constant (time-independent) coefficients, which are known from RES. As each subspace is linked to only up to 10 other subspaces, the system is sparse. We solve it with a fourth order Runge-Kutta scheme. The initial values of the variational parameters areThe sum over t > 0 the αm,n(t) are in general complex.For r according to As observables we consider the rescaled modulus of the magnetization according to For the interaction quenches the expectation values of the observables and their distributions are computed in the course of the single spin flip quantum Monte Carlo algorithm for the coefficients of the equations of motion at each time step, while for the field quenches there is a direct functional relationship to the variational parameters. For the modulus of the magnetization this relationship readswith the eigenvalues t are given bywith the eigenvalues andThe correlation function between spins that are not nearest neighbours can be computed according toThe To decide whether the system thermalizes or not the expectation values of the observables as well as their distributions in the stationary state have to be compared to their counterparts for the thermal system. As the exact computation in the diagonal ensemble would require the knowledge of the full spectrum of the Hamiltonian after the quench, we use time averages to approximate the expectation values and distributions in the stationary state according toandt and different t0 in the range of times that we can simulate  and found that the results are stable with respect to our tests.with Teff attributed to the quench, which are given byThe time averages of the observables are compared to the thermal expectation values of the system in equilibrium at the effective temperature The thermal distribution of the eigenvalues The expectation values and distributions for the system in thermal equilibrium are computed with a cluster Monte Carlo algorithm in continuous imaginary timeTeff with increasing quench strength. In order to quantify the deviations between the thermal expectation values and the time averages after the quench as a function of the quench parameters we compare in Teff attributed to the quench (black) and the time-averaged values of the exact time evolution (green) and of the rt-VMC time evolution (red). The relative error between the time-averaged expectation values of the exact and the rt-VMC time evolution is shown in (a) ii. and (b) ii.. For the interaction quenches we observe an excellent agreement while h/J ≈ 0.75 deviations increase, but even for h/J = 1.5 they do not exceed 1.5%. For both quench protocols our rt-VMC method thus allows us to compute the time averages of the observables as well as their distributions with high accuracy for a wide range of ratios h/J. We may derive two main results from our studies of the 4 × 4 system: First we observe that for this small system size significant deviations between time-averaged results and the thermal results exist when the system is quenched close to its phase transition. Second there is a very good agreement between the time-averaged values of the exact time evolution and the rt-VMC time evolution for a wide range of ratios h/J. This agreement does not just concern the time averages of the observables, but also the underlying distributions. Only for strong quenches deviations between the time averages of the exact time evolution and the rt-VMC time evolution can be observed, but these deviations are much smaller than the deviations from the values of the system in thermal equilibrium.Before we apply the rt-VMC algorithm to large system sizes, we consider a system of size 4 × 4, whose time evolution can also be computed with numerical integration of the Schrödinger equation, and compare the rt-VMC results to the exact results to benchmark the algorithm. We do this exemplarily for the rescaled modulus of the magnetization. Results are presented in L = 8, 12 and 16. Teff as a function of h/J after the quench. For the interaction quenches we confine us to values h/J > (h/J)crit, as for smaller ratios h/J the accuracy of the rt-VMC with the Jastrow ansatz decreases, while for the field quenches we study quenches with h/J after the quench we observe that the relative shape of the curves is preserved when the system size is increased. There is a good agreement between the time averages and the thermal expectation values for small quenches with continuously increasing deviations for stronger quenches. While for the interaction quenches the observed deviations are small, in case of the field quenches the deviations become significant when the system is quenched closer to the phase transition. Comparing the deviations for strong field quenches one observes that they do not decrease with the system size, so that we assume them not to be caused by finite size effects.We now apply the rt-VMC method to larger system sizes. To make predictions for the system in the thermodynamic limit, we computed results for the rescaled modulus of the magnetization and the correlation function between nearest neighbours after interaction quenches and after field quenches for systems of size n. The possible values of n depend on the number m of spin down. In the two-dimensional model the smallest possible value of n for a given m increases with m. For this reason small values of n are linked to small values of m. For small values of m the configurations with the highest possible number of kinks are much more likely than configurations with the smallest possible number of kinks. This causes the strong variations in the distributions of h/J after the quench is decreased, while for the field quenches it is increased. For small interaction quenches we observe a very good agreement between the thermal and the time-averaged distributions. Deviations increase when the system is quenched closer to the phase transition, but the shape of the thermal curves is well reproduced by the time averages after the quenches. Comparing the distributions for the 16 × 16 system to those of the 4 × 4 system in Up to this point we have only considered the time-dependent expectation values of the observables and compared their time averages to the thermal expectation values. To decide whether the stationary state of the system after the quenches can be described by the CGE not only the expectation values of the observables have to coincide, but also their distributions. The study of the distributions is thus especially important for the interaction quenches and small field quenches, for which we have found a good agreement between the thermal and the time-averaged expectation values of the rescaled modulus of the magnetization as well as of the correlation function between nearest neighbours. [The strong variations in the distributions of the correlation function between nearest neighbours for Up to this point we have only given a qualitative discussion of the distributions of the observables. In order to compare the deviations as a function of the system size and make predictions for the system in the thermodynamic limit we introduce the measure for the deviations between the thermal and the time-averaged distributionsL = 4, 8, 12 and 16 for the described quench protocols and did a finite size scaling to conclude to the system in the thermodynamic limit. N. We find that for the interaction quenches the deviations between the thermal and the time-averaged distribution decrease with increasing system size. We thus conclude that the observed deviations of the expectation values and the distributions between the thermal and the time-averaged system will further decrease with increasing system size and that the system will thermalize in the thermodynamic limit. Before we make a statement on the results of the finite size scaling for the field quenches we discuss the effect of the spontaneous symmetry breaking in the ferromagnetic phase for the system in the thermodynamic limit. For the finite system sizes considered in our numerical studies the initial state of the field quenches is the symmetric superposition of the two fully magnetized states according to For the rescaled modulus of the magnetization T = 0) of gapped quantum systems correlations decay exponentially with the distance83d between two sites is measured in the Manhattan metric, i.e. if r = R − R′ defines the relative position of the sites we have D the dimensionality (here D = 2). Thus for a given distance d there may be several r. The strength of the correlation between two sites depends on the number of shortest paths between them, which depends on r. We compare the time-averaged rt-VMC results after the quenches (red) to the correlation functions for the system in equilibrium at the temperature Teff attributed to the quench (black). For the 4 × 4 system we additionally show results of the exact time evolution (green). For the interaction quenches we do a least-square fit with a · ebd. We observe that the decay of the correlations in the thermal system as well as in the quenched system is well described by the fit curves in agreement with the analytic results. For the field quenches we have added the square of the rescaled modulus of the magnetization. We observe that within the numerical error its value and the correlation function for large d coincide as predicted. Comparing the decay of the correlations for the thermal system to the decay in the quenched system we find for the interaction quenches a good agreement both for the 4 × 4 and the 12 × 12 system. The curves show only small deviations and have the same shape going to 0 for large distances d. The rt-VMC results show a very good agreement to the exact results apart from the interaction quench  → . For the field quenches the differences between the time-averaged results after the quenches and the thermal results are more significant. For small quenches the shapes of the curves are very similar and the values coincide within the numerical error. For larger quenches we still observe a good agreement between the rt-VMC results and the exact results, but here larger deviations from the thermal curves arise. Although the absolute differences between the curves for a given value of d are not too large, the decay of the correlations as function of d is different for strong field quenches. After the field quenches the correlations decay faster with d than for the system in thermal equilibrium and the stationary long-range value of the correlations is reached at larger distances d and is lower than for the thermal system.As the last point of our studies we consider correlation functions between spins which are not nearest neighbours. Due to the long-range order in the ferromagnetic phase the correlation function of two spins will not vanish at large distances but reach a constant non-vanishing value given by the square of the expectation value of the rescaled modulus of the magnetization. In the paramagnetic phase on the other hand there is no long-range order and the Hamiltonian is gapped, i.e. there is an energy gap between the ground state and the first excited state. It has been shown that in the ground state 〉 for the rt-VMC in the ferromagnetic phase. For vanishing transverse field h the states of the x-basis are eigenstates of the Hamiltonian. Their energies are determined just by the number n of kinks. Turning on the transverse field, the Hamiltonian is not diagonal in the x-basis any more. The effect of the non-diagonal part of the Hamiltonian is to flip the orientation of spins. For small system sizes we computed the eigenstates of the Hamiltonian with exact diagonalization and found that for small ratios h/J basis states of the x-basis with the same number n of kinks and the same magnetization, which we determine by the number m of spin down, have in good approximation the same coefficients, so thatWe use m,n〉 according to h/J. For this reason we only apply it to field quenches starting from the completely ordered state and not leaving the ferromagnetic phase. Another important point for the accuracy of the ansatz is the spatial dimension D of the system, as for a fixed number N of sites the number of symmetries within the system increases in higher dimensions. With increasing number of symmetries the number of basis states which can be transformed into each other by symmetry transformations like translation, rotation or reflection, also increases. For basis states which can be transformed into each other by symmetry transformations the description with the same variational parameter is exact.with |ΨUsing the approximation in αm,n(t) are the variational parameters of the rt-VMC for the field quenches in the ferromagnetic phase.The As shown in the main part of the text by a comparison to results of exact time evolution for the 4 × 4 system, using the ansatz function (42) for the rt-VMC calculations allows an accurate description of the time evolution of the 2D-TFIM after a field quench starting from the completely ordered state.The exact time evolution of the variational state in and its variational time dynamicsThus inserting Obviously k′, multiplication with i and division through Summation over How to cite this article: Blaß, B. and Rieger, H. Test of quantum thermalization in the two-dimensional transverse-field Ising model. Sci. Rep.6, 38185; doi: 10.1038/srep38185 (2016).Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


We also discuss two approaches for achieving asymmetric scattering from metasurfaces, and realizing metasurfaces which overcome the performance limitations of single layer ultrathin metasurfaces.We present a set of universal relations which relate the local transmission, reflection, and polarization conversion coefficients of a general class of non-magnetic passive ultrathin metasurfaces. We show that these relations are a result of equal forward and backward scattering by single layer ultrathin metasurfaces, and they lead to confinement of the transmission, reflection, and polarization conversion coefficients to limited regions of the complex plane. Using these relations, we investigate the effect of the presence of a substrate, and show that the maximum polarization conversion efficiency for a transmissive metasurface decreases as the refractive index contrast between the substrate and cladding layer increases. Furthermore, we demonstrate that a single layer reflective metasurface can achieve full 2 Metamaterials are artificial materials with electromagnetic properties which are controllable by design. They are generally three dimensional periodic arrays of scatterers with deep subwavelength periods, and their electromagnetic properties can be fully represented by their permittivity and permeability tensors. Metasurfaces are two dimensional counterparts of the metamaterials. They are composed of an array of scatterers with sub-wavelength period which are located on a planar surface. Ultrathin metasurfaces are a class of metasurfaces composed of scatterers which are significantly thinner than the wavelength of the light13458910114513π phase coverage cannot be achieved using reflective ultrathin metasurfaces unless the metasurface is illuminated from the material with higher refractive index. We also show that the inability of the transmissive metasurfaces for achieving 2π phase coverage is conserved for lossy metasurfaces and asymmetric metasurfaces (i.e. when the substrate and cladding have different refractive indices). We also demonstrate that the maximum achievable polarization conversion efficiency depends on the refractive index contrast between the substrate and the cladding layer, and it decreases for transmissive metasurfaces as this refractive index contrast increases, while it increases beyond 25% for reflective metasurfaces illuminated from the material with higher refractive index. We show that the main source of these performance limitations is the lack of phase retardation as the light propagates through the metasurface layer, and we suggest two methods for realization of metasurfaces with higher efficiencies.Materials with plasmonic resonances such as gold and silver are popular choices for the metasurface layer. This is because these materials have large negative permittivity; therefore, even a thin layer of them can scatter the light significantly. One of the well-known drawbacks of using these materials is their substantial absorption loss which limits the efficiency of the diffractive elements. As a result, most of the work in this area has been limited to the near and mid-infrared, terahertz, and microwave wavelengths range where the absorption losses are smaller271517n2 and are surrounded by a cladding material with the refractive index of n1. We assume that the scatterers have a thickness of h which is much smaller than the wavelength of light in the cladding material (i.e. h ≪ λ1 = λ0/n1). The metasurface locally modifies the amplitude, phase, or polarization of an incident light either in transmission or reflection. For weakly coupled and gradually varying scatterers, the metasurface can be modeled as a surface with spatially dependent local reflection and transmission coefficients.We consider an ultrathin metasurface which is composed of an array of potentially dissimilar subwavelength passive and non-magnetic scatterers. A schematic illustration of such a metasurface is depicted in For each of the scatterers comprising the metasurface, we can form a periodic metasurface by arranging that scatterer on a periodic lattice similar to the lattice of the original metasurface in the vicinity of that scatterer. An example of such a periodic metasurface is shown in t|| and r||, respectively. For a linearly polarized light, t|| and r|| represent the transmission and reflection coefficients of the parts of the light with the same polarization as the incident light. These coefficients are the total transmission and reflection coefficients for a metasurface that does not modify the polarization. Polarization modification happens if the light reflected from or transmitted through the metasurface acquire some polarization component orthogonal to the polarization of a plane wave reflected from or transmitted through the bare interface. We represent the transmission and reflection coefficients for the parts of the light whose polarization has been converted to the orthogonal polarization by t⊥ and r⊥, respectively. We refer to t⊥ and r⊥ as polarization conversion coefficients. For example, if the incident plane wave is right handed circularly polarized then t|| and t⊥ are the transmission coefficients for the parts of the transmitted light which are right and left handed circularly polarized, respectively, while r|| and r⊥ are the reflection coefficients for the parts which are left and right handed circularly polarized, respectively.We assume that a plane wave is normally incident on the periodic metasurface as shown in Since the materials composing the periodic metasurface are non-magnetic, we can replace the entire metasurface by an equivalent volume electric current densityω is the angular frequency of the incident light, ε0 is the vacuum permittivity, nms is the complex refractive index of the metasurface materials, and E is the total electric fieldJe is nonzero only at the location of the metasurface, and is confined to the thickness h above the substrate as illustrated in Je in the presence of the substrate-cladding interface is equal to the light scattered by the metasurface. The volume current density Je is also periodic with the same period as the periodic metasurface it has replaced; therefore, it emits into the substrate and top cladding materials only along the interface normal directions  and below (Ez−) the slab region are given by ref. 20To find the amplitudes of the plane waves emitted by k1 = n12π/λ0, Z0 is the impedance of free space, Jt is the components of Je parallel to the xy plane, and k1z| < πh/λ1 ≪ 1, the where z, we can interpret the first term inside the parentheses in (3) as electric field emitted from an effective surface electric current and the second term as electric field emitted by an effective surface magnetic current. If Je along the ±z directions are equal to each other. As we mentioned earlier, the light does not accumulate significant phase as it propagates through an ultrathin metasurface which operates based on creating a phase discontinuity. Therefore, the phase of the electric field E inside the metasurface layer and, according to (1), the phase of Je does not vary significantly along the propagation direction inside ultrathin metasurfaces. As a result, Je toward the ±z directions would be equal to each other.Considering the polarity of the electric fields of the light emitted along ±Je toward the −z direction is reflected back at the substrate-cladding interface. We obtain the electric field amplitudes of the plane waves emitted by Je inside the substrate (Es−) and cladding (Es+) asSome of the light emitted by r0 = (n1 − n2)/(n1 + n2) and t0 = 2n1/(n1 + n2) are respectively the reflection and transmission coefficients of the bare substrate-cladding interface for normally incident plane waves. To obtain (5), we have used exp(ik1h) ≈ 1 and the relation t0 = 1 + r0. Since the electric fields of the optical waves scattered by the metasurface into the substrate and the cladding are equal to each other, we omit the + and − subscripts and represent both by Es. According to the superposition principleJe. As shown in Es is decomposed into two plane waves with orthogonal polarizations. The parts of Es which have the same polarizations as Es||, and the parts which have polarizations orthogonal to them are shown by Es⊥.where Since the periodic metasurface is passive, the sum of the transmitted and reflected powers is equal to, or smaller than (for lossy metasurfaces) the incident power, that isFor the bare interface we haveWe can express the transmission and reflection coefficients of the periodic metasurface in terms of the electric field amplitudes asUsing (6) to (11), we find the fundamental relations of ultrathin metasurfaces asθi with respect to the interface normal direction, relation (12) is still valid and metasurface relations (13) and (14) are modified asBy using a similar procedure, we can show see  that whefor a transverse electric (TE) polarized incident plane wave, and asθr is the angle of refraction and for a transverse electric (TM) polarized incident light. Here, L, where L is the fraction of the light absorbed by the metasurface.The transmission and reflection coefficients of any periodic ultrathin metasurface satisfy (12)–(14). As we mentioned earlier, the local transmission and reflection coefficients of a metasurface with gradually enough varying parameters is similar to those of a periodic metasurface without the gradual variations of the metasurface. Therefore, the relations (12)–(14) are valid for the local transmission, reflection and polarization conversion coefficients of an ultrathin metasurface. It should be noted that the limitations expressed by (12)–(14) are merely the results of deep subwavelength thickness of the metasurface, and are valid regardless of the potential absorption loss caused by metasurfaces. Material absorption loss will tighten the limit in (12) even further. More specifically, for a lossy metasurface the left hand side of (12) is equal to 1 − In the following, as two example cases, we discuss the implications that the relations (12)–(14) have on the performance of reflective and transmissive metasurfaces designed for shaping the phase front, and modifying the polarization of a normally incident light.Consider an ultrathin metasurface which is designed to shape the phase front of an incident beam to a desired form without disturbing its polarization. For such a metasurface, by using (12) and (13), we findWe can also express (19) in terms of reflection coefficient asπ/2, π/2) interval. Therefore, a transmissive ultrathin metasurface cannot provide full control over the phase of the transmitted light which has the same polarization as the incident light. A special case of this result for a lossless and reciprocal metasurface with n1 = n2 is previously presented in ref. n1 ≤ n2 , the attainable reflection coefficients also only cover a phase range smaller than π and therefore an arbitrary reflective phase mask cannot be implemented using a metasurface. When the light is incident from the material with larger refractive index (i.e. n1 > n2), the phase of the reflection coefficient might cover the full 2π range, but if n1 and n2 are of the same order of magnitude then the reflection efficiency cannot be large for all phases. For example, if the ultrathin metasurface is fabricated on a silicon substrate with refractive index of n1 = 3.48 and cladded with air, then the reflection efficiency of an infrared light which is incident from the silicon side and is reflected with zero phase is smaller than (n1 − n2)2/(n1 + n2)2 = 31%.The inequalities (19) and (20) limit the transmission and reflection coefficients to two circles in the complex plane as shown in Ultrathin metasurfaces can also be designed to modify the polarization of an incident light. Polarization modification is achieved by converting the polarization of all or part of the incident light to a polarization orthogonal to that of the incident light. For such metasurfaces, by using (12)–(14) we obtainThe right hand side of (21) can be simplified further ast|||, and the right hand side of the inequality represents the maximum of this quadratic function. By combining (21) and (22) we find the limit on the polarization conversion efficiency of the transmitted light asThe left hand side of the last inequality in (22) is a quadratic function of |and by using (14) we obtain the limit on the polarization conversion efficiency in reflection asn2 = 1.44 is about 24%. As another example, a transmissive metasurface quarter-wave plate fabricated on the fused silica substrate cannot be more than 48% efficient in converting the polarization from linear to circular. Similar limits can be established for efficiencies of any metasurface that modifies the polarization of light. We note that (23) reduces to |t⊥|2 = 0.25% which has been previously shown for a lossless and reciprocal metasurface when n1 = n2t⊥ and r⊥ can achieve. As this figure shows, the phases of t⊥ and r⊥ may take any values in the  interval, but the transmission and reflection efficiencies are limited to the values enforced by (23) and (24), respectively. Therefore, a limited efficiency transmissive phase mask which imposes an arbitrary phase profile might be implemented using a metasurface if we allow the polarization of the phase shifted transmitted light to be orthogonal to that of the incident light. Such metasurfaces which implement phase masks corresponding to a lens and an axicon have been previously reportedThe inequalities (23) and (24) limit the performance of a metasurface designed to manipulate the light’s polarization. For example, the maximum efficiency of a transmissive metasurface half-wave plate which rotates the polarization of a linearly polarized beam by 90 degrees, and is fabricated on a fused silica substrate with the refractive index of The maximum polarization conversion efficiencies for a metasurface as a function of the ratio of the substrate to cladding refractive indices have been plotted in The approximation of the local transmission and reflection coefficients of a gradient metasurface with those of a periodic one composed of the same shape scatterers has been successfully used in many metasurface designs reported so far. Although we have also used this assumption here, its accuracy is not essential in the derivation of the limits because we do not use this assumption directly. The main assumption used is the locality of the scattering from the metasurface which allows for defining local transmission and reflection coefficients and the local conservation of energy (i.e. (6)). We also note that, because of the continuity of the tangential electric field at the metasurface interface, nonlocal ultrathin metasurfaces also scatter light symmetrically  in the substrate and cladding; however, the results presented here are expressed in terms of local transmission and reflection coefficients and cannot be directly applied to them.I and thus Je to vary at least by π as a function of z within the thickness of the metasurface. As we discussed, this cannot happen for a single layer ultrathin metasurface. However, a multilayer metasurface might be designed such that the phase of I varies among different layers. This has been shown in the microwave regime162223242518272829k1h, for it to become comparable to the symmetric term, large values of effective current density are required which lead to large absorption losses. This is in agreement with the operation of the multilayer metasurfaces close to resonances where the values of the effective current density are large. We also note that the presented limits are valid for local reflection and transmission coefficients of an interface containing a single layer metasurface, and optical response of a structure containing multiple cascaded interfaces, in general, is not bound by these limits and can be computed using well-known techniques such as the transfer matrix methodThe limitations we presented here are results of the non-directionality of the scattering by the ultrathin metasurfaces. In other words, the scattered electric field of an ultrathin metasurface is equal in both the substrate and the cladding layers. Therefore, to overcome these limitations, it is essential to break this symmetric scattering. From (3), we see that the asymmetric emission can only be achieved if the anti-symmetric term (which corresponds to an effective magnetic current) becomes comparable to the symmetric term. For this to happen, Another approach for overcoming the ultrathin metasurface limits is to use thicker scatterers. Scatterers with a thickness on the order of a wavelength may scatter the light with unequal electric fields into the substrate and cladding layers; therefore, their operations are not limited by the fundamental ultrathin metasurface relations. If we use lossy materials such as gold or silver then thick scatterers absorb significantly, and the absorption limits the efficiency of such metastructures3335363738394041424344454647484950404144We showed that the local transmission, reflection and polarization conversion coefficients of a non-magnetic passive ultrathin metasurface satisfy a set of fundamental relations. These fundamental relations enforce some theoretical limitations on the complex values of these coefficients. We demonstrated that these theoretical limitations are the direct results of the lack of significant phase retardation during the light propagation through the metasurface layer. We can use two approaches to overcome the limitations of ultrathin single layer metasurfaces: cascading ultrathin metasurfaces with other ultrathin metasurfaces, or using high contrast transmitarrays. The latter approach not only is not restricted by the ultrathin metasurface fundamental relations but also does not suffer from the material absorption loss.How to cite this article: Arbabi, A. and Faraon, A. Fundamental limits of ultrathin metasurfaces. Sci. Rep.7, 43722; doi: 10.1038/srep43722 (2017).Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


We prove strong convergence of order  We consider time-homogeneous stochastic differential equations (SDEs) of the formThe Euler–Maruyama approximation with step-size mentclass2pt{minimmentclass2pt{minimmentclasspt{minimaIn applications, frequently SDEs with less regular coefficients appear. For example in stochastic control theory, whenever the optimal control is of bang–bang type, meaning that the strategy is of the form If  Zvonkin  and Vere Zvonkin , 26 prov Zvonkin  extends In  existencCurrently, research on numerical methods for SDEs with irregular coefficients is highly active. Hutzenthaler et al.  introducHalidias and Kloeden  show thaThe method introduced in  is the fFirst results in this direction are contained in a series of papers by Ngo and Taguchi. In  they shoContrasting the above, there are several delimiting results which state that even equations with infinitely often differentiable coefficients cannot always be solved approximately in finite time, even if the Euler–Maruyama method converges, cf. Hairer et al. , JentzenIn this paper we prove strong convergence of order Our convergence proof is based on estimating the difference between the Euler–Maruyama scheme and the scheme presented in . Close tThis paper’s result is the first one that gives strong convergence and also a strong convergence rate of a fully explicit scheme for multidimensional SDEs with discontinuous drift and degenerate diffusion coefficient, and the first one for multidimensional SDEs with discontinuous drift that does not satisfy a one-sided Lipschitz condition.In this section we first state the assumptions on the coefficients of SDE , under wWe want to prove strong convergence of the Euler–Maruyama method for SDEs with discontinuous drift coefficient. Instead of the usual requirement of Lipschitz continuity we only assume that the drift is a piecewise Lipschitz function on the intrinsic metricA is given by x to y.For a continuous curve f is intrinsic Lipschitz, if it is Lipschitz w.r.t. the intrinsic metric on A, i.e. if there exists a constant L such that Let ).Fix D in direction n(p(x)) in x is 1. From this and from  is orthogonal to Let b. Let For U is open and Since m or m continuously to Since b, then we could use the same mapping m continuously beyond Now, if We will need the following estimate on the local time of a one-dimensional Itô process.Let Then The claim is a special case of [From the Meyer–Tanaka formula [We are ready to prove the result of this section.Let X: let Y as follows:Next we decompose the path of Y with the property that the occupation time of Y in an environment of 0 is the same as the occupation time of X in an environment of We have thus constructed a continuous Y is an Itô process, we want to use Itô’s formula. For this we recognize that Y, depending on its proximity to Y are uniformly bounded. Therefore To show that Let The proof of convergence is based on a transformation that removes the discontinuity from the drift and makes the drift Lipschitz while preserving the Lipschitz property of the diffusion coefficient. A suitable transform is presented in . We recac andDefine c is chosen sufficiently small, see [G is invertible by [G and If With this we can define a process We are ready to formulate the main result.C such that for all Let Assumption In preparation of the proof of the main result, we prove two lemmas.C such that for sufficiently small step size Let Assumption By the definition of the Euler–Maruyama method  we have\For all C such that for all Let Assumption Now, we are ready to prove our main result.Z. It holds thatG applied to the Euler–Maruyama approximation of X and the Euler–Maruyama approximation of Z. Denote, for all d-dimensional Burkholder–Davis–Gundy inequality [C, for arbitrarily small Since mentclasspt{minima,ε,jc in , we get\mentclasspt{minimaWe ran simulations for several examples—ones of theoretical interest as well as an example coming from applications.When studying stochastic dynamical systems which include a noisy signal, then filtering this signal leads to a higher dimensional system with a degenerate diffusion coefficient. Stochastic control problems often lead to an optimal control policy which makes the drift of the system discontinuous. Examples are models with incomplete market information in mathematical finance where the rate with which cashflows are paid from a firm value process change systematically when the asset-liability ratio passes a certain threshold which then triggers a rating change.The class of equations studied here appears frequently in several areas of applied mathematics and the natural sciences.In the first example the drift is the step function In this example the drift has a discontinuity along the unit circle, and the diffusion coefficient is degenerate on the whole of In insurance mathematics, a well-studied problem is the maximization of the expected discounted future dividend payments until the time of ruin of an insurance company, a value which serves as a risk measure. In  the probely, see . The funWe need to check Assumption The Figure Even though the proven rate for the Euler–Maruyama method is lower than for the transformation-based method from , the calG, then For completeness, we remark that one can construct examples, where the transformation method is much faster while giving a smaller error. For example, start with prescribing the transform In this paper we have for the first time proven strong convergence and also a positive strong convergence rate for an explicit method (the Euler–Maruyama method) for multidimensional SDEs with discontinuous drift that has a degenerate diffusion coefficient, or with a discontinuous drift that does not satisfy a one-sided Lipschitz condition, or both. The Euler–Maruyama method has the advantage that it does not need the exact form of the set of discontinuities of the drift as an input, and that in practice, computation of one path is fast in comparison to the second method in the literature that can deal with this class of SDEs. Our numerical experiments suggest that in addition to these advantages, it even seems that the Euler–Maruyama method converges at a higher than the theoretically obtained rate for many examples and it will be a topic of future research to prove sharpness, or find a sharp bound.