Non-trivial Boundaries Research Articles

Mitigating density fluctuations in particle-based active nematic simulations

Understanding active matter has led to new perspectives on biophysics and non-equilibrium dynamics. However, the development of numerical tools for simulating active fluids capable of incorporating non-trivial boundaries or inclusions has lagged behind. Active particle-based methods, which typically excel at this, suffer from large density fluctuations that affect the dynamics of inclusions. To this end, we advance the Active-Nematic Multi-Particle Collision Dynamics algorithm, a particle-based method for simulating active nematics, by addressing the large density fluctuations that arise from activity. This paper introduces three activity formulations that mitigate the coupling between activity and local density. Local density fluctuations are decreased to a level comparable to the passive limit while retaining active nematic phenomenology and increasing the active turbulence regime four-fold in two dimensions. These developments extend the technique into a flexible tool for modeling active systems, including solutes and inclusions, with broad applications for the study of biophysical systems.

Controllable Floquet edge modes in a multifrequency driving system

A driven quantum system was recently studied in the context of nonequilibrium phase transitions and their responses. In particular, for a periodically driven system, its dynamics are described in terms of the multidimensional Floquet lattice with a lattice size depending on the number of driving frequencies and their rational or irrational ratio. So far, for a multifrequency driving system, the energy pumping between the sources of frequencies has been widely discussed as a signature of topologically nontrivial Floquet bands. However, the unique edge modes emerging in the Floquet lattice have not been explored yet. Here, we discuss how the edge modes in the Floquet lattice are controlled and result in the localization at particular frequencies, when multiple frequencies are present and their magnitudes are commensurate values. First, we construct the minimal model to exemplify our argument, focusing on a two-level system with two driving frequencies. For the strong frequency limit, one can describe the system as a quasi-one-dimensional Floquet lattice where the effective hopping between the neighboring sites depends on the relative magnitudes of the potential for two frequency modes. With multiple driving modes, nontrivial Floquet lattice boundaries always exist from controlling the frequencies, and this gives rise to states that are mostly localized at such Floquet lattice boundaries, i.e., particular frequencies. We suggest the time-dependent Creutz ladder model as a realization of our theoretical Hamiltonian and show the emergence of controllable Floquet edge modes.

Fast $L^2$ Optimal Mass Transport via Reduced Basis Methods for the Monge--Ampère Equation

Repeatedly solving the parameterized optimal mass transport (pOMT) problem is a frequent task in applications such as image registration and adaptive grid generation. It is thus critical to develop a highly efficient reduced solver that is equally accurate as the full order model. In this paper, we propose such a machine learning--like method for pOMT by adapting a new reduced basis (RB) technique specifically designed for nonlinear equations, the reduced residual reduced over-collocation (R2-ROC) approach, to the parameterized Monge--Ampère equation. It builds on top of a narrow-stencil finite difference method (FDM), a so-called truth solver, which we propose in this paper for the Monge--Ampère equation with the transport boundary. Together with the R2-ROC approach, it allows us to handle the strong and unique nonlinearity pertaining to the Monge--Ampère equation achieving online efficiency without resorting to any direct approximation of the nonlinearity. Several challenging numerical tests demonstrate the accuracy and high efficiency of our reduced solver for solving the parameterized Monge--Ampère equation, effectively transporting the nontrivial boundaries.

On weak lensing response functions

We introduce the response function approach to model the weak lensing statistics in the context of separate universe formalism. Numerical results for the RFs are presented for various semi-analytical models that includes perturbative modelling and variants of halo models. These results extend the recent studies of the Integrated Bispectrum and Trispectrum to arbitrary order. We find that due to the line-of-sight projection effects, the expressions for RFs are not identical to the squeezed correlation functions of the same order. We compute the RFs in three-dimensions using the spherical Fourier-Bessel formalism which provides a natural framework for incorporating photometric redshifts, and relate these expressions to tomographic and projected statistics. We generalise the concept of k-cut power spectrum to k-cut response functions. In addition to response functions, we also define their counterparts in real space, since they are easier to estimate from surveys with low sky-coverage and non-trivial survey boundaries.

One-dimensional lattice models for the boundary of two-dimensional Majorana fermion symmetry-protected topological phases: Kramers-Wannier duality as an exact Z2 symmetry

It is well known that symmetry-protected topological (SPT) phases host nontrivial boundaries that cannot be mimicked in a lower-dimensional system with a conventional realization of symmetry. However, for SPT phases of bosons (fermions) within the cohomology (supercohomology) classification the boundary can be recreated without the bulk at the cost of a non-onsite-symmetry action. This raises the following question: Can one also mimic the boundaries of SPT phases which lie outside the (super)cohomology classification? In this paper, we study this question in the context of $(2+1)$-dimensional $[(2+1)\mathrm{D}]$ fermion SPTs. We focus on the root SPT phase for the symmetry group $G={Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{f}$. Starting with an exactly solvable model for the bulk of this phase constructed by Tarantino and Fidkowski, we derive an effective one-dimensional (1D) lattice model for the boundary. Crucially, the Hilbert space of this 1D model does not have a local tensor product structure, but rather is obtained by placing a local constraint on a local tensor product Hilbert space. We derive the action of the ${Z}_{2}$ symmetry on this Hilbert space and find a simple three-site Hamiltonian that respects this symmetry. We study this Hamiltonian numerically using exact diagonalization and DMRG and find strong evidence that it realizes an Ising conformal field theory where the ${Z}_{2}$ symmetry acts as the Kramers-Wannier duality; this is the expected stable gapless boundary state of the present SPT. A simple modification of our construction realizes the boundary of the $(2+1)\mathrm{D}$ topological superconductor protected by time-reversal symmetry $\mathcal{T}$ with ${\mathcal{T}}^{2}={(\ensuremath{-}1)}^{\mathcal{F}}$.

Flat-floor bubbles, dark solitons, and vortices stabilized by inhomogeneous nonlinear media

We consider one- and two-dimensional (1D and 2D) optical or matter-wave media with a maximum of the local self-repulsion strength at the center, and a minimum at periphery. If the central area is broad enough, it supports ground states in the form of flat-floor \textquotedblleft bubbles", and topological excitations, in the form of dark solitons in 1D and vortices with winding number $m$ in 2D. Unlike bright solitons, delocalized bubbles and dark modes were not previously considered in this setting. The ground and excited states are accurately approximated by the Thomas-Fermi expressions. The 1D and 2D bubbles, as well as vortices with $m=1$, are completely stable, while the dark solitons and vortices with $m=2$ have nontrivial stability boundaries in their existence areas. Unstable dark solitons are expelled to the periphery, while unstable double vortices split in rotating pairs of unitary ones. Displaced stable vortices precess around the central point.

Quiescent Gap Solitons in Coupled Nonuniform Bragg Gratings with Cubic-Quintic Nonlinearity

We study the stability characteristics of zero-velocity gap solitons in dual-core Bragg gratings with cubic-quintic nonlinearity and dispersive reflectivity. The model supports two disjointed families of gap solitons (Type 1 and Type 2). Additionally, asymmetric and symmetric solitons exist in both Type 1 and Type 2 families. A comprehensive numerical stability analysis is performed to analyze the stability of solitons. It is found that dispersive reflectivity improves the stability of both types of solitons. Nontrivial stability boundaries have been identified within the bandgap for each family of solitons. The effects and interplay of dispersive reflectivity and the coupling coefficient on the stability regions are also analyzed.

Visualization of Thermal Transport within and between Carbon Nanotubes.

Despite the excellent thermal properties of individual carbon nanotubes (CNTs), the thermal characteristics of macroscopic CNT assemblies are poor. This is probably due to the presence of numerous nontrivial intertube boundaries. Currently, clarity on the inherent difference between intra- and inter-CNT thermal conductivities is not well-established. Herein, we report an in situ nanoscale observation on the anisotropic thermal transport of a single bundle of CNTs by monitoring evaporated gold nanoparticles as "thermomarkers". The experimental results indicate that even a small bundle shows colossal thermal anisotropy due to the intertube boundaries. The results are validated by finite element analysis that estimates an anisotropic thermal conductivity ratio greater than 100. The estimated value is much greater than most of the reported values measured on macroscopic specimens and matches with that of highly ordered pyrolytic graphite. Our study reveals the intrinsic thermal anisotropy of bundled CNTs and aids in visualizing nanoscale thermal transport.

PORTFOLIO OPTIMIZATION UNDER A QUANTILE HEDGING CONSTRAINT

We study a problem of portfolio optimization under a European quantile hedging constraint. More precisely, we consider a class of Markovian optimal stochastic control problems in which two controlled processes must meet a probabilistic shortfall constraint at some terminal date. We denote by [Formula: see text] the corresponding value function. Following the arguments introduced in the literature on stochastic target problems, we convert this problem into a state constraint one in which the constraint is defined by means of an auxiliary value function [Formula: see text] characterizing the reachable set. This set is therefore not given a priori but is naturally integrated in [Formula: see text] solving, in a viscosity sense, a nonlinear parabolic partial differential equation (PDE). Relying on the existing literature, we derive, in the interior of the domain, a Hamilton–Jacobi–Bellman characterization of [Formula: see text]. However, [Formula: see text] involves an additional controlled state variable coming from the diffusion of the probability of reaching the target and belonging to the compact set [Formula: see text]. This leads to nontrivial boundaries for [Formula: see text] that must be discussed. Our main result is thus the characterization of [Formula: see text] at those boundaries. We also provide examples for which comparison results exist for the PDE solved by [Formula: see text] on the interior of the domain.

Scenario to chaos and multistability in a modified Coullet system: effects of broken symmetry

Recently, the study of nonlinear systems with multiple attractors has drained the attention of researchers worldwide owing to its fundamental and technological relevance. In this work, we consider a modified Coullet system with a cubic polynomial nonlinearity in the form $$ \varphi_{m} \left( x \right) = x - mx^{2} - x^{3} $$ where $$ m $$ is a control parameter. The new system bridges the gap between chaotic systems with symmetry and those without symmetry. In fact for $$ m = 0 $$ , the system displays a perfect symmetry which is reflected on the location of equilibria, the type of attractors, and the shape of basins of attraction as well. In this special case, multistability involves a pair of mutually symmetric periodic or chaotic attractors, a pair of chaotic attractors with a pair of limit cycles, two pairs of limit cycles with a pair of chaotic attractors, and so on. For $$ m \ne 0 $$ , the system is non symmetric and more complex nonlinear phenomena are observed such as parallel bifurcation branches, coexisting multiple asymmetric attractors, and hysteresis. For instance, the coexistence of a stable fixed point with a limit cycle, two different non symmetric limit cycles or chaotic attractors, three or four different non symmetric periodic and chaotic attractors are reported when monitoring the system parameters and initial conditions as well. Basins of attraction with non-trivial basin boundaries structures are computed to visualize the magnetization of the state space in the presence of multiple attractors. Experimental results based on a suitable electronic implementation of the proposed jerk system are included.

Three-dimensional simulation of gas–solid–liquid flows using the DEM–VOF method

Gas–solid–liquid flows are widely found in chemical engineering, e.g. the processes of mixing, wet ball milling, and screw kneading, for which the numerical modeling is now a pressing research topic to help improve the design and investigate operational conditions. On the other hand, computational challenges are posed for existing methods in modeling the interphase interactions and complex boundaries within such three-phase systems. In this paper, an Eulerian–Lagrangian numerical model, specifically the DEM–VOF method, is presented for three-dimensional simulations of gas–solid–liquid flows. The fluid motion is solved by using a computational fluid dynamics (CFD) based approach with gas–liquid interface capturing provided by the volume-of-fluid (VOF) method. The particle phase is tracked by the discrete element method (DEM) as discrete entities. The fluid–particle coupling is achieved by the volume-averaging technique wherein a well-established empirical closure is adopted for the description of hydrodynamic forces. Particularly, the modeling of arbitrary-shaped walls and moving boundaries is addressed via the introduction of signed distance function (SDF) representation and immersed boundary (IB) method, which was proved to be highly efficient for gas–solid–liquid systems interacting with complex geometries. Special attentions when computing fluid–particle interactions near those boundaries are raised and their treatments are also discussed. Various model verifications and validation tests are performed in this study to show the validity and capability of the DEM–VOF method. By comparing with analytical solutions and experimental data, we generally find good agreements from the simulation results, thereby highlighting its potential in accurately modeling complicated gas–solid–liquid flows. To the best of our knowledge, the proposed method is the first report that successfully couples the DEM to a VOF solver with non-trivial wall boundaries.

Exploring torus universes in causal dynamical triangulations

Motivated by the search for new observables in nonperturbative quantum gravity, we consider causal dynamical triangulations in $2+1$ dimensions with the spatial topology of a torus. This system is of particular interest, because one can study not only the global scale factor, but also global shape variables in the presence of arbitrary quantum fluctuations of the geometry. Our initial investigation focuses on the dynamics of the scale factor and uncovers a qualitatively new behavior, which leads us to investigate a novel type of boundary conditions for the path integral. Comparing large-scale features of the emergent quantum geometry in numerical simulations with a classical minisuperspace formulation, we find partial agreement. By measuring the correlation matrix of volume fluctuations we succeed in reconstructing the effective action for the scale factor directly from the simulation data. Apart from setting the stage for the analysis of shape dynamics on the torus, the new setup highlights the role of nontrivial boundaries and topology.

Running boundary actions, Asymptotic Safety, and black hole thermodynamics

Previous explorations of the Asymptotic Safety scenario in Quantum Einstein Gravity (QEG) by means of the effective average action and its associated functional renormalization group (RG) equation assumed spacetime manifolds which have no boundaries. Here we take a first step towards a generalization for non-trivial boundaries, restricting ourselves to action functionals which are at most of second order in the derivatives acting on the metric. We analyze two examples of truncated actions with running boundary terms: full fledged QEG within the single-metric Einstein-Hilbert (EH) truncation, augmented by a scale dependent Gibbons-Hawking (GH) surface term, and a bi-metric truncation for gravity coupled to scalar matter fields. The latter contains 17 running couplings, related to both bulk and boundary terms, whose beta-functions are computed in the induced gravity approximation. We find that the bulk and the boundary Newton constant, pertaining to the EH and GH term, respectively, show opposite RG running; proposing a scale dependent variant of the ADM mass we argue that the running of both couplings is consistent with gravitational anti-screening. We describe a simple device for counting the number of field modes integrated out between the infrared cutoff scale and the ultraviolet. This method makes it manifest that, in an asymptotically safe theory, there are effectively no field modes integrated out while the RG trajectory stays in the scaling regime of the underlying fixed point. As an application, we investigate how the semiclassical theory of Black Hole Thermodynamics gets modified by quantum gravity effects and compare the new picture to older work on `RG-improved black holes' which incorporated the running of the bulk Newton constant only. We find, for instance, that the black hole's entropy vanishes and its specific heat capacity turns positive at Planckian scales.

Static solutions with nontrivial boundaries for the Einstein-Gauss-Bonnet theory in vacuum

The classification of certain class of static solutions for the Einstein-Gauss-Bonnet theory in vacuum is performed in $d\geq5$ dimensions. The class of metrics under consideration is such that the spacelike section is a warped product of the real line and an arbitrary base manifold. It is shown that for a generic value of the Gauss-Bonnet coupling, the base manifold must be necessarily Einstein, with an additional restriction on its Weyl tensor for $d>5$. The boundary admits a wider class of geometries only in the special case when the Gauss-Bonnet coupling is such that the theory admits a unique maximally symmetric solution. The additional freedom in the boundary metric enlarges the class of allowed geometries in the bulk, which are classified within three main branches, containing new black holes and wormholes in vacuum.

One-point functions in massive integrable QFT with boundaries

We consider the expectation value of a local operator on a strip with non-trivial boundaries in 1+1 dimensional massive integrable QFT. Using finite volume regularisation in the crossed channel and extending the boundary state formalism to the finite volume case we give a series expansion for the one-point function in terms of the exact form factors of the theory. The truncated series is compared with the numerical results of the truncated conformal space approach in the scaling Lee-Yang model. We discuss the relevance of our results to quantum quench problems.

The exact finite size ground state of the O(n = 1) loop model with open boundaries

We explicitly describe certain components of the finite size ground state of the inhomogeneous transfer matrixof the O(n = 1) loop model on a strip with non-trivial boundaries on both sides. In addition we computeexplicitly the ground state normalization which is given as a product of four symplecticcharacters.

Placement-Proximity-Based Voltage Island Grouping Under Performance Requirement

High power consumption not only leads to short battery life for hand-held devices but also causes on-chip thermal and reliability problems in general. As power consumption is proportional to the square of supply voltage, reducing supply voltage can significantly reduce power consumption. Multi-supply voltage (MSV) has previously been introduced to provide finer grain power and performance tradeoff. In this paper, we propose a methodology on top of a set of algorithms to exploit nontrivial voltage island boundaries for optimal power versus design-cost tradeoff under performance requirement. Our algorithms are efficient, robust, and error-bounded and can be flexibly tuned to optimize for various design objectives (e.g., minimal power within a given number of voltage islands, or minimal fragmentation in voltage islands within a given power bound) depending on the design requirement. Our experiment on real industry designs shows a tenfold improvement of our method over current logical-boundary-based industry approach.

Extension of Levi-flat hypersurfaces past CR boundaries

Local conditions on boundaries of C∞ Levi-flat hypersurfaces, in case the boundary is a generic submanifold, are studied. For nontrivial real analytic boundaries we get an extension and uniqueness result, which forces the hypersurface to be real analytic. This allows us to classify all real analytic generic boundaries of Levi-flat hypersurfaces in terms of their normal coordinates. For the remaining case of generic real analytic boundary we get a weaker extension theorem. We find examples to show that these two extension results are optimal. Further, a class of nowhere minimal real analytic submanifolds is found, which is never the boundary of even a C 2 Levi-flat hypersurface.

Optimised boundary compact finite difference schemes for computational aeroacoustics

A set of optimised boundary closure schemes is presented for use with compact central finite difference schemes in computational aeroacoustics (CAA) involving non-trivial boundaries. The boundary schemes are given in a form of non-central compact finite differences. They maintain fourth-order accuracy, a pentadiagonal matrix system and seven-point stencil which the main interior scheme employs. This paper introduces a new strategy to optimise the boundary schemes in the spectral domain and achieve the best resolution characteristics given a strict tolerance for the dispersion and dissipation errors. The boundary schemes are derived from sophisticated extrapolation of solutions outside the domain. The extrapolation functions are devised by combining polynomials and trigonometric series which contain extra control variables used to optimise the resolution characteristics. The differencing coefficients of the boundary schemes are determined in association with the existing coefficients of the interior scheme which is also optimised through an improved procedure in this paper. The accuracy of the proposed schemes is demonstrated by their application to CAA benchmark problems.

Ramp-induced transitions in traffic dynamics

We present the analytical results of ramp effects in asymmetric simple exclusion processes. Both on-ramp and off-ramp are included in between the two open boundaries. The ramps can be taken as the nontrivial boundaries to trigger the phase transitions. Exact phase diagrams are obtained analytically in the full parameter space. We find that the order of the two ramps is crucial. When the on-ramp is placed after the off-ramp along the traffic direction, there are only four distinct phases: free-free-free, free-free-jam, free-jam-jam, and jam-jam-jam. The other four phases from naive expectation cannot be realized, i.e., jam-free-free, jam-jam-free, jam-free-jam, and free-jam-free are all absent. The free flow will not follow the congestion. When the on-ramp is placed before the off-ramp, we observe an interesting phase: jam-max.-free. The bottleneck emerges as the flow in between the two ramps saturates to its maximum. We further show that the roadway configuration is equivalent to a nonstandard intersection. Applications to a traffic rotary are discussed.