Space-time Dependence Research Articles

All-loop Heavy-Heavy-Light-Light correlators in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super Yang-Mills theory

We study Heavy-Heavy-Light-Light (HHLL) correlators HHO2O2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\left\\langle \\mathcal{HH}{\\mathcal{O}}_2{\\mathcal{O}}_2\\right\\rangle $$\\end{document} in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super Yang-Mills theory with SU(N) gauge group at generic N. The light operator O2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{O}}_2 $$\\end{document} is the dimension two superconformal primary in the stress tensor multiplet and H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{H} $$\\end{document} is a general half-BPS superconformal primary operator with dimension (or R-charge) ∆H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\Delta }_{\\mathcal{H}} $$\\end{document}. We consider the large-charge ’t Hooft limit, where ∆H→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\Delta }_{\\mathcal{H}}\ o \\infty $$\\end{document} with fixed ’t Hooft-like coupling λ≔∆HgYM2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\lambda := {\\Delta }_{\\mathcal{H}}{g}_{\ extrm{YM}}^2 $$\\end{document}. We show that the L-loop contribution to the HHLL correlators in the leading large-charge limit is universal for any choice of the heavy operator H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{H} $$\\end{document}, given as λL∑ℓ=0LΦℓΦL−ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\lambda}^L{\\sum}_{\\ell =0}^L{\\Phi}^{\\left(\\ell \\right)}{\\Phi}^{\\left(L-\\ell \\right)} $$\\end{document} with an SU(N) colour factor coefficient, where Φ(ℓ) is the ladder Feynman integral, which is known to all loops. The dependence on the explicit form of the heavy operator lies only in the colour factor coefficients. We determine such colour factors for several classes of heavy operators, and show that the large charge limit leads to minimal powers of N. For the special class of “canonical heavy operators”, one can even resum the all-loop ladder integrals and determine the correlators at finite λ. Furthermore, upon integrating over the spacetime dependence the resulting integrated HHLL correlators agree with the existing results derived from supersymmetric localisation. Finally, as an application of the all-loop analytic results, we derive exact expressions for the structure constants of two heavy operators and the Konishi operator, finding intriguing connections with the integrated HHLL correlators.

Nontriviality of dynamical Chern-Simons gravity and the standard model

Given the growing interest in gravitational-wave and cosmological parity-violating effects in dynamical Chern-Simons (dCS) gravity, it is crucial to investigate whether the scalar-gravitational Pontryagin term in dCS gravity persists when formulated in the context of the U(1)B−L anomaly in the standard model (SM). In particular, it has been argued that dCS gravity can be reduced to Einstein gravity after “rotating away” the gravitational-Pontryagin coupling into the phase of the Weinberg operator—analogous to the rotation of the axion zero mode into the quark mass matrix. We find that dCS gravity is nontrivial if the scalar field ϕ has significant spacetime dependence from dynamics. We provide a comprehensive consideration of the dCS classical and quantum symmetries relevant for embedding a dCS sector in the SM. We find that, because of the baryon/lepton number chiral gravitational anomaly, the scalar-Pontryagin term cannot be absorbed by a field redefinition. Assuming a minimal extension of the SM, we also find that a coupling of the dCS scalar with right-handed neutrinos induces both the scalar-Pontryagin coupling and an axionlike phase in the dimension-five Weinberg operator. We comment on the issue of gauging the U(1)B−L, the observational effects with these two operators present for upcoming experiments, and the origin of dCS gravity in string theory. Published by the American Physical Society 2024

Exact results for giant graviton four-point correlators

We study the four-point correlator O2O2DD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\left\\langle {\\mathcal{O}}_2{\\mathcal{O}}_2\\mathcal{DD}\\right\\rangle $$\\end{document} in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super Yang-Mills theory (SYM) with SU(N) gauge group, where O2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{O}}_2 $$\\end{document} represents the superconformal primary operator with dimension two, while D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{D} $$\\end{document} denotes a determinant operator of dimension N, which is holographically dual to a giant graviton D3-brane extending along S5. We analyse the integrated correlator associated with this observable, obtained after integrating out the spacetime dependence over a supersymmetric invariant measure. Similarly to other classes of integrated correlators in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM, this integrated correlator can be computed through supersymmetric localisation on the four-sphere. Employing matrix-model recursive techniques, we demonstrate that the integrated correlator can be reformulated as an infinite sum of protected three-point functions with known coefficients. This insight allows us to circumvent the complexity associated with the dimension-N determinant operator, significantly streamlining the large-N expansion of the integrated correlator. In the planar limit and beyond, we derive exact results for the integrated correlator valid for all values of the ’t Hooft coupling, and investigate the resurgent properties of their strong coupling expansion. Additionally, in the large-N expansion with fixed (complexified) Yang-Mills coupling, we deduce the SL(2, ℤ) completion of these results in terms of the non-holomorphic Eisenstein series. The proposed modular functions are confirmed by explicit instanton calculations from the matrix model, and agree with expectations from the holographic dual picture of known results in type IIB string theory.

Quasiparticle cosmology

We consider thermodynamics of the Universe within a quasiparticle approach where the collective dynamics of a system is governed by the thermal mass of the constituents. The spacetime dependence of this thermal mass leads to a negative contribution to the pressure and a positive contribution to the energy density, similar to the effect due to dark energy. We propose a mechanism based on thermodynamic arguments to quantify this contribution from the thermal vacuum of the system. For a sufficiently large spacetime variation of the thermal mass, the effective pressure can become negative and could mimic a dark energy equation of state. We validate our framework using results from renormalizable interacting scalar field theory and demonstrate an application on QCD axion. Published by the American Physical Society 2024

Charged participants and their electromagnetic fields in an expanding fluid

We investigate the space-time dependence of electromagnetic fields produced by charged participants in an expanding fluid. To address this problem, we need to solve the Maxwell's equations coupled to the hydrodynamics conservation equation, specifically the relativistic magnetohydrodynamics (RMHD) equations, since the charged participants move with the flow. To gain analytical insight, we approximate the problem by solving the equations in a fixed background Bjorken flow, onto which we solve Maxwell's equations. The dynamical electromagnetic fields interact with the fluid's kinematic quantities such as the shear tensor and the expansion scalar, leading to additional non-trivial coupling. We use mode decomposition of Green's function to solve the resulting non-linear coupled wave equations. We then use this function to calculate the electromagnetic field for two test cases: a point source and a transverse charge distribution. The results show that the resulting magnetic field vanishes at very early times, grows, and eventually falls at later times.

High-resolution global precipitation downscaling with latent Gaussian models and non-stationary stochastic partial differential equation structure

Abstract Obtaining high-resolution maps of precipitation data can provide key insights to stakeholders to assess a sustainable access to water resources at urban scale. Mapping a non-stationary, sparse process such as precipitation at very high spatial resolution requires the interpolation of global datasets at the location where ground stations are available with statistical models able to capture complex non-Gaussian global space–time dependence structures. In this work, we propose a new approach based on capturing the spatially varying anisotropy of a latent Gaussian process via a locally deformed stochastic partial differential equation (SPDE) with a buffer allowing for a different spatial structure across land and sea. The finite volume approximation of the SPDE, coupled with integrated nested Laplace approximation ensures feasible Bayesian inference for tens of millions of observations. The simulation studies showcase the improved predictability of the proposed approach against stationary and no-buffer alternatives. The proposed approach is then used to yield high-resolution simulations of daily precipitation across the United States.

Generating functions and large-charge expansion of integrated correlators in \U0001d4a9 = 4 supersymmetric Yang-Mills theory

We recently proved that, when integrating out the spacetime dependence with a certain integration measure, four-point correlators leftlangle {mathcal{O}}_2{mathcal{O}}_2{mathcal{O}}_p^{(i)}{mathcal{O}}_p^{(i)}rightrangle in \U0001d4a9 = 4 supersymmetric Yang-Mills theory with SU(N) gauge group are governed by a universal Laplace-difference equation. Here {mathcal{O}}_p^{(i)} is a superconformal primary with charge p and degeneracy i. These physical observables, called integrated correlators, are modular-invariant functions of Yang-Mills coupling τ. The Laplace-difference equation is a recursion relation that relates integrated correlators of operators with different charges. In this paper, we introduce the generating functions for these integrated correlators that sum over the charge. By utilising the Laplace-difference equation, we determine the generating functions for all the integrated correlators, in terms of the initial data of the recursion relation. We show that the transseries of the integrated correlators in the large-p (i.e. large-charge) expansion for a fixed N consists of three parts: 1) is independent of τ, which behaves as a power series in 1/p, plus an additional log(p) term when i = j; 2) is a power series in 1/p, with coefficients given by a sum of the non-holomorphic Eisenstein series; 3) is a sum of exponentially decayed modular functions in the large-p limit, which can be viewed as a generalisation of the non-holomorphic Eisenstein series. When i = j, there is an additional modular function of τ that is independent of p and is fully determined in terms of the integrated correlator with p = 2. The Laplace-difference equation was obtained with a reorganisation of the operators that means the large-charge limit is taken in a particular way here. From these SL(2, ℤ)-invariant results, we also determine the generalised ’t Hooft genus expansion and the associated large-p non-perturbative corrections of the integrated correlators by introducing λ = p {g}_{YM}^2 . The generating functions have subtle differences between even and odd N, which have important consequences in the large-charge expansion and resurgence analysis. We also consider the generating functions of the integrated correlators for some fixed p by summing over N, and we study their large-N behaviour, as well as comment on the similarities and differences between the large-p expansion and the large-N expansion.

Laplace-difference equation for integrated correlators of operators with general charges in mathcal{N} = 4 SYM

We consider the integrated correlators associated with four-point correlation functions leftlangle {mathcal{O}}_2{mathcal{O}}_2{mathcal{O}}_p^{(i)}{mathcal{O}}_p^{(j)}rightrangle in four-dimensional mathcal{N} = 4 supersymmetric Yang-Mills theory (SYM) with SU(N) gauge group, where {mathcal{O}}_p^{(i)} is a superconformal primary with charge (or dimension) p and the superscript i represents possible degeneracy. These integrated correlators are defined by integrating out spacetime dependence with a certain integration measure, and they can be computed via supersymmetric localisation. They are modular functions of complexified Yang-Mills coupling τ. We show that the localisation computation is systematised by appropriately reorganising the operators. After this reorganisation of the operators, we prove that all the integrated correlators for any N, with some crucial normalisation factor, satisfy a universal Laplace-difference equation (with the laplacian defined on the τ-plane) that relates integrated correlators of operators with different charges. This Laplace-difference equation is a recursion relation that completely determines all the integrated correlators, once the initial conditions are given.

Short-Term Passenger Flow Forecasting for Rail Transit considering Chaos Theory and Improved EMD-PSO-LSTM-Combined Optimization

This paper proposes a prediction method based on chaos theory and an improved empirical-modal-decomposition particle-swarm-optimization long short-term-memory (EMD-PSO-LSTM)-combined optimization process for passenger flow data with high nonlinearity and dynamic space-time dependence, using EMD to process the original passenger flow data and generate several eigenmodal functions (IMFs) and residuals with different characteristic scales. Based on the chaos theory, each component of the PSO algorithm was improved by introducing an inertia factor to facilitate the adjustment of its search capability to improve optimization. Each subsequence of the phase-space reconstruction was built into an improved PSO-LSTM prediction model, and the output of each prediction model was summed to determine the final output. Experimental studies were performed using data from the North Railway Station of Chengdu Rail Transit, and the results showed that the proposed model can generate better prediction results. The proposed model obtained root mean square error (RMSE) and mean absolute error (MAE) of 16.0908 and 11.3704, respectively. Compared with the LSTM, the improved PSO-LSTM, the improved EMD-PSO-LSTM, and the model proposed in this paper improved the RMSE values by 25.53%, 29.97%, and 58.76%, respectively, and the MAE values by 30.41%, 40.13%, and 63.08%, respectively, of the prediction results.

Toverline{T} deformed scattering happens within matrices

We show the Toverline{T} deformation of two-dimensional quantum field theories is equivalent to replacing the spacetime dependence of the fields with dependence on the indices of infinitely large matrices. We show how this correspondence explains the CDD phase dressing of the S-matrix and the general formula for the deformation of arbitrary correlation functions. We also describe how the Moyal deformation of self-dual gravity is a Toverline{T} deformation of the theory described by the Chalmers-Siegel action, where the Toverline{T} deformation is defined on the two-dimensional plane of interactions.

An Analytic Solution for 2D Heat Conduction Problems with General Dirichlet Boundary Conditions

This paper proposed a closed-form solution for the 2D transient heat conduction in a rectangular cross-section of an infinite bar with the general Dirichlet boundary conditions. The boundary conditions at the four edges of the rectangular region are specified as the general case of space–time dependence. First, the physical system is decomposed into two one-dimensional subsystems, each of which can be solved by combining the proposed shifting function method with the eigenfunction expansion theorem. Therefore, through the superposition of the solutions of the two subsystems, the complete solution in the form of series can be obtained. Two numerical examples are used to investigate the analytic solution of the 2D heat conduction problems with space–time-dependent boundary conditions. The considered space–time-dependent functions are separable in the space–time domain for convenience. The space-dependent function is specified as a sine function and/or a parabolic function, and the time-dependent function is specified as an exponential function and/or a cosine function. In order to verify the correctness of the proposed method, the case of the space-dependent sinusoidal function and time-dependent exponential function is studied, and the consistency between the derived solution and the literature solution is verified. The parameter influence of the time-dependent function of the boundary conditions on the temperature variation is also investigated, and the time-dependent function includes harmonic type and exponential type.

Transient longitudinal waves on impact excited viscoelastic rods with lateral inertia.

An analytical and numerical Fourier transform based approach is presented to investigate the space-time dependence for the longitudinal velocity resulting from the longitudinal impact force excitation of viscoelastic rods with lateral inertia. A one-dimensional dissipative Rayleigh-Love wave equation including material memory, dissipation, and/or transverse effects due to Poisson coupling is developed from a generic model of the time dependent stress strain relationship for the rod material. A Gaussian signal with a suitable time scale is used to represent the hammer impact forces. Fourier transform, standing wave, and modal methods are utilized to develop analytical solutions for the longitudinal velocity resulting from the impact excitation of semi-infinite and finite length free-free elastic rods with no transverse effects to provide a baseline and segue to analogous results for viscoelastic rods with transverse effects. Numerical results from an FFT approach are presented to illustrate the effects of material losses and transverse effects on attenuation and dispersion in viscoelastic rods using the Kelvin-Voigt model for the constitutive equation of the rods.

Quantifying causal influence in quantum mechanics

We extend Pearl's definition of causal influence to the quantum domain, where two quantum systems $A, B$ with finite-dimensional Hilbert space are embedded in a common environment $C$ and propagated with a joint unitary $U$. For a finite-dimensional Hilbert space $C$, we find the necessary and sufficient condition on $U$ for a causal influence of $A$ on $B$ and vice versa. We introduce an easily computable measure of the causal influence and use it to study the causal influence of different quantum gates, its mutuality, and quantum superpositions of different causal orders. For two two-level atoms dipole-interacting with a thermal bath of electromagnetic waves, the space-time dependence of causal influence almost perfectly reproduces the one of reservoir-induced entanglement.

Understanding the transmission dynamics of a large-scale measles outbreak in Southern Vietnam

ObjectivesSouthern Vietnam experienced a large measles outbreak of over 26,000 cases during 2018-2020. We aimed to understand and quantify the measles spread in space-time dependence and the transmissibility during the outbreak. MethodsMeasles surveillance reported cases between January 2018 and June 2020, vaccination coverage, and population data at provinicial level were used. To illustrate the spatio-temporal pattern of disease spread, we employed the endemic-epidemic multivariate time series model decomposing measles risk additively into autoregressive, spatio-temporal, and endemic components. Likelihood-based estimation procedures were performed to determine the time-varying reproductive number Re of measles. ResultsOur analysis showed that the incidence of measles was associated with vaccination coverage heterogeneity and spatial interaction between provincial units. The risk of infections was dominated by between-province transmission (36.1% to 78.8%), followed by local endogenous transmission (4.1% to 61.5%). In contrast, the endemic behavior had a relatively small contribution (2.4% to 33.4%) across provinces. In the exponential phase of the epidemic, Re was above the threshold with a maximum value of 2.34 (95% CI: 2.20-2.46). ConclusionLocal vaccination coverage and human mobility are important factors contributing to the measles dynamics in Southern Vietnam, and the high risk of inter-provincial transmission is of most concern. Strengthening the disease surveillance is recommended, and further research is essential to understand the relative contribution of population immunity and control measures in measles epidemics.

Space of initial conditions and universality in nonequilibrium quantum dynamics

We study analytically the role of initial conditions in nonequilibrium quantum dynamics considering the one-dimensional ferromagnets in the regime of spontaneously broken symmetry. We analyze the expectation value of local operators for the infinite-dimensional space of initial conditions of domain wall type, generally intended as initial conditions spatially interpolating between two different ground states. At large times the unitary time evolution takes place inside a light cone produced by the spatial inhomogeneity of the initial condition. In the innermost part of the light cone the form of the space-time dependence is universal, in the sense that it is specified by data of the equilibrium universality class. The global limit shape in the variable x/t changes with the initial condition. In systems with more than two ground states the tuning of an interaction parameter can induce a transition which is the nonequilibrium quantum analog of the interfacial wetting transition occurring in classical systems at equilibrium. We illustrate the general results through the examples of the Ising, Potts and Ashkin-Teller chains.

Beer–Lambert law in photochemistry: A new approach

Obtaining a reliable formulation of the Beer–Lambert law in photochemistry requires knowledge on the role that the space–time dependence of the absorbance plays on the system. Spatial memory due to correlation between obstacles can be modeled, among other methods, by using a fractional calculus approach. Here we present a generalized Beer–Lambert law, based on Mittag-Leffler extinction of radiation, which is derived through probabilistic arguments, by assuming that the number of extinction events follows a fractional Poisson distribution. We applied such an approach in photochemistry by using a mathematical model that involves fractional derivative with respect to another function that accounts for the role of both space-dependence of the absorbance and spatial memory. We finally provide a discussion of the utility and implications of this new approach.

Space-Time Trend Detection and Dependence Modeling in Extreme Event Approaches by Functional Peaks-Over-Thresholds: Application to Precipitation in Burkina Faso

In this paper, we propose a new method for estimating trends in extreme spatiotemporal processes using both information from marginal distributions and dependence structure. We combine two statistical approaches of an extreme value theory: the temporal and spatial nonstationarities are handled via a tail trend function in the marginal distributions. The spatial dependence structure is modeled by a latent spatial process using generalized ℓ -Pareto processes. This methodology for trend analysis of extreme events is applied to precipitation data from Burkina Faso. We show that a significant increasing trend for the 50 and 100 year return levels in some parts of the country. We also show that extreme precipitation is spatially correlated with distance for a radius of approximately 200 km.

Quantum generalized hydrodynamics of the Tonks–Girardeau gas: density fluctuations and entanglement entropy

We apply the theory of quantum generalized hydrodynamics (QGHD) introduced in (2020 Phys. Rev. Lett. 124 140603) to derive asymptotically exact results for the density fluctuations and the entanglement entropy of a one-dimensional trapped Bose gas in the Tonks–Girardeau (TG) or hard-core limit, after a trap quench from a double well to a single well. On the analytical side, the quadratic nature of the theory of QGHD is complemented with the emerging conformal invariance at the TG point to fix the universal part of those quantities. Moreover, the well-known mapping of hard-core bosons to free fermions, allows to use a generalized form of the Fisher–Hartwig conjecture to fix the non-trivial spacetime dependence of the ultraviolet cutoff in the entanglement entropy. The free nature of the TG gas also allows for more accurate results on the numerical side, where a higher number of particles as compared to the interacting case can be simulated. The agreement between analytical and numerical predictions is extremely good. For the density fluctuations, however, one has to average out large Friedel oscillations present in the numerics to recover such agreement.

A Concise Review of State Estimation Techniques for Partial Differential Equation Systems

While state estimation techniques are routinely applied to systems represented by ordinary differential equation (ODE) models, it remains a challenging task to design an observer for a distributed parameter system described by partial differential equations (PDEs). Indeed, PDE systems present a number of unique challenges related to the space-time dependence of the states, and well-established methods for ODE systems do not translate directly. However, the steady progresses in computational power allows executing increasingly sophisticated algorithms, and the field of state estimation for PDE systems has received revived interest in the last decades, also from a theoretical point of view. This paper provides a concise overview of some of the available methods for the design of state observers, or software sensors, for linear and semilinear PDE systems based on both early and late lumping approaches.

Efficient Computation of Spatially Discrete Traveling-Wave Modulated Structures

Traveling-wave modulation is a form of space-time modulation which has been shown to enable unique electromagnetic phenomena such as non-reciprocity, beam-steering, frequency conversion, and amplification. In practice, traveling-wave modulation is achieved by applying a staggered time-modulation signal to a spatially-discrete array of unit cells. Therefore, the capability to accurately simulate spatially-discrete traveling-wave modulated structures is critical to design. However, simulating these structures is challenging due to the complex space-time dependence of the constituent unit cells. In this paper, a field relation (referred to as the interpath relation) is derived for spatially-discrete traveling-wave modulated structures. The interpath relation reveals that the field within a single time-modulated unit cell (rather than an entire spatial period) is sufficient to determine the field solution throughout space. It will be shown that the interpath relation can be incorporated into existing periodic method of moments solvers simply by modifying the source basis functions. As a result, the computational domain is reduced from an entire spatial period to a single time-modulated unit cell, dramatically reducing the number of unknowns. In the context of traveling-wave modulation, this enables researchers to efficiently simulate both complex structures with patterned unit cells in addition to continuous structures with infinitesimal unit cells.