Higher-order Interaction Terms Research Articles

Higher-Order Network Interactions Through Phase Reduction for Oscillators with Phase-Dependent Amplitude

Coupled oscillator networks provide mathematical models for interacting periodic processes. If the coupling is weak, phase reduction—the reduction of the dynamics onto an invariant torus—captures the emergence of collective dynamical phenomena, such as synchronization. While a first-order approximation of the dynamics on the torus may be appropriate in some situations, higher-order phase reductions become necessary, for example, when the coupling strength increases. However, these are generally hard to compute and thus they have only been derived in special cases: This includes globally coupled Stuart–Landau oscillators, where the limit cycle of the uncoupled nonlinear oscillator is circular as the amplitude is independent of the phase. We go beyond this restriction and derive second-order phase reductions for coupled oscillators for arbitrary networks of coupled nonlinear oscillators with phase-dependent amplitude, a scenario more reminiscent of real-world oscillations. We analyze how the deformation of the limit cycle affects the stability of important dynamical states, such as full synchrony and splay states. By identifying higher-order phase interaction terms with hyperedges of a hypergraph, we obtain natural classes of coupled phase oscillator dynamics on hypergraphs that adequately capture the dynamics of coupled limit cycle oscillators.

Loop contributions to the scalar power spectrum due to quartic order action in ultra slow roll inflation

The investigation of the theory of inflation beyond the linearorder in perturbations is important both for theoretical consistency andpotential observables.In the contemporary literature, the calculation of modifications to theinflationary scalar power spectrum due to the loops from the higher orderinteraction terms in the Hamiltonian have led to an interesting discussionregarding the validity of perturbation theory and the robustness of itspredictions.Recently, there have been many efforts to examine the contributions to thescalar power spectrum due to the loops arising from the cubic order termsin the action describing the perturbations, specifically in inflationaryscenarios that permit an epoch of ultra slow roll (USR).A brief phase of USR during inflation is known to lead to interesting featuresin the scalar power spectrum which in turn has significant observationalconsequences, such as the copious production of primordial black holes.In this work, we consider the loop contributions to the scalar power spectrumin a scenario of USR inflation arising due to the quartic order termsin the action describing the scalar perturbations.We compute the loop contributions to the scalar power spectrum due to thedominant term in the action at the quartic order in a scenario wherein ashort phase of USR is sandwiched between two stages of slow roll (SR) inflation.We analyze the behaviour of the loop contributions in terms of the parametersthat characterize the non-trivial inflationary dynamics, viz. the onset andduration of USR, and the smoothness of transitions between the USR and SR phases.We examine three different cases of the scenario — the late, intermediate andearly epochs of USR during inflation, each of which affects the scalar powerspectrum over different ranges of wave numbers.In the inflationary scenario involving a late phase of USR, for reasonable choicesof the parameters, we show that the loop corrections are negligible for the entirerange of wave numbers.In the intermediate case, the contributions from the loops prove to be scaleinvariant over large scales and, we find that these contributions can amountto 30% of the leading order (i.e. the Gaussian) power spectrum.In the case wherein USR sets in early, we find that the loop contributions couldbe negative and can dominate the power spectrum at the leading order, whichindicates a breakdown of the validity of the perturbative expansion.We discuss the origin of the negative sign and the divergences that arise in theloop contributions to the power spectrum.We conclude with a brief summary and outlook.

Spatial dynamics of higher order rock-paper-scissors and generalisations

We introduce and study the spatial replicator equation with higher order interactions and both infinite (spatially homogeneous) populations and finite (spatially inhomogeneous) populations. We show that in the special case of three strategies (rock–paper–scissors) higher order interaction terms allow travelling waves to emerge in non-declining finite populations. We show that these travelling waves arise from diffusion stabilisation of an unstable interior equilibrium point that is present in the aspatial dynamics. Based on these observations and prior results, we offer two conjectures whose proofs would fully generalise our results to all odd cyclic games, both with and without higher order interactions, assuming a spatial replicator dynamic. Intriguingly, these generalisations for N⩾5 strategies seem to require declining populations, as we show in our discussion.

Annealing for prediction of grand canonical crystal structures: Implementation of n -body atomic interactions

We propose an annealing scheme usable on modern Ising machines for crystal structures prediction (CSP) by taking into account the general n-body atomic interactions and in particular three-body interactions which are necessary to simulate covalent bonds. The crystal structure is represented by discretizing a unit cell and placing binary variables which express the existence or nonexistence of an atom on every grid point. The resulting quadratic unconstrained binary optimization (QUBO) or higher-order unconstrained binary optimization (HUBO) problems implement the CSP problem and is solved using simulated and quantum annealing. Using the example of Lennard-Jones clusters we show that it is not necessary to include the target atom number in the formulation allowing for simultaneous optimization of both the particle density and the configuration and argue that this is advantageous for use on annealing machines as it reduces the total amount of interactions. We further provide a scheme that allows for reduction of higher-order interaction terms that is inspired by the underlying physics. We show for a covalently bonded monolayer MoS2 crystal that we can simultaneously optimize for the particle density as well as the crystal structure using simulated annealing. We also show that we reproduce ground states of the interatomic potential with high probability that are not represented on the initial discretization of the unit cell. Published by the American Physical Society 2024

Improved mean-field dynamical equations are able to detect the two-step relaxation in glassy dynamics at low temperatures

We study the stochastic relaxation dynamics of the Ising p-spin model on a random graph, which is a well-known model with glassy dynamics at low temperatures. We introduce and discuss a new closure scheme for the master equation governing the continuous-time relaxation of the system, which translates into a set of differential equations for the evolution of local probabilities. The solution to these dynamical mean-field equations describes the out-of-equilibrium dynamics at high temperatures very well, notwithstanding the key observation that the off-equilibrium probability measure contains higher-order interaction terms not present in the equilibrium measure. In the low-temperature regime, the solution to the dynamical mean-field equations shows the correct two-step relaxation (a typical feature of glassy dynamics), but with a too-short relaxation timescale. We propose a solution to this problem by identifying the range of energies where entropic barriers play a key role and defining a renormalized microscopic timescale for the dynamical mean-field solution. The final result perfectly matches the complex out-of-equilibrium dynamics computed through extensive Monte Carlo simulations.

Quantum decay of an optical soliton

Optical solitons are known to be classically stable objects which are robust to perturbations. In this work, we show that due to quantum mechanical effects, an optical soliton that is initially in a classical soliton coherent state will shed photons into the continuum and hence decay. The standard formulation of the quantized soliton uses the linearized version of the quantum nonlinear Schrodinger equation in the background of the classical soliton, and the quantized soliton remains stable in this approximation. We show that if higher-order interaction terms are taken into account, the soliton is no longer stable, and its photon number decreases quadratically as a function of the number of soliton cycles. We compute the power spectrum for the continuum radiation and find a narrow band that is localized about the initial soliton momentum with a cut-off that is inversely proportional to the initial soliton width.

Rate coefficients for rotational state-to-state transitions in H2O + H2O collisions for cometary and planetary applications, as predicted by mixed quantum-classical theory

Aims. We present new calculations of collision cross sections for state-to-state transitions between the rotational states in an H2O + H2O system, which are used to generate a new database of collisional rate coefficients for cometary and planetary applications. Methods. Calculations were carried out using a mixed quantum-classical theory approach that is implemented in the code MQCT. The large basis set of rotational states used in these calculations permits us to predict thermally averaged cross sections for 441 transitions in para- and ortho-H2O in a broad range of temperatures. Results. It is found that all state-to-state transitions in the H2O + H2O system split into two well-defined groups, one with higher cross-section values and lower energy transfer, which corresponds to the dipole-dipole driven processes. The other group has smaller cross sections and higher energy transfer, driven by higher-order interaction terms. We present a detailed analysis of the theoretical error bars, and we symmetrized the state-to-state transition matrixes to ensure that excitation and quenching processes for each transition satisfy the principle of microscopic reversibility. We also compare our results with other data available from the literature for H2O + H2O collisions.

A human body model based method for injury risk prediction considering occupant stature and posture

This study developed a framework to generate injury prediction functions in order to evaluate the safety of occupants in traffic accidents, using human body model simulations to analyse the influences of crash severity, occupant stature, occupant posture, and restraint stiffness. The THUMS AM50-O Human Body Model (HBM) was used in a sled test setup. Six influencing parameters were included in injury prediction: impact velocity, occupant stature, Body Mass Index (BMI), torso angle, seating position, and seatbelt restraint force. Impact velocity was found as a significant influencing factor on skull injury, while BMI mainly affected chest injury. Sequential sampling method was used for generation of simulation matrix, and high-order interaction terms were identified. As injury prediction functions, quadratic regressions correlate better with the simulation results than other forms of regressions, the combinations of parameters in which also enhanced the robustness of injury prediction when considering uncertainty of input parameters. Using numerical models to generate injury prediction functions can better characterize occupant variations, such as sitting posture, than using traffic accident reconstruction. Such obtained functions can provide reasonable estimation on occupant injury risk based on accident information, and we also hope it may benefit vehicle safety improvement and triage protocol in medical treatments of traffic accident victims.

Approximating compact objects in bootstrapped Newtonian gravity: use of the canonical potential

We consider compact objects in a classical and non-relativistic generalisation of Newtonian gravity, dubbed bootstrapped Newtonian theory, which includes higher-order derivative interaction terms of the kind generically present in the strong-field regime of gravity. By means of a field redefinition, the original bootstrapped Newtonian action is written in a canonical Newtonian form with non-linear source terms. Exact analytic solutions remain unattainable, but we show that perturbative solutions of the canonical theory can be efficiently used to derive approximate descriptions of compact objects. In particular, using the canonical potential, we can more directly and generally show that the Arnowitt–Deser–Misner mass differs from the (Newtonian) proper mass due to the non-linear couplings in the theory. A few examples of sources with different density profiles are explicitly reanalysed in this framework.

Cumulant theory of the unitary Bose gas: Prethermal and Efimovian dynamics

We study the quench of a degenerate ultracold Bose gas to the unitary regime, where interactions are as strong as allowed by quantum mechanics. We lay the foundations of a cumulant theory able to capture simultaneously the three-body Efimov effect and ergodic evolution. After an initial period of rapid quantum depletion, a universal prethermal stage is established characterized by a kinetic temperature and an emergent Bogoliubov dispersion law while the microscopic degrees of freedom remain far-from-equilibrium. Integrability is then broken by higher-order interaction terms in the many-body Hamiltonian, leading to a momentum-dependent departure from power law to decaying exponential behavior of the occupation numbers at large momentum. We find also signatures of the Efimov effect in the many-body dynamics and make a precise identification between the observed beating phenomenon and the binding energy of an Efimov trimer. Throughout the work, our predictions for a uniform gas are quantitatively compared with experimental results for quenched unitary Bose gases in uniform potentials.

Self-induction and magnetic effects in electron transport through a photon cavity

We explore higher order dynamical effects in the transport through a two-dimensional nanoscale electron system embedded in a three-dimensional far-infrared photon cavity. The nanoscale system is considered to be a short quantum wire with a single circular quantum dot defined in a GaAs heterostructure. The whole system, the external leads and the central system are placed in a constant perpendicular magnetic field. The Coulomb interaction of the electrons, the para- and diamagnetic electron-photon interactions are all treated by a numerically exact diagonalization using step-wise truncations of the appropriate many-body Fock spaces. We focus on the difference in transport properties between a description within an electric dipole approximation and a description including all higher order terms in a single photon mode model. We find small effects mostly caused by an electrical quadrupole and a magnetic dipole terms that depend strongly on the polarization of the cavity field with respect to the transport direction and the photon energy. When the polarization is aligned along the transport direction we find indications of a weak self-induction that we analyze and compare to the classical counterpart, and the self-energy contribution of high-order interaction terms to the states the electrons cascade through on their way through the system. Like expected the electron-photon interaction is well described in the dipole approximation when it is augmented by the lowest order diamagnetic part for a nanoscale system in a cavity in an external magnetic field.

A new global sensitivity measure based on derivative-integral and variance decomposition and its application in structural crashworthiness

For the traditional variance-based global sensitivity analysis, the total effect of individual variable commonly involves the interactions with other variables. To further decompose the interactive effects, this paper proposes a new global sensitivity measure based on derivative-integral and variance decomposition. Firstly, the first-order sensitivity index only relating to individual variable and the high-order sensitivity indices involving with other interactive variables are analyzed through analysis of variance (ANOVA) representation. Then, the partial derivatives of high-order interaction terms of ANOVA representation with respect to individual variable and the integrals of the squares of partial derivative functions are calculated in the whole variable space. Consequently, to measure the contribution of individual variable to each high-order interaction term, the ratio of the square root of each integral to their sum is defined as the sensitivity weight factor. A high-order sensitivity index can be further decomposed into a series of sensitivity sub-indices by using the defined sensitivity weight factors. Accordingly, a new global sensitivity measure for individual variable is proposed by combining the first-order sensitivity index with the decomposed sensitivity sub-indices. Finally, three numerical examples and an engineering application are investigated to demonstrate the reasonability and superiority of the proposed sensitivity measure.

RFSA: An R Package for Finding Best Subsets and Interactions.

Herein we present the R package rFSA, which implements an algorithm for improved variable selection. The algorithm searches a data space for models of a user-specified form that are statistically optimal under a measure of model quality. Many iterations afford a set of feasible solutions (or candidate models) that the researcher can evaluate for relevance to his or her questions of interest. The algorithm can be used to formulate new or to improve upon existing models in bioinformatics, health care, and myriad other fields in which the volume of available data has outstripped researchers' practical and computational ability to explore larger subsets or higher-order interaction terms. The package accommodates linear and generalized linear models, as well as a variety of criterion functions such as Allen's PRESS and AIC. New modeling strategies and criterion functions can be adapted easily to work with rFSA.

Prethermalization to thermalization crossover in a dilute Bose gas following an interaction ramp

The dynamics of a weakly interacting Bose gas at low temperatures is close to integrable due to the approximate quadratic nature of the many-body Hamiltonian. While the short-time physics after an abrupt ramp of the interaction constant is dominated by the integrable dynamics, integrability is broken at longer times by higher-order interaction terms in the Bogoliubov Hamiltonian, in particular by Beliaev-Landau scatterings involving three quasiparticles. The two-stage relaxation process is highlighted in the evolution of local observables such as the density-density correlation function: a dephasing mechanism leads the system to a prethermal stage, followed by true thermalization conveyed by quasiparticle collisions. Our results bring the crossover from prethermalization to thermalization within reach of current experiments with ultracold atomic gases.

Microwave study of internal rotation in para-tolualdehyde: Local versus global symmetry effects at the methyl-rotor site

The rotational spectrum of para-tolualdehyde (CH3-C6H4-CHO) has been measured using three different microwave spectrometers, with the goal of quantifying the influence of the aldehyde group at the top of the benzene ring on the internal rotation barrier seen by the methyl group at the bottom of the ring. This barrier consists of a six-fold component, which results from the local C2v symmetry of the benzene ring seen by the methyl top (as in toluene), and an additional three-fold component, which results from information on the non-C2v symmetry at the aldehyde site at the top of the ring being transmitted to the methyl-group site at the bottom of the ring. The nearly-free internal rotation of the methyl group splits each of the rotational transitions into two components, one of A and one of E symmetry. Assignment and fit of 786 A-state and E-state transitions to an internal rotation Hamiltonian containing barrier terms of three-fold (V3 = 28.111(1) cm−1) and six-fold (V6 = −4.768(7) cm−1) symmetry with respect to the internal rotation angle, as well as the three rotational constants and a number of higher-order torsion-rotation interaction terms, resulted in residuals equal to experimental measurement uncertainty. Isotopic data from all eight mono-substituted 13C species and the one 18O species were obtained in natural abundance and used to determine an rs substitution structure. Various chemical and physical implications of this structure and the two barrier parameters are discussed.

A Moment-Based Maximum Entropy Model for Fitting Higher-Order Interactions in Neural Data.

Correlations in neural activity have been demonstrated to have profound consequences for sensory encoding. To understand how neural populations represent stimulus information, it is therefore necessary to model how pairwise and higher-order spiking correlations between neurons contribute to the collective structure of population-wide spiking patterns. Maximum entropy models are an increasingly popular method for capturing collective neural activity by including successively higher-order interaction terms. However, incorporating higher-order interactions in these models is difficult in practice due to two factors. First, the number of parameters exponentially increases as higher orders are added. Second, because triplet (and higher) spiking events occur infrequently, estimates of higher-order statistics may be contaminated by sampling noise. To address this, we extend previous work on the Reliable Interaction class of models to develop a normalized variant that adaptively identifies the specific pairwise and higher-order moments that can be estimated from a given dataset for a specified confidence level. The resulting “Reliable Moment” model is able to capture cortical-like distributions of population spiking patterns. Finally, we show that, compared with the Reliable Interaction model, the Reliable Moment model infers fewer strong spurious higher-order interactions and is better able to predict the frequencies of previously unobserved spiking patterns.

Correlation effects and many-body interactions in water clusters.

Background: The quantum-chemical description of the interactions in water clusters is an essential basis for deriving accurate and physically sound models of the interaction potential for water to be used in molecular simulations. In particular, the role of many-body interactions beyond the two-body interactions, which are often not explicitly taken into account by empirical force fields, can be accurately described by quantum chemistry methods on an adequate level, e.g., random-phase approximation electron correlation methods. The relative magnitudes of the different interaction energy contributions obtained by accurate ab initio calculations can therefore provide useful insights that can be exploited to develop enhanced force field methods.Results: In line with earlier theoretical studies of the interactions in water clusters, it has been found that the main contribution to the many-body interactions in clusters with a size of up to N = 13 molecules are higher-order polarisation interaction terms. Compared to this, many-body dispersion interactions are practically negligible for all studied sytems. The two-body dispersion interaction, however, plays a significant role in the formation of the structures of the water clusters and their stability, since it leads to a distinct compression of the cluster sizes compared to the structures optimized on an uncorrelated level. Overall, the many-body interactions amount to about 13% of the total interaction energy, irrespective of the cluster size. The electron correlation contribution to these, however, amounts to only about 30% to the total many-body interactions for the largest clusters studied and is repulsive for all structures considered in this work.Conclusion: While this shows that three- and higher-body interactions can not be neglected in the description of water complexes, the electron correlation contributions to these are much smaller in comparison to the two-body electron correlation effects. Efficient quantum chemistry approaches for describing intermolecular interactions between water molecules may therefore describe higher-body interactions on an uncorrelated Hartree–Fock level without a serious loss in accuracy.

Bayesian variable selection with strong heredity constraints

In this paper, we propose a Bayesian variable selection method for linear regression models with high-order interactions. Our method automatically enforces the heredity constraint, that is, a higher order interaction term can exist in the model only if both of its parent terms are in the model. Based on the stochastic search variable selection George and McCulloch (1993), we propose a novel hierarchical prior that fully considers the heredity constraint and controls the degree of sparsity simultaneously. We develop a Markov chain Monte Carlo (MCMC) algorithm to explore the model space efficiently while accounting for the heredity constraint by modifying the shotgun stochastic search algorithm Hans et al. (2007). The performance of the new model is demonstrated through comparisons with other methods. Numerical studies on both real data analysis and simulations show that our new method tends to find relevant variable more effectively when higher order interaction terms are considered.

First-order transition in XY model with higher-order interactions

The effect of inclusion of higher-order interactions in the XY model on critical properties is studied by Monte Carlo simulations. It is found that an increasing number of the higher-order terms in the Hamiltonian modifies the shape of the potential, which beyond a certain value leads to the change of the nature of the transition from continuous to first order. The evidence for the first-order transition is provided in the form of the finite-size scaling and the energy histogram analysis. A rough phase diagram is presented as a function of the number of the higher-order interaction terms.

Higher-order baryon number susceptibilities: Interplay between the chiral and the nuclear liquid-gas transitions

We use an improved version of the SU(3) flavour parity-doublet quark-hadron model to investigate the higher order baryon number susceptibilities near the chiral and the nuclear liquid-gas transitions. The parity-doublet model has been improved by adding higher-order interaction terms of the scalar fields in the effective mean field Lagrangian, resulting in a much-improved description of nuclear ground-state properties, in particular the nuclear compressibility. The resulting phase diagram of the model agrees qualitatively with expectations from lattice QCD, i.e., it shows a crossover at zero net baryo-chemical potential and a critical point at finite density. Using this model, we investigate the dependence of the higher-order baryon number susceptibilities as function of temperature and chemical potential. We observe a strong interplay between the chiral and liquid-gas transition at intermediate baryo chemical potentials. Due to this interplay between the chiral and the nuclear liquid-gas transitions, the experimentally measured cumulants of the net baryon number may show very different beam energy dependence, subject to the actual freeze-out temperature.