Linear Response Function Research Articles

Extended system-bath entanglement theorem with multiple baths in the presence of external fields.

The system-bath entanglement theorem (SBET) was established in terms of linear response functions [Du et al., J. Chem. Phys. 152, 034102 (2020)] and generalized to correlation functions [Su et al., J. Chem. Phys. 160, 084104 (2024)] in our previous studies. This theorem connects the entangled system-bath properties to the local system and bare-bath ones. In this work, we extend the SBET to field-dressed conditions with multiple baths at different temperatures. As in reality, the external fields may interact with not only the system but also environments. The extended SBET facilitates, for example, photo-acoustic, photo-thermal, pump-probe related studies. The theorem under the field-free condition (multiple baths) and its counterpart in the classical limit is also presented.

Adaptive Variational Quantum Computing Approaches for Green's Functions and Nonlinear Susceptibilities.

We present and benchmark quantum computing approaches for calculating real-time single-particle Green's functions and nonlinear susceptibilities of Hamiltonian systems. The approaches leverage adaptive variational quantum algorithms for state preparation and propagation. Using automatically generated compact circuits, the dynamical evolution is performed over sufficiently long times to achieve adequate frequency resolution of the response functions. We showcase accurate Green's function calculations using a statevector simulator on classical hardware for Fermi-Hubbard chains of 4 and 6 sites, with maximal ansatz circuit depths of 65 and 424 layers, respectively, and for the molecule LiH with a maximal ansatz circuit depth of 81 layers. Additionally, we consider an antiferromagnetic quantum spin-1 model that incorporates the Dzyaloshinskii-Moriya interaction to illustrate calculations of the third-order nonlinear susceptibilities, which can be measured in two-dimensional coherent spectroscopy experiments. These results demonstrate that real-time approaches using adaptive parametrized circuits to evaluate linear and nonlinear response functions can be feasible with near-term quantum processors.

Hyperhardness and hypersoftness of atoms and their ions

ContextThe theory of reactivity based on cDFT has been supplemented with the new method of calculating the atomic and local indices. With the use of previously derived relationship of the electron density gradient to the softness kernel and to the linear response function, we deliver theoretical analysis to obtain significant reactivity indices—the electron density derivatives: local softness and local hypersoftness together with the global hyperhardness index and the derivative of the global softness with respect to the number of electrons. The local derivatives have been applied in the calculation of responses of atoms to perturbation by an external potential by the alchemical approach. The vital role of the local softness has been confirmed; the potential role of the hypersoftness has been indicated.MethodOur original theoretical scheme has been numerically illustrated with the results obtained with electron density calculations with B3LYP method implemented in Gaussian 16 package. The aug-cc-pvqz basis set has been routinely applied, except for the Ca atom (cc-pvqz). Using the pVTZ basis set recommended by Sadlej was necessary for the potassium atom.Graphical

Computing linear optical spectra in the presence of nonadiabatic effects on graphics processing units using molecular dynamics and tensor-network approaches.

We compare two recently developed strategies, implemented in open source software packages, for computing linear optical spectra in condensed phase environments in the presence of nonadiabatic effects. Both approaches rely on computing excitation energy and transition dipole fluctuations along molecular dynamics (MD) trajectories, treating molecular and environmental degrees of freedom on the same footing. Spectra are then generated in two ways: in the recently developed Gaussian non-Condon theory, the linear response functions are computed in terms of independent adiabatic excited states, with non-Condon effects described through spectral densities of transition dipole fluctuations. For strongly coupled excited states, we instead parameterize a linear vibronic coupling Hamiltonian directly from spectral densities of energy fluctuations and diabatic couplings computed along the MD trajectory. The optical spectrum is then calculated using powerful, numerically exact tensor-network approaches. Both the electronic structure calculations to sample system fluctuations and the quantum dynamics simulations using tensor-network methods are carried out on graphics processing units, enabling rapid calculations on complex condensed phase systems. We assess the performance of the approaches using model systems in the presence of a conical intersection and the pyrazine molecule in different solvent environments.

Assessment of hybrid machine learning models for non-linear system identification of fatigue test rigs

The prediction of system responses for a given fatigue test bench drive signal is a challenging problem, since highly dynamic loads from measurement campaigns must be reproduced accurately. Linear frequency response function models are commonly used for this system identification task, but energy intensive experimental iterations are required to account for system non-linearities. Two novel hybrid modeling strategies are suggested, which augment existing approaches using non-linear Long Short-Term Memory networks. These are trained and deployed on short subsequences of measurement data and recombined using a windowing technique, which enables their application to measurement data with high sampling rates. In addition to fatigue test bench commissioning, these hybrid models can also be employed in the field of virtual sensing. The approach is tested using non-linear experimental data from a servo-hydraulic test rig and this dataset is made publicly available. A variety of metrics in time and frequency domains, as well as fatigue strength under variable amplitudes, are employed in the evaluation. It is shown, that hybrid models can successfully use frequency response function models as a linear baseline estimate, which is further improved by Long Short-Term Memory networks to enable non-linear predictions.

Overcoming mechanistic limitations of process-based phenological models: A data clustering method for large-scale applications

Process-based phenological models use thermal requirement (TR) defined by planting date, temperature and photoperiod to predict crop developmental stages. The TR value for a specific developmental stage for a given variety is often presumed to be constant regardless of environmental conditions. We calibrated and compared 12 phenological models using 27-year of observation data of one unique rice (Oryza sativa L.) variety ('Shanyou63') from 46 sites in southern China. Our findings indicated that shifts in environmental conditions significantly affected TR values, e.g., standard deviations of TR at physiological maturation ranged from 83 to 167 °C d. Clustering sites together minimized environmental heterogeneity, and thus minimized the differences in TR for different phenological stages. When the increased from 1 to 24, simulation errors for the 12 models showed a significant decrease across all developmental stages, from 1.8 to 1.4 days for tillering, from 5.3 to 3.7 days for jointing, from 5.6 to 3.9 days for booting, from 4.7 to 3.3 days for heading, and from 6.6 to 3.7 days for physiological maturation. Furthermore, our findings indicate that models featuring a three-segment piecewise linear temperature response function provide a more precise prediction. In contrast, models incorporating a Beta temperature response function have not performed well. This difference is attributed to different mechanisms used to describe the response of crop development rate to temperature, particularly at non-optimum temperatures. The impact of the photoperiod response function on prediction accuracy became significant with the expansion of scale. Our results demonstrate that clustering method effectively compensates for the lack of crop adaptation processes in common phenological models, leading to significantly improved phenological prediction accuracy in regions with environmental diversity.

Neural Networks to Find the Optimal Forcing for Offsetting the Anthropogenic Climate Change Effects

Abstract Of great relevance to climate engineering is the systematic relationship between the radiative forcing to the climate system and the response of the system, a relationship often represented by the linear response function (LRF) of the system. However, estimating the LRF often becomes an ill-posed inverse problem due to high-dimensionality and nonunique relationships between the forcing and response. Recent advances in machine learning make it possible to address the ill-posed inverse problem through regularization and sparse system fitting. Here, we develop a convolutional neural network (CNN) for regularized inversion. The CNN is trained using the surface temperature responses from a set of Green’s function perturbation experiments as imagery input data together with data sample densification. The resulting CNN model can infer the forcing pattern responsible for the temperature response from out-of-sample forcing scenarios. This promising proof of concept suggests a possible strategy for estimating the optimal forcing to negate certain undesirable effects of climate change. The limited success of this effort underscores the challenges of solving an inverse problem for a climate system with inherent nonlinearity. Significance Statement Predicting the climate response for a given climate forcing is a direct problem, while inferring the forcing for a given desired climate response is often an inverse, ill-posed, problem, posing a new challenge to the climate community. This study makes the first attempt to infer the radiative forcing for a given target pattern of global surface temperature response using a deep learning approach. The resulting deeply trained convolutional neural network inversion model shows promise in capturing the forcing pattern corresponding to a given surface temperature response, with a significant implication on the design of an optimal solar radiation management strategy for curbing global warming. This study also highlights the technical challenges that future research should prioritize in seeking feasible solutions to the inverse climate problem.

Second-order phase transitions and divergent linear response in dynamical mean-field theory

Second-order phase transitions appear as a divergence in one of the linear response functions. For a system of correlated electrons, the relevant divergent response can and does involve many-particle observables, most famously the double occupancy. Generally, evaluating the linear response function of many-particle observables requires a many-particle generalization of the Bethe-Salpeter equation. However, here I show that the divergence of linear response functions in dynamical mean-field theory is governed by a -particle Bethe-Salpeter equation, even for many-particle observables. The reason for this is that the divergence at the second-order phase transition is produced by the self-consistent feedback of the dynamical mean field. Published by the American Physical Society 2024

Measurement-based Detector Characteristics for Digital Twins in aRTist

Various software products for the simulation of industrial X-ray radiography have been developed in recent years (e.g., aRTist 2, CIVA CT, Scorpius XLab, SimCT, Wilcore) and their application potential has been shown in numerous works. However, full systematic approaches to characterise a specific CT system for these simulation software products to obtain a truthful digital twin are still missing. In this contribution, we want to present two approaches to obtain realistic grey values in X-ray projections in aRTist 2 simulations based on measured projections. In aRTist 2, the displayed grey value of a pixel is based on the energy density incident on that pixel. The energy density is calculated based on the X-ray tube spectrum, the attenuation between source and detector as well as an energy-dependent sensitivity curve of the detector. The first approach presented in this contribution uses the sensitivity curve as a free modelling parameter. We measured the signal response at different thicknesses of Al EN-AW6082 at different tube voltages (i.e., different tube spectra). We then regarded the grey values displayed by these projections as a data regression respectively an optimisation problem and obtained the sensitivity curve that is best able to reproduce the measured behaviour in aRTist 2. The resulting sensitivity curve does not necessarily hold physical meaning but is able to simulate the real system behaviour in the simulation software. The second approach presented in this contribution is to estimate the sensitivity curve based on assumptions about the characteristics of the scintillation detector (e.g., scintillator material, scintillator thickness and signal processing characteristics). For this approach, a linear response function (linear relationship between the deposited energy per pixel and the resulting grey value) is assumed. If the detector characteristics, which affect the simulated deposited energy, are properly modelled, the slope and offset of the response function to match the measured grey values should be the same for different tube spectra. As the offset is constant and given by the grey values measured at no incident radiation, the slope is the remaining parameter to evaluate the success of the detector modelling. We therefore adapted the detector characteristics by changing the detector setup until the slope was nearly the same for all measured tube spectra. We are aware that the resulting parameters of the scintillator material and thickness might not be the real ones, but with those modelling parameters we are able to simulate realistic grey values in aRTist 2. Both of those approaches could potentially be a step forward to a full systematic approach for a digital twin of a real CT system in aRTist 2.

Fluctuation-response relation as a probe of long-range correlations in nonequilibrium quantum and classical fluids.

The absence of a simple fluctuation-dissipation theorem is a major obstacle for studying systems that are not in thermodynamic equilibrium. We show that for a fluid in a nonequilibrium steady state characterized by a constant temperature gradient the commutator correlation functions are still related to response functions; however, the relation is to the bilinear response of products of two observables, rather than to a single linear response function as is the case in equilibrium. This modified fluctuation-response relation holds for both quantum and classical systems. It is both motivated and informed by the long-range correlations that exist in such a steady state and allows for probing them via response experiments. This is of particular interest in quantum fluids, where the direct observation of fluctuations by light scattering would be difficult. In classical fluids it is known that the coupling of the temperature gradient to the diffusive shear velocity leads to correlations of various observables, in particular temperature fluctuations, that do not decay as a function of distance, but rather extend over the entire system. We investigate the nature of these correlations in a fermionic quantum fluid and show that the crucial coupling between the temperature gradient and velocity fluctuations is the same as in the classical case. Accordingly, the nature of the long-ranged correlations in the hydrodynamic regime also is the same. However, as one enters the collisionless regime in the low-temperature limit the nature of the velocity fluctuations changes: they become ballistic rather than diffusive. As a result, correlations of the temperature and other observables are still singular in the long-wavelength limit, but the singularity is weaker than in the hydrodynamic regime.

Predicting early auditory evoked potentials using a computational model of auditory-nerve processing.

Non-invasive electrophysiological measures, such as auditory evoked potentials (AEPs), play a crucial role in diagnosing auditory pathology. However, the relationship between AEP morphology and cochlear degeneration remains complex and not well understood. Dau [J. Acoust. Soc. Am. 113, 936-950 (2003)] proposed a computational framework for modeling AEPs that utilized a nonlinear auditory-nerve (AN) model followed by a linear unitary response function. While the model captured some important features of the measured AEPs, it also exhibited several discrepancies in response patterns compared to the actual measurements. In this study, an enhanced AEP modeling framework is presented, incorporating an improved AN model, and the conclusions from the original study were reevaluated. Simulation results with transient and sustained stimuli demonstrated accurate auditory brainstem responses (ABRs) and frequency-following responses (FFRs) as a function of stimulation level, although wave-V latencies remained too short, similar to the original study. When compared to physiological responses in animals, the revised model framework showed a more accurate balance between the contributions of auditory-nerve fibers (ANFs) at on- and off-frequency regions to the predicted FFRs. These findings emphasize the importance of cochlear processing in brainstem potentials. This framework may provide a valuable tool for assessing human AN models and simulating AEPs for various subtypes of peripheral pathologies, offering opportunities for research and clinical applications.

Linear Time-Invariant Models of a Large Cumulus Ensemble

Abstract Methods in system identification are used to obtain linear time-invariant state-space models that describe how horizontal averages of temperature and humidity of a large cumulus ensemble evolve with time under small forcing. The cumulus ensemble studied here is simulated with cloud-system-resolving models in radiative–convective equilibrium. The identified models extend steady-state linear response functions used in past studies and provide accurate descriptions of the transfer function, the noise model, and the behavior of cumulus convection when coupled with two-dimensional gravity waves. A novel procedure is developed to convert the state-space models into an interpretable form, which is used to elucidate and quantify memory in cumulus convection. The linear problem studied here serves as a useful reference point for more general efforts to obtain data-driven and interpretable parameterizations of cumulus convection.

Thermodynamic Correlation Inequality.

Trade-off relations place fundamental limits on the operations that physical systems can perform. This Letter presents a trade-off relation that bounds the correlation function, which measures the relationship between a system's current and future states, in Markov processes. The obtained bound, referred to as the thermodynamic correlation inequality, states that the change in the correlation function has an upper bound comprising the dynamical activity, a thermodynamic measure of the activity of a Markov process. Moreover, by applying the obtained relation to the linear response function, it is demonstrated that the effect of perturbation can be bounded from above by the dynamical activity.

Generalized system-bath entanglement theorem for Gaussian environments.

The entanglement between system and bath often plays a pivotal role in complex systems spanning multiple orders of magnitude. A system-bath entanglement theorem was previously established for Gaussian environments in J. Chem. Phys. 152, 034102 (2020) regarding linear response functions. This theorem connects the entangled responses to the local system and bare bath properties. In this work, we generalize it to correlation functions. Key steps in derivations involve using the generalized Langevin dynamics for hybridizing bath modes and the Bogoliubov transformation that maps the original finite-temperature reservoir to an effective zero-temperature vacuum by employing an auxiliary bath. The generalized theorem allows us to evaluate the system-bath entangled correlations and the bath mode correlations in the total composite space, as long as we know the bare-bath statistical properties and obtain the reduced system correlations. To demonstrate the cross-scale entanglements, we utilize the generalized theorem to calculate the solvation free energy of an electron transfer system with intramolecular vibrational modes.

Vibration and force properties of an actuator formed from a piezoelectric stack within a frame structure

This paper examines the vibration and force properties of an actuator formed from a piezoelectric stack within a frame structure. Such configuration avoids the use of the traditional “earth-connected” support and allows the convenient attachment of the actuator to structures, such as power transformer tanks. The steady-state sensitivity of the actuator is analyzed, which comprised a linear frequency response function of the actuator system and a steady-state line frequency spectra of the transducer force at the frequency of the input voltage and its odd harmonics. The sensitivity is first analyzed by examining the forces transmitted to a rigid base structure through the stack and two legs of the frame when the actuator is excited by a single-frequency voltage input. The magnitude of the steady-state sensitivity at the frequency of the input voltage is approximately 15 dB higher than that at the other frequencies. The force transmitted through the piezoelectric stack is approximately twice as large as that through the two frame legs and with opposite phase. Next, the properties of transmitted forces when the rigid base structure is replaced by a beam and a plate of infinite size are also investigated. The interaction between the actuator and the elastic base structure has little effect on the magnitudes of the transmitted forces except at the resonace frequency of the actuator structure. However, the induced structural response is greatly affected and should be considered when describing the excitation capacity of the actuator. This capacity could be illustrated by the power flow between the actuator and the base structure and by the spatial distribution of the structural vibration induced by the actuator.

A Nonparametric Regularization for Spectrum Estimation of Time-Varying Output-Only Measurements

In this work, an advanced 2D nonparametric correlogram method is presented to cope with output-only measurements of linear (slow) time-varying systems. The proposed method is a novel generalization of the kernel function-based regularization techniques that have been developed for estimating linear time-invariant impulse response functions. In the proposed system identification technique, an estimation method is provided that can estimate the time-varying auto- and cross-correlation function and indirectly, the time-varying auto- and cross-correlation power spectrum estimates based on real-life measurements without measuring the perturbation signals. The (slow) time-varying behavior means that the dynamic of the system changes as a function of time. In this work, a tailored regularization cost function is considered to impose assumptions such as smoothness and stability on the 2D auto- and cross-correlation function resulting in robust and uniquely determined estimates. The proposed method is validated on two examples: a simulation to check the numerical correctness of the method, and a flutter test measurement of a scaled airplane model to illustrate the power of the method on a real-life challenging problem.

Extending the Scope of Conceptual Density Functional Theory with Second Order Analytical Methodologies.

In the context of the growing impact of conceptual density functional theory (DFT) as one of the most successful chemical reactivity theories, response functions up to second order have now been widely applied; in recent years, among others, particular attention has been focused on the linear response function and also extensions to higher order have been put forward. As the larger part of these studies have been carried using a finite difference approach to compute these concepts, we now embarked on (an extension of) an analytical approach to conceptual DFT. With the ultimate aim of providing a complete set of analytically computable second order properties, including the softness and hardness kernels, the hardness as the simplest second order response function is scrutinized again with numerical results highlighting the difference in nature between the analytical hardness (referred to as hardness condition) and the Parr-Pearson absolute chemical hardness. The hardness condition is investigated for its capability to gauge the (de)localization error of density functional approximations (DFAs). The analytical Fukui function, besides overcoming the difficulties in the finite difference approach in treating negatively charged systems, also showcases the errors of deviating from the straight-line behavior using fractional occupation number calculations. Subsequently, the softness kernel and its atom-condensed inverse, the hardness matrix, are accessed through the Berkowitz-Parr relation. Revisiting the softness kernel confirms and extends previous discussions on how Kohn's Nearsightedness of Electronic Matter principle can be retrieved and identified as the physicist's version of the chemist's "transferability of functional groups" concept. The accurate, analytical hardness matrix evaluation on the other hand provides further support for the basics of Nalewajski's charge sensitivity analysis. Based on Parr and Liu's functional expansion of the energy functional, a new energy decomposition is introduced with an order of magnitude analysis of the different terms for a series of simple molecules both at their equilibrium geometry and upon variation in bond length and dihedral angle. Finally, for the first time, the perturbation expansion of the energy functional is studied numerically up to second order now that all response functions and integration techniques are at hand. The perturbation expansion energies are in excellent agreement with those obtained directly from DFA calculations giving confidence in the convergence of the perturbation series and its use in judging the importance of the different terms in reactivity investigations.

Solid-that-Flows Picture of Glass-Forming Liquids.

This perspective article reviews arguments that glass-forming liquids are different from those of standard liquid-state theory, which typically have a viscosity in the mPa·s range and relaxation times on the order of picoseconds. These numbers grow dramatically and become 1012 - 1015 times larger for liquids cooled toward the glass transition. This translates into a qualitative difference, and below the "solidity length" which is roughly one micron at the glass transition, a glass-forming liquid behaves much like a solid. Recent numerical evidence for the solidity of ultraviscous liquids is reviewed, and experimental consequences are discussed in relation to dynamic heterogeneity, frequency-dependent linear-response functions, and the temperature dependence of the average relaxation time.

Alternative streamflow-based approach to estimate catchment response time in medium to large catchments: case study in Primary Drainage Region X, South Africa

Event-based estimates of the design flood in ungauged catchments are normally based on a single catchment response time parameter expressed as either the time of concentration (TC), lag time (TL) and/or time to peak (TP). In small, gauged catchments, a simplified convolution process between a single observed hyetograph and hydrograph is generally used to estimate these time parameters. In medium to large heterogeneous, gauged catchments, such a simplification is neither practical nor applicable, given that the variable antecedent soil moisture status resulting from previous rainfall events and spatially non-uniform rainfall hyetographs can result in multi-peaked hydrographs. In ungauged catchments, time parameters are estimated using either empirical or hydraulic methods. In South Africa (SA), unfortunately, the majority of the empirical methods recommended for general use were developed and verified in catchments ≤ 0.45 km² without using any local data. This paper presents the further development and verification of the streamflow-based approach developed by Gericke (2016) to estimate observed TP values and to derive a regional empirical TP equation in Primary Drainage Region X, SA. A semi-automated hydrograph analysis tool was developed to extract and analyse complete hydrographs for time parameter estimation using primary streamflow data from 51 flow-gauging sites. The observed TP values were estimated using three methods: (i) duration of total net rise of a multi-peaked hydrograph, (ii) triangular-shaped direct runoff hydrograph approximations, and (iii) linear catchment response functions. The combined use of these methods incorporated the high variability of event-based time parameters, and Method (iii), in conjunction with an ensemble-event approach sampled from the time parameter distributions, should replace the event-based approaches to enable the improved calibration of empirical time parameter equations. The conceptual approach used to derive the regional empirical TP equation should also be adopted when regional equations need to be derived at a national scale in SA.

Early impairment of magnocellular visual pathways mediated by isolated-check visual evoked potentials in primary open-angle glaucoma: a cross-sectional study

ObjectiveTo explore different performances in the magnocellular (MC) and parvocellular (PC) visual pathways in patients with primary open-angle glaucoma (POAG) and to objectively assess impairment in early stage of POAG.Methods...