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This article introduces an advanced Koopman mode decomposition (KMD) technique—coined Featurized Koopman Mode Decomposition (FKMD)—that uses delay embedding and a learned Mahalanobis distance to enhance analysis and prediction of high-dimensional dynamical systems. The delay embedding expands the observation space to better capture underlying manifold structures, while the Mahalanobis distance adjusts observations based on the system's dynamics. This aids in featurizing KMD in cases where good features are not a priori known. We show that FKMD improves predictions for a high-dimensional linear oscillator, a high-dimensional Lorenz attractor that is partially observed, and a cell signaling problem from cancer research. [ABSTRACT FROM AUTHOR]

An outstanding property of any Hamiltonian system is the symplecticity of its flow, namely, the continuous trajectory preserves volume in phase space. Given a symplectic but discrete trajectory generated by a transition matrix applied at a fixed time-increment (τ > 0), it was generally believed that there exists a unique Hamiltonian producing a continuous trajectory that coincides at all discrete times (t = nτ with n integers) as long as τ is small enough. However, it is now exactly demonstrated that, for any given discrete symplectic dynamics of a harmonic oscillator, there exist an infinite number of real-valued Hamiltonians for any small value of τ and an infinite number of complex-valued Hamiltonians for any large value of τ. In addition, when the transition matrix is similar to a Jordan normal form with the supradiagonal element of 1 and the two identical diagonal elements of either 1 or −1, only one solution to the Hamiltonian is found for the case with the diagonal elements of 1, but no solution can be found for the other case. [ABSTRACT FROM AUTHOR]

In non-perturbative non-Markovian open quantum systems, reaching either low temperatures with the hierarchical equations of motion (HEOM) or high temperatures with the Thermalized Time Evolving Density Operator with Orthogonal Polynomials Algorithm (T-TEDOPA) formalism in Hilbert space remains challenging. We compare different ways of modeling the environment. Sampling the Fourier transform of the bath correlation function, also called temperature dependent spectral density, proves to be very effective. T-TEDOPA [Tamascelli et al., Phys. Rev. Lett. 123, 090402 (2019)] uses a linear chain of oscillators with positive and negative frequencies, while HEOM is based on the complex poles of an optimized rational decomposition of the temperature dependent spectral density [Xu et al., Phys. Rev. Lett. 129, 230601 (2022)]. Resorting to the poles of the temperature independent spectral density and of the Bose function separately is an alternative when the problem due to the huge number of Bose poles at low temperatures is circumvented. Two examples illustrate the effectiveness of the HEOM and T-TEDOPA approaches: a benchmark pure dephasing case and a two-bath model simulating the dynamics of excited electronic states coupled through a conical intersection. We show the efficiency of T-TEDOPA to simulate dynamics at a finite temperature by using either continuous spectral densities or only all the intramolecular oscillators of a linear vibronic model calibrated from ab initio data of a phenylene ethynylene dimer. [ABSTRACT FROM AUTHOR]

We present a general and practical theoretical framework to investigate how energy is dissipated in open quantum system dynamics. This is performed by quantifying the contributions of individual bath components to the overall dissipation of the system. The framework is based on the Nakajima–Zwanzig projection operator technique, which allows us to express the rate of energy dissipation into a specific bath degree of freedom by using traces of operator products. The approach captures system-bath interactions to all orders, but is based on second-order perturbation theory on the off-diagonal subsystem's couplings and a Markovian description of the bath. The usefulness of our theory is demonstrated by applying it to various models of open quantum systems involving harmonic oscillators or spin baths and connecting the outcomes to existing results such as our previously reported formula derived for locally coupled harmonic baths [Kim and Franco, J. Chem. Phys. 154, 084109 (2021)]. We also prove that the dissipation calculated by our theory rigorously satisfies thermodynamic principles such as energy conservation and detailed balance. Overall, the strategy can be used to develop the theory and simulation of dissipation pathways to interpret and engineer the dynamics of open quantum systems. [ABSTRACT FROM AUTHOR]

Environmental effects in excitation energy transfer have mostly been modeled by baths of harmonic oscillators, but to what extent such modeling provides a reliable description of actual interactions between molecular systems and environments remains an open issue. We address this issue by investigating fluctuations in the excitation energies of the light harvesting 2 complex using a realistic all-atomistic simulation of the potential energy surface. Our analyses reveal that molecular motions exhibit significant anharmonic features, even for underdamped intramolecular vibrations. In particular, we find that the anharmonicity contributes to the broadening of spectral densities and substantial overlaps between neighboring peaks, which complicates the meaning of mode frequencies constituting a bath model. Thus, we develop a strategy to construct a minimally underdamped harmonic bath that has a clear connection to all-atomistic dynamics by utilizing actual normal modes of molecules but optimizing their frequencies such that the resulting bath model can best reproduce the all-atomistic simulation results. By subtracting the underdamped contribution from the entire fluctuations, we also show that identifying a residual spectral density representing all other contributions with overdamped behavior is possible. We find that this can be fitted well with a well-established analytic form of a spectral density function or, alternatively, modeled as explicit time dependent fluctuations with muti-exponential or power law type correlation functions. We provide an assessment and the implications of these possibilities. The approach presented here can also serve as a general strategy to construct a simplified bath model that can effectively represent the underlying all-atomistic bath dynamics. [ABSTRACT FROM AUTHOR]

The combination of vibrational perturbation theory with the replacement of the harmonic oscillator quantization condition along the reaction coordinate with an imaginary action to be used in the uniform semiclassical approximation for the transmission probability has been shown in recent years to be a practical method for obtaining thermal reaction rates. To date, this theory has been developed systematically only up to second order in perturbation theory. Although it gives the correct leading order term in an ℏ2 expansion, its accuracy at lower temperatures, where tunneling becomes important, is not clear. In this paper, we develop the theory to fourth order in the action. This demands developing the quantum perturbation theory up to sixth order. Remarkably, we find that the fourth order theory gives the correct ℏ4 term in the expansion of the exact thermal rate. The relative magnitude of the fourth order correction as compared to the second order term objectively indicates the accuracy of the second order theory. We also extend the previous modified second order theory to the fourth order case, creating an ℏ2 modified potential for this purpose. The resulting theory is tested on the standard examples—symmetric and asymmetric Eckart potentials and a Gaussian potential. The modified fourth order theory is remarkably accurate for the asymmetric Eckart potential. [ABSTRACT FROM AUTHOR]

We ran molecular dynamics simulations of folded graphene sheets and present a procedure to measure the sliding friction in these systems based on the rate of decay of a damped harmonic oscillator. This procedure allowed us to study the effect that the size, geometry, and the temperature of the graphene sheet had on the ability to propagate the initial fold and the rate at which it settles to a final "fully folded" equilibrium state. We offer simple rationalizations for the relationships between the initial geometries of our simulations and the friction values that emerge. [ABSTRACT FROM AUTHOR]

Hydrogen clathrate hydrates are ice-like crystalline substances in which hydrogen molecules are trapped inside polyhedral cages formed by the water molecules. Small cages can host only a single H2 molecule, while each large cage can be occupied by up to four H2 molecules. Here, we present a neutron scattering study on the structure of the sII hydrogen clathrate hydrate and on the low-temperature dynamics of the hydrogen molecules trapped in its large cages, as a function of the gas content in the samples. We observe spectral features at low energy transfer (between 1 and 3 meV), and we show that they can be successfully assigned to the rattling motion of a single hydrogen molecule occupying a large water cage. These inelastic bands remarkably lose their intensity with increasing the hydrogen filling, consistently with the fact that the probability of single occupation (as opposed to multiple occupation) increases as the hydrogen content in the sample gets lower. The spectral intensity of the H2 rattling bands is studied as a function of the momentum transfer for partially emptied samples and compared with three distinct quantum models for a single H2 molecule in a large cage: (i) the exact solution of the Schrödinger equation for a well-assessed semiempirical force field, (ii) a particle trapped in a rigid sphere, and (iii) an isotropic three-dimensional harmonic oscillator. The first model provides good agreement between calculations and experimental data, while the last two only reproduce their qualitative trend. Finally, the radial wavefunctions of the three aforementioned models, as well as their potential surfaces, are presented and discussed. [ABSTRACT FROM AUTHOR]

When used as pump pulses in transient absorption spectroscopy measurements, femtosecond laser pulses can produce oscillatory signals known as quantum beats. The quantum beats arise from coherent superpositions of the states of the sample and are best studied in the Fourier domain using Femtosecond Coherence Spectroscopy (FCS), which consists of one-dimensional amplitude and phase plots of a specified oscillation frequency as a function of the detection frequency. Prior works have shown ubiquitous amplitude nodes and π phase shifts in FCS from excited-state vibrational wavepackets in monomer samples. However, the FCS arising from vibronic-exciton states in molecular aggregates have not been studied theoretically. Here, we use a model of vibronic-exciton states in molecular dimers based on displaced harmonic oscillators to simulate FCS for dimers in two important cases. Simulations reveal distinct spectral signatures of excited-state vibronic-exciton coherences in molecular dimers that may be used to distinguish them from monomer vibrational coherences. A salient result is that, for certain relative orientations of the transition dipoles, the key resonance condition between the electronic coupling and the frequency of the vibrational mode may yield strong enhancement of the quantum-beat amplitude and, perhaps, also cause a significant decrease of the oscillation frequency to a value far lower than the vibrational frequency. Future studies using these results will lead to new insights into the excited-state coherences generated in photosynthetic pigment–protein complexes. [ABSTRACT FROM AUTHOR]

In this article, we show that the collective light–matter strong coupling regime where N molecular emitters couple to the photon mode of an optical cavity can be mapped to a quantum impurity model where the photon is the impurity that is coupled to a bath of anharmonic transitions. In the thermodynamic limit where N ≫ 1, we argue that the bath can be replaced with an effective harmonic bath, leading to a dramatic simplification of the problem into one of the coupled harmonic oscillators. We derive simple analytical expressions for linear optical spectra (transmission, reflection, and absorption) where the only molecular input required is the molecular linear susceptibility. This formalism is applied to a series of illustrative examples, showing the role of temperature, disorder, vibronic coupling, and optical saturation of the molecular ensemble, explaining that it is useful even when describing an important class of nonlinear optical experiments. For completeness, we provide Appendixes A–C that include a self-contained derivation of the relevant spectroscopic observables for arbitrary anharmonic systems (for both large and small N) within the rotating-wave approximation. While some of the presented results herein have already been reported in the literature, we provide a unified presentation of the results as well as new interpretations that connect powerful concepts in open quantum systems and linear response theory with molecular polaritonics. [ABSTRACT FROM AUTHOR]

We investigate the endofullerene system 3He@C60 with a four-dimensional potential energy surface (PES) to include the three He translational degrees of freedom and C60 cage radius. We compare second order Møller–Plesset perturbation theory (MP2), spin component scaled-MP2, scaled opposite spin-MP2, random phase approximation (RPA)@Perdew, Burke, and Ernzerhof (PBE), and corrected Hartree–Fock-RPA to calibrate and gain confidence in the choice of electronic structure method. Due to the high cost of these calculations, the PES is interpolated using Gaussian Process Regression (GPR), owing to its effectiveness with sparse training data. The PES is split into a two-dimensional radial surface, to which corrections are applied to achieve an overall four-dimensional surface. The nuclear Hamiltonian is diagonalized to generate the in-cage translational/vibrational eigenstates. The degeneracy of the three-dimensional harmonic oscillator energies with principal quantum number n is lifted due to the anharmonicity in the radial potential. The (2l + 1)-fold degeneracy of the angular momentum states is also weakly lifted, due to the angular dependence in the potential. We calculate the fundamental frequency to range between 96 and 110 cm−1 depending on the electronic structure method used. Error bars of the eigenstate energies were calculated from the GPR and are on the order of ∼±1.5 cm−1. Wavefunctions are also compared by considering their overlap and Hellinger distance to the one-dimensional empirical potential. As with the energies, the two ab initio methods MP2 and RPA@PBE show the best agreement. While MP2 has better agreement than RPA@PBE, due to its higher computational efficiency and comparable performance, we recommend RPA as an alternative electronic structure method of choice to MP2 for these systems. [ABSTRACT FROM AUTHOR]

Tracer diffusion in polymer networks and hydrogels is relevant in biology and technology, while it also constitutes an interesting model process for the dynamics of molecules in fluctuating, heterogeneous soft matter. Here, we systematically study the time-dependent dynamics and (non-Markovian) memory effects of tracers in polymer networks based on (Markovian) implicit-solvent Langevin simulations. In particular, we consider spherical tracer solutes at high dilution in regular, tetrafunctional bead-spring polymer networks and control the tracer–network Lennard-Jones (LJ) interactions and the polymer density. Based on the analysis of the memory (friction) kernels, we recover the expected long-time transport coefficients and demonstrate how the short-time tracer dynamics, polymer fluctuations, and the viscoelastic response are interlinked. Furthermore, we fit the characteristic memory modes of the tracers with damped harmonic oscillations and identify LJ contributions, bond vibrations, and slow network relaxations. Tuned by the LJ interaction parameter, these modes enter the kernel with an approximately linear to quadratic scaling, which we incorporate into a reduced functional form for convenient tracer memory interpolation and extrapolation. This eventually leads to highly efficient simulations utilizing the generalized Langevin equation, in which the polymer network acts as an additional thermal bath with a tunable intensity. [ABSTRACT FROM AUTHOR]

Traditional theories of the nuclear magnetic resonance (NMR) autocorrelation function for intra-molecular dipole pairs assume a single-exponential decay, yet the calculated autocorrelation of realistic systems displays a rich, multi-exponential behavior, resulting in anomalous NMR relaxation dispersion (i.e., frequency dependence). We develop an approach to model and interpret the multi-exponential intra-molecular autocorrelation using simple, physical models within a rigorous statistical mechanical development that encompasses both rotational diffusion and translational diffusion in the same framework. We recast the problem of evaluating the autocorrelation in terms of averaging over a diffusion propagator whose evolution is described by a Fokker–Planck equation. The time-independent part admits an eigenfunction expansion, allowing us to write the propagator as a sum over modes. Each mode has a spatial part that depends on the specified eigenfunction and a temporal part that depends on the corresponding eigenvalue (i.e., correlation time) with a simple, exponential decay. The spatial part is a probability distribution of the dipole pair, analogous to the stationary states of a quantum harmonic oscillator. Drawing inspiration from the idea of inherent structures in liquids, we interpret each of the spatial contributions as a specific molecular mode. These modes can be used to model and predict the NMR dipole–dipole relaxation dispersion of fluids by incorporating phenomena on the molecular level. We validate our statistical mechanical description of the distribution in molecular modes with molecular dynamics simulations interpreted without any relaxation models or adjustable parameters: the most important poles in the Padé–Laplace transform of the simulated autocorrelation agree with the eigenvalues predicted by the theory. [ABSTRACT FROM AUTHOR]

By using the recently derived universal discrete imaginary-time propagator of the harmonic oscillator, both thermodynamic and Hamiltonian energies can be given analytically and evaluated numerically at each imaginary time step for any short-time propagator. This work shows that, using only currently known short-time propagators, the Hamiltonian energy can be optimized to the twelfth-order, converging to the ground state energy of the harmonic oscillator in as few as three beads. This study makes it absolutely clear that the widely used second-order primitive approximation propagator, when used in computing thermodynamic energy, converges extremely slowly with an increasing number of beads. [ABSTRACT FROM AUTHOR]

The time-dependent rotational and vibrational temperatures were measured to study the shock-heated thermal nonequilibrium behaviors of CO with Ar, He, and H2 as collision partners. Three interference-free transition lines in the fundamental vibrational band of CO were applied to the fast, in situ, and state-specific measurements. Vibrational relaxation times of CO were summarized over a temperature range of 1110–2820 K behind reflected shocks. The measured rotational temperature instantaneously reached an equilibrium state behind shock waves. The measured vibrational temperature experienced a relaxation process before reaching the equilibrium state. The measured vibrational temperature time histories were compared with predictions based on the Landau–Teller model and the state-to-state approach. The state-to-state approach treats the vibrational energy levels of CO as pseudo-species and accurately describes the detailed thermal nonequilibrium processes behind shock waves. The datasets of state-specific inelastic rate coefficients of CO–Ar, CO–He, CO–CO, and CO–H2 collisions were calculated in this study using the mixed quantum-classical method and the semiclassical forced harmonic oscillator model. The predictions based on the state-to-state approach agreed well with the measured data and nonequilibrium (non-Boltzmann) vibrational distributions were found in the post-shock regions, while the Landau–Teller model predicted slower vibrational temperature time histories than the measured data. Modifications were applied to the Millikan–White vibrational relaxation data of the CO–Ar and CO–H2 systems to improve the performance of the Landau–Teller model. In addition, the thermal nonequilibrium processes behind incident shocks, the acceleration effects of H2O on the relaxation process of CO, and the characterization of vibrational temperature were highlighted. [ABSTRACT FROM AUTHOR]

It has been assumed that the negative divergence of all stress tensors in common use yields the same force. This work finds that this is untrue, and, in fact, can vary wildly. We demonstrate this for the hydrogen atom, the one-particle isotropic harmonic oscillator, and a particle in an infinite spherical well where the exact density, pair-density, and the first order reduced density matrix are known for ground and excited states without any approximation. The Ehrenfest stress-tensor is introduced as any stress-tensor whose negative divergence will yield the corresponding Ehrenfest force for the same system when the exact wave-function is utilized. Stress-tensors within the literature are examined to show those that are Ehrenfest stress-tensors. Those that differ are demonstrated by how they differ within an exact formulation. The proof that the negative divergence of an Ehrenfest stress-tensor yields the Ehrenfest force is summarized. [ABSTRACT FROM AUTHOR]

The performance of Hamiltonian Monte Carlo simulations crucially depends on both the integration timestep and the number of integration steps. We present an adaptive general-purpose framework to automatically tune such parameters based on a local loss function that promotes the fast exploration of phase space. We show that a good correspondence between loss and autocorrelation time can be established, allowing for gradient-based optimization using a fully differentiable set-up. The loss is constructed in such a way that it also allows for gradient-driven learning of a distribution over the number of integration steps. Our approach is demonstrated for the one-dimensional harmonic oscillator and alanine dipeptide, a small protein commonly used as a test case for simulation methods. Through the application to the harmonic oscillator, we highlight the importance of not using a fixed timestep to avoid a rugged loss surface with many local minima, otherwise trapping the optimization. In the case of alanine dipeptide, by tuning the only free parameter of our loss definition, we find a good correspondence between it and the autocorrelation times, resulting in a > 100 fold speedup in the optimization of simulation parameters compared to a grid search. For this system, we also extend the integrator to allow for atom-dependent timesteps, providing a further reduction of 25% in autocorrelation times. [ABSTRACT FROM AUTHOR]

The time-dependent rotational and vibrational temperatures were measured to study the shock-heated thermal nonequilibrium behaviors of CO with Ar, He, and H2 as collision partners. Three interference-free transition lines in the fundamental vibrational band of CO were applied to the fast, in situ, and state-specific measurements. Vibrational relaxation times of CO were summarized over a temperature range of 1110–2820 K behind reflected shocks. The measured rotational temperature instantaneously reached an equilibrium state behind shock waves. The measured vibrational temperature experienced a relaxation process before reaching the equilibrium state. The measured vibrational temperature time histories were compared with predictions based on the Landau–Teller model and the state-to-state approach. The state-to-state approach treats the vibrational energy levels of CO as pseudo-species and accurately describes the detailed thermal nonequilibrium processes behind shock waves. The datasets of state-specific inelastic rate coefficients of CO–Ar, CO–He, CO–CO, and CO–H2 collisions were calculated in this study using the mixed quantum-classical method and the semiclassical forced harmonic oscillator model. The predictions based on the state-to-state approach agreed well with the measured data and nonequilibrium (non-Boltzmann) vibrational distributions were found in the post-shock regions, while the Landau–Teller model predicted slower vibrational temperature time histories than the measured data. Modifications were applied to the Millikan–White vibrational relaxation data of the CO–Ar and CO–H2 systems to improve the performance of the Landau–Teller model. In addition, the thermal nonequilibrium processes behind incident shocks, the acceleration effects of H2O on the relaxation process of CO, and the characterization of vibrational temperature were highlighted. [ABSTRACT FROM AUTHOR]

Vibrational heat-bath configuration interaction (VHCI)—a selected configuration interaction technique for vibrational structure theory—has recently been developed in two independent works [J. H. Fetherolf and T. C. Berkelbach, J. Chem. Phys. 154, 074104 (2021); A. U. Bhatty and K. R. Brorsen, Mol. Phys. 119, e1936250 (2021)], where it was shown to provide accuracy on par with the most accurate vibrational structure methods with a low computational cost. Here, we eliminate the memory bottleneck of the second-order perturbation theory correction using the same (semi)stochastic approach developed previously for electronic structure theory. This allows us to treat, in an unbiased manner, much larger perturbative spaces, which are necessary for high accuracy in large systems. Stochastic errors are easily controlled to be less than 1 cm−1. We also report two other developments: (i) we propose a new heat-bath criterion and an associated exact implicit sorting algorithm for potential energy surfaces expressible as a sum of products of one-dimensional potentials; (ii) we formulate VHCI to use a vibrational self-consistent field (VSCF) reference, as opposed to the harmonic oscillator reference configuration used in previous reports. Our tests are done with quartic and sextic force fields, for which we find that with VSCF, the minor improvements to accuracy are outweighed by the higher computational cost associated the matrix element evaluations. We expect VSCF-based VHCI to be important for more general potential representations, for which the harmonic oscillator basis function integrals are no longer analytic. [ABSTRACT FROM AUTHOR]

Endohedral fullerenes, or endofullerenes, are chemical systems of fullerene cages encapsulating single atoms or small molecules. These species provide an interesting challenge of Potential Energy Surface determination as examples of non-covalently bonded, bound systems. While the majority of studies focus on C60 as the encapsulating cage, introducing some anisotropy by using a different fullerene, e.g., C70 can unveil a double well potential along the unique axis. By approximating the potential as a pairwise Lennard-Jones (LJ) summation over the fixed C cage atoms, the parameter space of the Hamiltonian includes three tunable variables: (M, ɛ, σ) representing the mass of the trapped species, the LJ energy, and length scales respectively. Fixing the mass and allowing the others to vary can imitate the potentials of endohedral species trapped in more elongated fullerenes. We choose to explore the LJ parameter space of an endohedral atom in C70 with ɛ ∈ [20, 150 cm−1], and σ ∈ [2.85, 3.05 Å]. As the barrier height and positions of these wells vary between [1, 264 cm−1] and [0.35, 0.85 Å] respectively, using a 3D direct product basis of 1D harmonic oscillator (HO) wavefunctions centred at the origin where there is a local maximum is unphysical. Instead we propose the use of a non-orthogonal basis set, using 1D HO wavefunctions centred in each minimum and compare this to other choices. The ground state energy of the X@C70 is tracked across the LJ parameter space, along with its corresponding nuclear translational wavefunctions. A classification of the wavefunction characteristics, namely the prolateness and "peanut-likeness" based on its statistical moments is also proposed. Excited states of longer fullerenes are assigned quantum numbers, and the fundamental transitions of Ne@C70 are tracked across the parameter space. [ABSTRACT FROM AUTHOR]

Chirality is a concept that one object is not superimposable on its mirror image by translation and rotation. In particular, chiral plasmonics have been widely investigated due to their excellent optical chiral properties, and have led to numerous applications such as optical polarizing element etc. In this study, we develop a model based on the concept of the interaction between harmonic oscillators to investigate and explain the optical chiral mechanisms of strongly coupled metal nanoparticles (MNPs). The chirality of the scattering, absorption, and photoluminescence spectra are carefully discussed in detail. The results show that the chirality of the system originates not only from the orientations of the MNPs, but also from the different eigen parameters between them. Specifically, the derived three factors contribute to the chirality: the symmetry, the coupling strength, and the coherent superposition of the emitted electric field. This work provides a deeper understanding on the chiral plasmonics and may guide relevant applications in theory. [ABSTRACT FROM AUTHOR]

The direct integration of the harmonic oscillator path integral obscures the fundamental structure of its discrete, imaginary time propagator (density matrix). This work, by first proving an operator identity for contracting two free propagators into one in the presence of interaction, derives the discrete propagator by simple algebra without doing any integration. This discrete propagator is universal, having the same two hyperbolic coefficient functions for all short-time propagators. Individual short-time propagator only modifies the coefficient function's argument, its portal parameter, whose convergent order is the same as the thermodynamic energy. Moreover, the thermodynamic energy can be given in a closed form for any short-time propagator. Since the portal parameter can be systematically optimized by matching the expansion of the product of the two coefficients, any short-time propagator can be optimized sequentially, order by order, by matching the product coefficient's expansion alone, without computing the energy. Previous empirical findings on the convergence of fourth and sixth-order propagators can now be understood analytically. An eight-order convergent short-time propagator is also derived. [ABSTRACT FROM AUTHOR]

Several approaches to quantum gravity lead to nonlocal modifications of fields' dynamics. This, in turn, can give rise to nonlocal modifications of quantum mechanics at non-relativistic energies. Here, we analyze the nonlocal Schrödinger evolution of a quantum harmonic oscillator in one such scenario, where the problem can be addressed without the use of perturbation theory. We demonstrate that although deviations from standard quantum predictions occur at low occupation numbers, where they could potentially be detected or constrained by high-precision experiments, the classical limits of quantum probability densities and free energy remain unaffected up to energies comparable with the nonlocality scale. These results provide an example of nonlocal quantum dynamics compatible with classical predictions, suggesting massive quantum objects as a promising avenue for testing some phenomenological aspects of quantum gravity. [ABSTRACT FROM AUTHOR]

An experimental and numerical investigation is conducted on the thermo-acoustic stability of a bluff-body turbulent combustor, considering variations in airflow rate and fuel composition. The focus is on mapping the behavior of the combustor in terms of the Reynolds number (Re). The stability of the combustor is assessed for three different fuels: pure hydrogen (PH), synthesis natural gas (SNG), and syngas (SG). The experimental analysis was carried out for selected fuel compositions using simultaneous, unsteady pressure measurement and OH* chemiluminescence across the variation of Reynolds number, Re , over the 2289–8009 range. The numerical modelling of bluff body stabilized turbulent diffusion flames has been performed by the ANSYS-Fluent. The standard k-ɛ model of turbulent flow, and the PDF (Probability Density Function)/Mixture Fraction combustion model for non-premixed combustion have been used in the combustion modelling. The experimental results reveal at low Re and the fundamental duct acoustic mode, SNG shows stable combustion, while SG exhibits high amplitude oscillations. When the Re is increased, SNG experiences significant oscillations with high amplitude nearer to duct acoustic mode, whereas SG shows comparatively lower amplitude oscillations at the third harmonic. When studying PH, it is interesting to note the presence of significant oscillations at higher Re at the first harmonic and highest amplitude oscillations at fundamental mode at lower Re. The combustion dynamics of the fuels are affected by the global equivalence ratio (ϕ) and the momentum flux ratio (MFR) fuel to air. A clear distinction in flame anchoring has been observed among the selected fuels. The PH flame is anchored at the injection location, and SNG flame is found in the shear layer and wake of the bluff body, while SG flame is anchored near the walls of the combustor and shear layer of the bluff body. In addition, the average flow fields and temperature contours for the listed fuels are also presented in this study. • Stability of combustor is mapped for three hydrogen enriched fuels across the Re. • CH 4 in H 2 excites fundamental acoustic mode, while CO excites higher frequencies. • Equivalence ratio and momentum ratio dictates the flame anchoring mechanism. • Flame-driven vorticities and flow effect are responsible for the size of vortices. • Macro flame structures across fuels and flow conditions dictates spectral content. [ABSTRACT FROM AUTHOR]

This paper presents an investigation into one- and three-dimensional harmonic oscillators with time-dependent mass and frequency, within the framework of the Dunkl formalism, which is constituted by replacing the ordinary derivative with the Dunkl derivative. To ascertain a general form of the wave functions the Lewis-Riesenfeld method was employed. Subsequently, an exponentially changing mass function in time was considered and the parity-dependent quantum phase, energy eigenvalues, and the corresponding wave functions were derived in one dimension. The findings revealed that the mirror symmetries affect the wave functions, thus the associated probabilities. Finally, the investigation was extended to the three-dimensional case, where it was demonstrated that, as with the solution of the radial equation, the solutions of the angular equation could be classified according to their mirror symmetries. [ABSTRACT FROM AUTHOR]

In this paper, we construct a smooth vector bundle over the deformation to the normal cone DNC(V,M) through a rescaling of a vector bundle E → V, which generalizes the construction of the spinor rescaled bundle over the tangent groupoid by Nigel Higson and Zelin Yi. We also provide an equivariant version of their construction. As the main application, we recover the Kirillov character formula for the equivariant index of Dirac-type operators. As another application, we get an equivariant generalization of the description of the Witten and the Novikov deformations of the de Rham-Dirac operator using the deformation to the normal cone obtained recently by O. Mohsen. [ABSTRACT FROM AUTHOR]

The harmonic balance (HB) method is efficient in predicting the limit cycle oscillations for many dynamic systems including very high dimensional systems in fluid dynamics. Recently, some variations and extensions of the HB method have been developed and employed in the computation of large systems. This study focuses on a new development of the HB method. The new approach, the hybrid harmonic balance method (HHB), is developed based on the ideas in the conventional HB approach and the high dimensional harmonic balance (HDHB) approach. To demonstrate the applications and advantages, the new method is employed on a prototypical dynamic system in comparison with the results from both the HB and HDHB approaches. It is theoretically proved that when twice the number of harmonics in HB are included in the HHB derivation the results from HHB achieve the accuracy of the HB method for the cubic nonlinearity. Numerical simulations reveal that HHB combines the advantages of both the conventional HB and HDHB methods: Ease of implementation for very high dimensional systems regardless of the complexity of the nonlinearities and generation of meaningful solutions while the spurious solutions (from HDHB) are diminished. This new method is more accurate when compared with the HDHB method and much more computationally efficient when compared with the conventional HB method. [ABSTRACT FROM AUTHOR]

Perylene diimide (PDI) dimers and higher aggregates are key components in organic molecular photonics and photovoltaic devices, supporting singlet fission and symmetry breaking charge separation. Detailed understanding of their excited states is thus important. This has proven challenging because interchromophoric coupling is a strong function of dimer architecture. Recently, a macrocyclic PDI dimer was reported in which excitonic coupling could be turned on and off simply by changing the solvent. This presents a useful case where coupling is modified without synthetic changes to tune supramolecular structure. Here we present a detailed study of solvent dependent excited state dynamics in this dimer by means of coherent multidimensional spectroscopy. Spectral analysis resolves the different coupling strengths, which are consistent with solvent dependent changes in dimer conformation. The strongly coupled conformer forms an excimer within 300 fs. The low‐frequency Raman active modes recovered from two‐dimensional electronic spectra reveal frequencies characteristic of exciton coupling. These are assigned to modes modulating the coupling from the corresponding DFT calculations. Further analysis reveals a time dependent frequency during excimer formation. Analysis of two‐dimensional "beatmaps" reveals features in the coupled dimer which are not predicted by the displaced harmonic oscillator model and are assigned to vibronic coupling. [ABSTRACT FROM AUTHOR]

An impact of an electronic structure or force field method, gas‐phase thermodynamic correction, and continuum solvation model on organic carbonate clusters (S)n conformational and binding energies is explored. None of the tested force field (GFN‐FF, GAFF, MMFF94) and standard semiempirical methods (PM3, AM1, RM1, PM6, PM6‐D3, PM6‐D3H4, PM7) can reproduce reference RI‐SCS‐MP2 conformational energies. Tight‐binding GFNn‐xTB methods provide more realistic conformational energies which are accurate enough to discard the least stable conformers. The effect of thermodynamic correction is moderate and can be ignored if the gas phase conformational stability ranking is a goal. The influence of continuum solvation is stronger, especially if reinforced with the Gibbs free energy thermodynamic correction, and results in the reduced spread of conformational energies. The cluster formation binding energies strongly depend on a particular approach to vibrational thermochemistry with the difference between traditional harmonic and modified scaled rigid – harmonic oscillator approximations reaching 10 kcal mol−1. [ABSTRACT FROM AUTHOR]

The High-Static-Low-Dynamic (HSLD) stiffness vibration isolators have been exploited in many engineering applications due to its capability in having a wider isolation bandwidth, while maintaining the high static load capacities. However, it will lead to a large payload oscillation at the static equilibrium position, if the source of vibration is an oscillating force originating within the payload. In this case, the considerably large resultant motion of the payload will change the system nonlinearity. An active stiffness control for reducing the displacement amplitude of the payload oscillation subjected to a harmonic force excitation is proposed in this paper. The dynamic model of an actively stiffened HSLD stiffness isolator is introduced, and the approximate analytical expression for forced response is obtained using the Harmonic Balance Method (HBM). The obtained forced response curve has demonstrated that the active stiffness control is able to reduce the system's force response, particularly at low frequencies with an approximation of 50%. The nonlinearity of the system becomes smaller as the active stiffness control is applied. [ABSTRACT FROM AUTHOR]

This work introduces a geometrical object that generalizes the quantum geometric tensor; we call it N -bein. Analogous to the vielbein (orthonormal frame) used in the Cartan formalism, the N -bein behaves like a 'square root' of the quantum geometric tensor. Using it, we present a quantum geometric tensor of two states that measures the possibility of moving from one state to another after two consecutive parameter variations. This new tensor determines the commutativity of such variations through its anti-symmetric part. In addition, we define a connection different from the Berry connection, and combining it with the N -bein allows us to introduce a notion of torsion and curvature à la Cartan that satisfies the Bianchi identities. Moreover, the torsion coincides with the anti-symmetric part of the two-state quantum geometric tensor previously mentioned, and thus, it is related to the commutativity of the parameter variations. We also describe our formalism using differential forms and discuss the possible physical interpretations of the new geometrical objects. Furthermore, we define different gauge invariants constructed from the geometrical quantities introduced in this work, resulting in new physical observables. Finally, we present two examples to illustrate these concepts: a harmonic oscillator and a generalized oscillator, both immersed in an electric field. We found that the new tensors quantify correlations between quantum states that were unavailable by other methods. [ABSTRACT FROM AUTHOR]

Chemical reduction of a vertically expanded pentacene, TIPS-peri-pentacenopentacene (TIPS-PPP), with sodium metal in THF readily afforded a doubly-reduced product isolated as [{Na+(THF)3}2(TIPS-PPP2−)]. Single-crystal X-ray diffraction revealed the formation of a π-complex of TIPS-PPP2− with two {Na+(THF)3} moieties. The sodium ion is coordinated to three C-atoms at the most negatively charged edge sites and at the lateral ethynyl group. The delocalisation of the charge on the ethynyl function is accompanied by its notable elongation (Δ = 0.015 Å) coupled with the contraction of the adjacent single C–C bond (Δ = 0.033 Å). Bond length analysis supported by Harmonic Aromaticity Oscillator Model (HOMA) shows that the dianion can be written with four aromatic sextets in agreement with the Clar representation. Although TIPS-PPP2− has 36 π-electrons (4n), it does not show a global magnetic paratropic response. The rings with the most negative charge density have a very low paratropic response, while the other rings have a low diatropic response. Overall, the dianion is a non-aromatic molecule. [ABSTRACT FROM AUTHOR]

Terahertz communication technology holds promise for solving the challenging "blackout" problem. Research has revealed that the spectrum exhibits new peaks when terahertz waves propagate through plasma. In the realm of communications, elucidating the underlying physical mechanisms responsible for these novel peaks is of paramount importance. In this paper, a combination of theoretical analysis and simulation is utilized to investigate the causes of these spectral changes. Theoretically, the expression for the relative dielectric constant of plasma has been refined by incorporating collision terms. Drawing on radiation theory, a particle whose motion is interrupted by collisions can be considered equivalent to a damped harmonic oscillator. Thus, the frequency expression of the new peak was derived. Furthermore, the simulation involved the interaction of a terahertz wave, which had a frequency of 1 THz and lasted for 2 ps, and plasma characterized by a central electron density of 1021 m−3 and a collision frequency of 0.114 THz. The simulation results are consistent with the frequency expression of the new peak. In sum, this study found that collisions between electrons and neutral particles within plasma cause a shift in the frequency of terahertz waves, and has derived the frequency expression. This plasma model can be used to estimate the frequency range of terahertz waves that are generated as they propagate through collisional plasmas. [ABSTRACT FROM AUTHOR]

The method of quantum dynamics is employed to investigate the mean strategy in the swarm intelligence algorithm. The physical significance of the population mean point is explained as the location where the optimal solution with the highest likelihood can be found once a quantum system has reached a ground state. Through the use of the double well function and the CEC2013 test suite, controlled experiments are conducted to perform a comprehensive performance analysis of the mean strategy. The empirical results indicate that implementing the mean strategy not only enhances solution diversity but also yields accurate, efficient, stable, and effective outcomes for finding the optimal solution. [ABSTRACT FROM AUTHOR]

Continuous-time modeling using differential equations is a promising technique to model change processes with longitudinal data. Among ways to fit this model, the Latent Differential Structural Equation Modeling (LDSEM) approach defines latent derivative variables within a structural equation modeling (SEM) framework, thereby allowing researchers to leverage advantages of the SEM framework for model building, estimation, inference, and comparison purposes. Still, a few issues remain unresolved, including performance of multilevel variations of the LDSEM under short time lengths (e.g., 14 time points), particularly when coupled multivariate processes and time-varying covariates are involved. Additionally, the possibility of using Bayesian estimation to facilitate the estimation of multilevel LDSEM (M-LDSEM) models with complex and higher-dimensional random effect structures has not been investigated. We present a series of Monte Carlo simulations to evaluate three possible approaches to fitting M-LDSEM, including: frequentist single-level and two-level robust estimators and Bayesian two-level estimator. Our findings suggested that the Bayesian approach outperformed other frequentist approaches. The effects of time-varying covariates are well recovered, and coupling parameters are the least biased especially using higher-order derivative information with the Bayesian estimator. Finally, an empirical example is provided to show the applicability of the approach. [ABSTRACT FROM AUTHOR]

In this paper, we examine the role of the Jacobi last multiplier in the context of two-dimensional oscillators. We first consider two-dimensional unit-mass oscillators admitting a separable Hamiltonian description, i.e., H = H 1 + H 2 , where H 1 and H 2 are the Hamiltonians of two one-dimensional unit-mass oscillators. It is shown that there exists a third functionally-independent first integral Θ , ensuring superintegrability. Various examples are explicitly worked out. We then consider position-dependent-mass oscillators and the Bateman pair, where the latter consists of a pair of dissipative linear oscillators. Quite remarkably, the Bateman pair is found to be superintegrable, despite admitting a Hamiltonian which cannot be separated into two isolated (non-interacting) one-dimensional oscillators. [ABSTRACT FROM AUTHOR]

We study solutions of the collisionless Boltzmann equation (CBE) in a functional Koopman representation. This facilitates the use of linear spectral techniques characteristic of the analysis of Schrödinger-type equations. For illustrative purposes, we consider the classical phase mixing of a non-interacting distribution function in a quartic potential. Solutions are determined perturbatively relative to a harmonic oscillator. We impose a form of coarse-graining by choosing a finite-dimensional basis to represent the distribution function and time evolution operators, which sets a minimum length-scale on phase space structure. We observe a relationship between the dimension of the representation and the multiplicity of the harmonic oscillator eigenvalues. System dynamics are understood in terms of degenerate subspaces of the linear operator spectra. Each subspace is associated with a mode of the harmonic oscillator, the first two being bending and breathing structures. The quartic potential splits the degenerate eigenvalues within each subspace. This facilitates the formation of spiral structure as deformations from the harmonic oscillator modes. We ultimately argue that this construction provides a promising avenue for study of self-interacting systems experiencing phase mixing, which is an outstanding problem in the context of the Gaia DR2 vertical phase space spirals. [ABSTRACT FROM AUTHOR]

We present an algebraic approach for the construction of the model space for the few-body systems. The approach is suitable for the observable calculation. The model utilizes the translationally invariant harmonic oscillator basis. We extensively use the symmetric group apparatus for the fractional parentage coefficient calculation. [ABSTRACT FROM AUTHOR]

Enlightened by Lemma 1.7 in Liang and Luo (J Differ Equ 270:343–389, 2021), we prove a similar lemma which is based upon oscillatory integrals and Langer's turning point theory. From it we show that the Schrödinger equation i ∂ t u = - ∂ x 2 u + x 2 u + ϵ ⟨ x ⟩ μ ∑ k ∈ Λ a k (ω t) sin (k | x | β) + b k (ω t) cos (k | x | β) u , u = u (t , x) , x ∈ R , β > 1 , can be reduced in H 1 (R) to an autonomous system for most values of the frequency vector ω , where Λ ⊂ R \ { 0 } , | Λ | < ∞ and ⟨ x ⟩ : = 1 + x 2 . The functions a k (θ) and b k (θ) are analytic on T σ n and μ ≥ 0 will be chosen according to the value of β . Comparing with Liang and Luo (J Differ Equ 270:343–389, 2021), the novelty is that the phase functions of oscillatory integral are more degenerate when β > 1 . [ABSTRACT FROM AUTHOR]

In the present paper, we establish an infinite dimensional Kolmogorov–Arnold–Moser (KAM) theorem for reversible systems with double normal frequencies. Applying it, we prove the existence of quasi-periodic solutions for one dimensional coupled nonlinear quantum harmonic oscillators (QHO) with a natural reversible structure. To compensate the lack of smoothing effect of perturbation, we introduce a class of vector fields with polynomial decay which extends the works of Grébert and Thomann (Commun Math Phys 307(2):383–427, 2011) for Hamiltonian QHO. To deal with the reversible, coupled perturbations in the equations, we also introduce a new class of generating vector fields during the KAM iteration. Moreover, the quasi-periodic solutions we obtain may not be linearly stable. This is obviously different from the result in Grébert and Thomann (2011) for Hamiltonian QHO. [ABSTRACT FROM AUTHOR]

This document provides a historical overview of the development of the concept of intrinsic angular momentum, or "spin," of the photon. It discusses the contributions of Bose and Einstein in postulating the spin of the photon and the quantization of energy, momentum, and angular momentum in particles and atoms. The document also explores the understanding of field energy and momentum in physics, including the arguments made by Maxwell, Thomson, and Heaviside. It further discusses the electromagnetic momentum density and the transfer of field angular momentum to mechanical angular momentum. Additionally, the document mentions the Feynman disk paradox and the quantization of angular momentum. [Extracted from the article]

Ultrafast optical spectroscopy is an effective experimental technique for accessing electronic and atomic motions in materials at their fundamental timescales and studying their responses to external perturbations. Despite the important insights that ultrafast techniques can provide on the microscopic physics of solids, undergraduate students' exposure to this area of research is still limited. In this article, we describe an ultrafast optical pump-probe spectroscopy experiment for the advanced undergraduate instructional laboratory, in which students can measure coherently excited vibrations of the crystal lattice and connect their observations to the microscopic properties of the investigated materials. We designed a simple table-top apparatus based on a commercial Er-fiber oscillator emitting 50-fs pulses at 1560 nm and at 100 MHz repetition rate. We split the output into two beams, using one of them as an intense "pump" to coherently excite phonons in selected crystals, and the other as a weaker, delayed "probe" to measure the transient reflectivity changes induced by the pump. We characterize the ultrafast laser pulses via intensity autocorrelation measurements and detect coherent phonon oscillations in the reflectivity of Bi, Sb, and 1T-TaS2. We then discuss the oscillation amplitude, frequency, and damping in terms of microscopic properties of these systems. Editor's Note: This paper provides an introduction to modern time-domain spectroscopy using ultrafast lasers, an experimental technique used to observe fundamentally important phenomena in condensed matter physics as well as in other fields. As a specific example, the authors demonstrate the generation and detection of coherent phonons in three solids. The paper provides an undergraduate-accessible discussion of the theory of coherent phonon spectroscopy in terms of classical harmonic oscillator physics. Then, a relatively low-cost ultrafast pump-probe spectroscopy system and the experimental procedures that the authors use in their experiment are described. Finally, typical experimental data and analysis methods are presented. This laboratory experiment is a welcome addition to the undergraduate physics curriculum, introducing a system that is suitable for examining a variety of ultrafast and nonequilibrium phenomena in an advanced instructional laboratory. [ABSTRACT FROM AUTHOR]

Cracks, common indicators of deterioration in bridge frameworks, frequently stem from wear and rust, leading to increased local flexibility and changes in the structure's dynamic behavior. This study examines how these cracks affect the dynamics of footbridges when subjected to loads generated by walking individuals. The pedestrian is modeled as a linear oscillator, while the cracked bridge is represented by a simply supported beam following Euler–Bernoulli's theory. The use of the Dirac delta function allows for the precise representation of the localized stiffness reduction at the crack location, facilitating the calculation of analytical expressions for the beam's vibration modes. The research suggests that the presence of cracks minimally affects the bridge's mid-span displacement. However, with a limited depth of cracks, the appearance of cracks notably amplifies the mid-span acceleration amplitude of the bridge, leading to a pronounced concentration of energy at the third natural frequency of the bridge in the acceleration spectrum. As the depth and number of cracks increase, the acceleration amplitude continues to decrease, but the corresponding spectrum remains almost unchanged. The study's outcomes enhance the comprehension of how cracks affect the performance of bridge structures when subjected to loads from pedestrians, offering insights for the monitoring and evaluation of the condition of cracked footbridges. [ABSTRACT FROM AUTHOR]

We attempt to construct a (two-dimensional) statistical mechanics description of the scenario in which the experiment in [T. Westphal et al., Nature 591, 225 (2021)] was made. It includes two masses that, simultaneously, (1) interact classically via the Newton gravitation and (2) harmonically vibrate. Within the strictures of Gibbs canonical ensemble at the temperature T, our treatment is exact. [ABSTRACT FROM AUTHOR]

We investigate the optimal performance of the quantum Otto engine and refrigeration cycles of a time-dependent harmonic oscillator under a trade-off figure of merit for both adiabatic and nonadiabatic (sudden-switch) frequency modulations. For heat engines (refrigerators), the chosen trade-off figure of merit is an objective function defined by the product of efficiency (coefficient of performance) and work output (cooling load), thus representing a compromise between them. We obtain analytical expressions for the efficiency and coefficient of performance of the harmonic Otto cycle for the optimal performance of the thermal machine in various operational regimes. Particularly, in the sudden-switch regime, we discuss the implications of the nonadiabatic driving on the performance of the thermal machine under consideration and obtain analytic expressions for the maximum achievable efficiency and coefficient of performance of the harmonic Otto thermal machine. Particularly, we show that the quantum harmonic Otto cycle driven by sudden-switch protocol cannot work as a heat engine or refrigerator in the low-temperature limit. Finally, we show that in the high-temperature limit, the frictional effects give rise to a richer structure of the phase diagram of the harmonic Otto cycle. We identify the parametric regime for the operation of the Otto cycle as a heat engine, refrigerator, accelerator, and heater. [ABSTRACT FROM AUTHOR]

This work utilizes an analytic expression for a model of acoustic propagation in a two‐layered, inhomogeneous atmosphere developed by the authors. The model is used to study the atmospheres of Earth, Mars, and Titan. In particular, vertical wave propagation in these atmospheres is studied. The effect(s) of a two‐layered, inhomogeneous atmosphere on vertical, acoustic propagation due to a time‐harmonic, point source are examined. An adiabatic atmosphere is used for the bottom layer (troposphere) and an isothermal one for the top (stratosphere). The derived, analytic solution is expressed in terms of the acoustic pressure fluctuations. For the adiabatic layers, the solutions satisfy Bessel's equation for orders of χ=−3.5,−4.45$\chi =-3.5, -4.45$, and −3.63$-3.63$ for Earth, Mars, and Titan, respectively. The Bessel function's argument is 2Ωτ$2 \Omega \tau$, where Ω$\Omega$ and τ$\tau$ are dimensionless frequency and height, respectively. For the isothermal layer, the solution represents a damped, harmonic oscillator with a cutoff value of Ωc$\Omega _{c}$. Only values greater than Ωc$\Omega _{c}$ are considered. The analysis and results are reported for combinations of single‐ and double‐layer atmospheres in the presence of a source on given boundaries. Acoustic propagation and transmission loss results are shown and discussed for all three planetary bodies: Earth, Mars, and Titan. [ABSTRACT FROM AUTHOR]

Hamiltonian Neural Networks (HNNs) provide structure-preserving learning of Hamiltonian systems. In this paper, we extend HNNs to structure-preserving inversion of stochastic Hamiltonian systems (SHSs) from observational data. We propose the quadrature-based models according to the integral form of the SHSs' solutions, where we denoise the loss-by-moment calculations of the solutions. The integral pattern of the models transforms the source of the essential learning error from the discrepancy between the modified Hamiltonian and the true Hamiltonian in the classical HNN models into that between the integrals and their quadrature approximations. This transforms the challenging task of deriving the relation between the modified and the true Hamiltonians from the (stochastic) Hamilton–Jacobi PDEs, into the one that only requires invoking results from the numerical quadrature theory. Meanwhile, denoising via moments calculations gives a simpler data fitting method than, e.g., via probability density fitting, which may imply better generalization ability in certain circumstances. Numerical experiments validate the proposed learning strategy on several concrete Hamiltonian systems. The experimental results show that both the learned Hamiltonian function and the predicted solution of our quadrature-based model are more accurate than that of the corrected symplectic HNN method on a harmonic oscillator, and the three-point Gaussian quadrature-based model produces higher accuracy in long-time prediction than the Kramers–Moyal method and the numerics-informed likelihood method on the stochastic Kubo oscillator as well as other two stochastic systems with non-polynomial Hamiltonian functions. Moreover, the Hamiltonian learning error ε H arising from the Gaussian quadrature-based model is lower than that from Simpson's quadrature-based model. These demonstrate the superiority of our approach in learning accuracy and long-time prediction ability compared to certain existing methods and exhibit its potential to improve learning accuracy via applying precise quadrature formulae. [ABSTRACT FROM AUTHOR]

We investigated a time-delayed optoelectronic oscillator (OEO) that displays a wide range of complex dynamic behavior under small time delay. The phase-space trajectory distributions in different dynamic regimes were compared which brings a new perspective on the underlying mechanism of the transition process. It was found that bifurcation is always possible no matter how small the time delay is even if the universal adiabatic approximation model is invalid. Hereby we proposed a versatile simple oscillator which has a potential capacity as memory carrier and high-dimensional state spatial mapping ability that brings 1000 times computing-efficiency improvements of reservoir computing over the large time delay one. Furthermore, we demonstrated a new approach for a tunable optoelectronic pulse generator (repetition rate at 0.2 MHz and 0.25 GHz) which depends critically on time-delayed input electrical pulse. The proposed oscillator is also a promising system for the applications of fast chaos-based communication. [ABSTRACT FROM AUTHOR]

The Gross–Pitaevskii (GP) equation is a model for the description of the dynamics of Bose–Einstein condensates. Here, we consider the GP equation in a two‐dimensional setting with an external periodic potential in the x$x$‐direction and a harmonic oscillator potential in the y$y$‐direction in the so‐called tight‐binding limit. We prove error estimates which show that in this limit the original system can be approximated by a discrete nonlinear Schrödinger equation. The paper is a first attempt to generalize the results from [19] obtained in the one‐dimensional setting to higher space dimensions and more general interaction potentials. Such a generalization is a non‐trivial task due to the oscillations in the external periodic potential which become singular in the tight‐binding limit and cause some irregularity of the solutions which are harder to handle in higher space dimensions. To overcome these difficulties, we work in anisotropic Sobolev spaces. Moreover, additional non‐resonance conditions have to be satisfied in the two‐dimensional case. [ABSTRACT FROM AUTHOR]