Multiple Potential Wells Research Articles

Response of Gaussian white noise excited oscillators with inertia nonlinearity based on the RBFNN method

Although stochastic averaging methods have proven effective in solving the responses of nonlinear oscillators with a strong stiffness term under broadband noise excitations, these methods appear to be ineffective when dealing with oscillators that have a strong inertial nonlinearity term (also known as coordinate-dependent mass) or multiple potential wells. To address this limitation, a radial basis function neural network (RBFNN) algorithm is applied to predict the responses of oscillators with both a strong inertia nonlinearity term and multiple potential wells. The well-known Gaussian functions are chosen as radial basis functions in the model. Then, the approximate stationary probability density function (PDF) is expressed as the sum of Gaussian basis functions (GBFs) with weights. The squared error of the approximate solution for the Fokker-Plank-Kolmogorov (FPK) function is minimized using the Lagrange multiplier method to determine optimal weight coefficients. Three examples are presented to demonstrate how inertia nonlinearity terms and potential wells affect the responses. The mean square errors between Monte Carlo simulations (MCS) and RBFNN predictions are provided. The results indicate that RBFNN predictions align perfectly with those obtained from MCS.

Response of Gaussian color noise excited oscillators with inertia nonlinearity based on the radial basis function neural network method

Given the widespread utilization of oscillators featuring inertial nonlinear terms across engineering disciplines and the omnipresence of noise excitations, comprehending their response to noise excitation holds paramount importance. In contrast to Gaussian white noise, colored noise stands out as a more adept mathematical model for simulating ambient noise scenarios. The widely used stochastic averaging method is somewhat ineffective in dealing with the response of oscillators with inertial nonlinearity under Gaussian colored noise excitation, as it cannot display the saddle shaped characteristics of the steady-state probability density function (PDF) caused by inertial nonlinearity. To overcome the shortcomings, this paper adopts a semi-analytical method called a radial basis function neural network method (RBFNN) to address this problem. This method represents the solution of the Fokker-Plank-Kolmogorov (FPK) equation of the system as the sum of a series of weighted radial basis functions, and obtains the optimal weight coefficients through the Lagrange multiplier method. This article investigates two examples, one of which has inertial nonlinearity and the other has multiple potential wells. The effects of inertial nonlinearity coefficients and the delay coefficients of colored noise on response are explored. The mean square errors between Monte Carlo (MC) and RBF are presented, which indicates the RBF results perfectly agree with the MC predictions.

Integrated Hybrid Tweezer for Particle Trapping with Combined Optical and Acoustic Forces

We propose an on-chip integrated hybrid tweezer that can simultaneously apply optical and acoustic forces on particles to control their motions. Multiple potential wells can be formed to trap particles, and the acoustic force generated by an interdigital transducer can balance the optical force induced by an optical waveguide. For example, by driving the waveguide with an optical power of 100 mW and the interdigital transducer with a voltage of 1.466 V, the particle with a refractive index of 1.4 and a diameter of 5 μm (similar to yeast cells) can be stably trapped on the waveguide surface, and its trapping position is controllable by changing the optical power or voltage.

Reaction dynamics of P(4S) + O2(X3Σ−g) → O(3P) + PO(X2Π) on a global CHIPR potential energy surface of PO2(X2A1): implications for atmospheric modelling

Abstract. The reaction dynamics of P(4S) + O2(X3Σg-) → O(3P) + PO(X2Π) are thought to be important in atmospheric and interstellar chemistry. Based on the state-of-the-art ab initio energy points, we analytically constructed a global potential energy surface (PES) for the ground-state PO2(X2A1) using the combined-hyperbolic-inverse-power-representation (CHIPR) method. A total of 6471 energy points were computed by the multireference configuration interaction method with the Davidson correction and aug-cc-pV5Z basis set. The analytical CHIPR PES reproduces ab initio energies accurately with a root-mean-square deviation of 91.5 cm−1 (or 0.262 kcal mol−1). The strongly bound valence region of the PES has complicated topographical features with multiple potential wells and barriers. The attributes of the important intermediates are carefully validated with our geometry optimization results, as well as previous experimental and computational results. Finally, the reaction probability, integral cross sections, and rate constants for P(4S) + O2(X3Σg-) → O(3P) + PO(X2Π) are calculated using the quasi-classical trajectory and time-dependent wave packet methods. The trends of probability and integral cross section versus the collision energy can be divided into three stages, which are governed by the entrance barriers or exothermicity of the reaction. The rate constant demonstrates strong Arrhenius linear behaviour at relatively low temperatures but deviates from this pattern at high temperatures. The calculated cross sections and rate constants are helpful for modelling the phosphorus chemistry in atmospheric and interstellar media.

Achiral nanoparticle trapping and chiral nanoparticle separating with quasi-BIC metasurface.

Dielectric metasurfaces based on quasi-bound states in the continuum (quasi-BICs) are a promising approach for manipulating light-matter interactions. In this study, we numerically demonstrate the potential of silicon elliptical tetramer dielectric metasurfaces for achirality nanoparticle trapping and chiral nanoparticle separation. We first analyze a symmetric tetramer metasurface, which exhibits dual resonances (P1 and P2) with high electromagnetic field intensity enhancement and a high-quality factor (Q-factor). This metasurface can trap achiral nanoparticles with a maximum optical trapping force of 35 pN for 20 nm particles at an input intensity of 100 mW. We then investigate an asymmetric tetramer metasurface, which can identify and separate enantiomers under the excitation of left-handed circularly polarized (LCP) light. Results show that the chiral optical force can push one enantiomer towards regions of the quasi-BIC system while removing the other. In addition, the proposed asymmetric tetramer metasurface can provide multiple Fano resonances (ranging from R1 to R5) and high trap potential wells of up to 33 kBT. Our results demonstrate that the proposed all-dielectric metasurface has high performance in nanoparticle detection, with potential applications in biology, life science, and applied physics.

Static and dynamic nonlinear behavior of a multistable structural system consisting of two coupled von Mises trusses

Recent years have witnessed a growing interest in multistable structures. In most cases it is attained by a sequence of bistable elements. However, little is known on their nonlinear static and dynamic responses. In this work a detailed static and dynamic nonlinear analysis of a multistable structural system consisting of two coupled bistable von Mises trusses is conducted to enlighten the complex behavior of these systems. For this, the exact nonlinear equilibrium equations and equations of motion in their dimensionless forms are obtained through the principle of stationary potential energy and Hamilton’s principle, respectively, considering a linear elastic material. Using continuation algorithms, the nonlinear equilibrium paths are obtained. The stability of each configuration is analyzed using the principle of minimum potential energy. Multiple equilibrium paths of the coupled system are identified, leading to several coexisting stable and unstable solutions, bifurcations and potential wells which are closely connected to the symmetries of the system. It is shown that the number of coexisting solutions increases exponentially with the number of bistable units. The effect of unavoidable imperfections is also clarified. The nonlinear dynamics and bifurcations of the system under harmonic forcing and static pre-load are then studied. Bifurcation diagrams, Poincaré maps and cross-sections of the basins of attraction are used to study the influence of coexisting attractors due to multiple potential wells, resonances and bifurcations. The coexisting responses can merge giving rise to several types of cross-well motions, the threshold value being closely connected with the limit point load of the coupled system. The increase in coexisting solutions leads to increasingly complex basins of attraction with broad fractal regions. On the one hand, the complexity of the bifurcation scenarios seems valuable in several applications to encode information, to respond to external forcing and to switch between different states. On the other hand, multiple attractors and their fractal basins can lead to the loss of global stability and dynamic integrity due to the small uncorrupted basins surrounding each attractor. Thus, the knowledge of the static and dynamic behavior of multistable systems is of great significance to either inducing or avoiding this phenomenon or making use of it.

Exact quantum dynamics developments for floppy molecular systems and complexes.

Molecular rotation, vibration, internal rotation, isomerization, tunneling, intermolecular dynamics of weakly and strongly interacting systems, intra-to-inter-molecular energy transfer, hindered rotation and hindered translation over surfaces are important types of molecular motions. Their fundamentally correct and detailed description can be obtained by solving the nuclear Schrödinger equation on a potential energy surface. Many of the chemically interesting processes involve quantum nuclear motions which are 'delocalized' over multiple potential energy wells. These 'large-amplitude' motions in addition to the high dimensionality of the vibrational problem represent challenges to the current (ro)vibrational methodology. A review of the quantum nuclear motion methodology is provided, current bottlenecks of solving the nuclear Schrödinger equation are identified, and solution strategies are reviewed. Technical details, computational results, and analysis of these results in terms of limiting models and spectroscopically relevant concepts are highlighted for selected numerical examples.

Global Bifurcation Behaviors and Control in a Class of Bilateral MEMS Resonators

The investigation of global bifurcation behaviors the vibrating structures of micro-electromechanical systems (MEMS) has received substantial attention. This paper considers the vibrating system of a typical bilateral MEMS resonator containing fractional functions and multiple potential wells. By introducing new variations, the Melnikov method is applied to derive the critical conditions for global bifurcations. By engaging in the fractal erosion of safe basin to depict the phenomenon pull-in instability intuitively, the point-mapping approach is used to present numerical simulations which are in close agreement with the analytical prediction, showing the validity of the analysis. It is found that chaos and pull-in instability, two initial-sensitive phenomena of MEMS resonators, can be due to homoclinic bifurcation and heteroclinic bifurcation, respectively. On this basis, two types of delayed feedback are proposed to control the complex dynamics successively. Their control mechanisms and effect are then studied. It follows that under a positive gain coefficient, delayed position feedback and delayed velocity feedback can both reduce pull-in instability; nevertheless, to suppress chaos, only the former can be effective. The results may have some potential value in broadening the application fields of global bifurcation theory and improving the performance reliability of capacitive MEMS devices.

An elliptical rail–mass–spring mechanism to realize multi-stable circulation motion for electromagnetic-energy harvesting

This study proposes an elliptical rail–mass–spring mechanism to realize multi-stable circulation motion for electromagnetic energy harvesting. Unlike the conventional bistable vibration model, the moveable body can jump between multiple potential energy wells. Correspondingly, the elliptical rail–mass–spring prototype is deliberately exploited, and experimental investigation indicates that the response angular displacement range of the moveable body can be expanded to [0°–630°] as a circulation elliptical motion and the response angular displacement range can be expanded if the low-frequency vibrational excitation is continued. Comparing the average power obtained by the elliptical rail–mass–spring mechanism, 17.33 mW was obtained for the random signal, 45.40 mW was obtained for the periodic signal with the largest motion response of 0.8 Hz, and 77.99 mW was obtained when the two signals were combined. The average power obtained by a combined signal is greater than the sum of the results obtained by a separate signal, which confirms that the elliptical rail–mass–spring mechanism noticeably enhances power-generation efficiency.

Design, fabrication and characterization of a micro-fabricated stacked-wafer segmented ion trap with two X-junctions

We describe the implementation of a three-dimensional Paul ion trap fabricated from a stack of precision-machined silica glass wafers, which incorporates a pair of junctions for two-dimensional ion transport. The trap has 142 dedicated electrodes which can be used to define multiple potential wells in which strings of ions can be held. By supplying time-varying potentials, this also allows for transport and re-configuration of ion strings. We describe the design, simulation, fabrication and packaging of the trap, including explorations of different parameter regimes and possible optimizations and design choices. We give results of initial testing of the trap, including measurements of heating rates and junction transport.

Kinetic characteristics of ions in an inertial electrostatic confinement device.

The kinetic analyses are quite important when it comes to understanding the particle behavior in any device as they start to deviate from a continuum nature. In the present study, kinetic simulations are performed using the particle-in-cell method to analyze the behavior of ions inside a cylindrical inertial electrostatic confinement fusion (IECF) device which is being developed as a tabletop neutron source. Here, the lighter ions, like deuterium, are accelerated by applying an electrostatic field between the chamber wall (anode) and the cathode (cylindrical gridded wire), placed at the center of the device. The plasma potential profiles obtained from the simulated results indicate the formation of multiple potential well structures inside the cathode grid depending upon the applied cathode potential (from -1 to -5 kV). The ion density at the core region of the device is found to be of the order of 10^{16}m^{-3}, which closely resembles the experimental observations. Spatial variation of ion energy distribution function has been measured in order to observe the characteristics of ions at different cathode voltages. Finally, the simulated results are compared and found to be in good agreement with the experimental profiles. The present analysis can serve as a reference guide to optimize the technological parameters of the discharge process in IECF devices.

Studies on virtual electrode and ion sheath characteristics in a cylindrical inertial electrostatic confinement fusion device

An experiment on the formation of virtual electrode and ion sheath characteristics has been carried out in a hot cathode discharge plasma produced inside a cylindrical inertial electrostatic confinement fusion device. The plasma parameters such as electron temperature and plasma density are evaluated by using a Langmuir probe. Transition from a single potential well to multiple potential wells, i.e., virtual electrodes inside the cathode grid, is observed when the bias voltage applied to the cathode is increased from –1000 to –5000 V. An emissive probe has been used for the measurement of plasma potential due to its greater accuracy than the conventional Langmuir probe. Ion sheath potential structures and presheath characteristics for different cathode potentials have been investigated using the emissive probe and are found to be consistent with the Debye sheath model. A detailed discussion on the obtained results has been presented in this paper.

Dissection of the multichannel reaction of acetylene with atomic oxygen: from the global potential energy surface to rate coefficients and branching dynamics.

The O(3P) + C2H2 reaction is the first step in acetylene oxidation. The accurate kinetic data and the understanding of the reaction dynamics is of great importance. To this end, a full-dimensional global potential energy surface (PES) for the ground triplet state of the O(3P) + C2H2 reaction is constructed based on approximately 85 000 ab initio points calculated at the level of explicitly correlated unrestricted coupled cluster single, double, and perturbative triple excitations with the explicitly correlated polarized valence triple zeta basis set (UCCSD(T)-F12b/VTZ-F12). The PES is fit using the permutation invariant polynomial-neural network (PIP-NN) approach with a total root mean square error of 0.21 kcal mol-1. The key topographic features of the PES, including multiple potential wells and saddle points along different reaction pathways, are well represented by this fit PES. The kinetics and dynamics of the O(3P) + C2H2 reaction are investigated using the quasi-classical trajectory (QCT) method. The calculated rate coefficients are in good agreement with experimental data over a wide temperature range, especially when the temperature is lower than 1500 K. The product branch ratio has also been determined, which indicates the H + HCCO channel as the dominant reaction pathway at 298-3000 K, accounting for 80-90% of the overall rate coefficient, in agreement with experimental observations. The dynamics of the reaction is analyzed in detail.

Bimolecular Master Equations for a Single and Multiple Potential Wells with Analytic Solutions.

The analytic solutions, that is, populations, are derived for the K-adiabatic and K-active bimolecular master equations, separately, for a single and multiple potential wells and reaction channels, where K is the component of the total angular momentum J along the axis of least moment of inertia of the recombination products at a given energy E. The analytic approach provides the functional dependence of the population of molecules on its K-active or K-adiabatic dissociation, association rate constants and the intermolecular energy transfer, where the approach may complement the usual numerical approaches for reactions of interest. Our previous work, Part I, considered the solutions for a single potential well, whereby an assumption utilized there is presently obviated in the derivation of the exact solutions and farther discussed. At the high-pressure limit, the K-adiabatic and K-active bimolecular master equations may each reduce, respectively, to the K-adiabatic and K-active bimolecular Rice-Ramsperger-Kassel-Marcus theory (high-pressure limit expressions) for bimolecular recombination rate constant, for a single potential well, and augmented by isomerization terms when multiple potential wells are present. In the low-pressure limit, the expression for population above the dissociation limit, associated with a single potential well, becomes equivalent to the usual presumed detailed balance between the association and dissociation rate constants, where the multiple well case is also considered. When the collision frequency of energy transfer, ZLJ, between the chemical intermediate and bath gas is sufficiently less than the dissociation rate constant kd( E' J' K') for postcollision ( E' J' K), then the solution for population, g( EJK)+, above the critical energy further simplifies such that depending on ZLJ, the dissociation and association rate constant kr( EJK), as g( EJK)+ = kr( EJK)A·BC/[ ZLJ+ kd( EJK)], where A and BC are the reactants, for example, relevant for O3 formation from O + O2 + Ar up to ∼100 bar; otherwise, additional contributions from postcollision are present and especially relevant at high pressures. In the aforementioned regime ZLJ < kd( E' J' K) the physical connection of recombination rate constants, krec based on either utilizing population from the master equation approach or a collision based bimolecular RRKM theory is traced and elucidated analytically that the rate constants are equal. Recombination rate constants, krec, based on the population, are also given and considered for an adiabatic or active K. For example, for O3 formation in Ar bath gas, the K-adiabatic-based krec for O3 yields 4.0 × 10-34 cm6 molecule-2 s-1 at T = 300 K and 1 bar, in agreement with the experimental value, where the contribution from the population of metastable ozone is discussed and the adiabaticity of K highlighted. A facile numerical implementation of the formalism for g( EJK) and krec for O3 is noted. The application of the expressions to ozone recombination as a function of pressure and temperature is given elsewhere, beyond the selection considered here, for studying the physical features, including the contributions from the K and intermolecular energy transfer to the krec, and the puzzles reported from experiments for this reaction.

Passage through a potential barrier and multiple wells

Consider the semiclassical limit, as the Planck constant $\hbar\ri 0$, of bound states of a one-dimensional quantum particle in multiple potential wells separated by barriers. We show that, for each eigenvalue of the Schr\"odinger operator, the Bohr-Sommerfeld quantization condition is satisfied at least for one potential well. The proof of this result relies on a study of real wave functions in a neighborhood of a potential barrier. We show that, at least from one side, the barrier fixes the phase of wave functions in the same way as a potential barrier of infinite width. On the other hand, it turns out that for each well there exists an eigenvalue in a small neighborhood of every point satisfying the Bohr-Sommerfeld condition.

Intrinsic photo-conductance triggered by the plasmonic effect in graphene for terahertz detection

Terahertz (THz) technology is becoming more eminent for applications in diverse areas including biomedical imaging, communication, security and astronomy. However, THz detection still has some challenges due to the lack of sources and detectors despite decades of considerable effort. The appearance of graphene and its gapless spectrum enable their applications in sensitive detection of light over a very wide energy spectrum from ultraviolet, infrared to terahertz. Several mechanisms in graphene for THz detection have been proposed, such as photo-thermoelectric, Dyakonov–Shur (DS) and bolometric effects. Here, we propose a photoconductive mechanism assisted by plasma wave in a graphene field-effect transistor (FET). Sensitive response to THz radiation can be realized far below the interband transition at room temperature. The response is due to the contributions of both plasma drag and convection effects. The two effects can both trigger multiple potential wells along the channel, which are different from other quantum-transition mechanisms. The photoconductive effects can be explored in both periodic and non-periodic systems and can be substantially enhanced under the electric field. They could reduce the burden of structural complexity compared to other mechanisms like unilateral thermoelectric and DS detection. This paves the way for more judicious photo-detector design for versatile THz applications.

Multiple Duffing problem in a folding structure with hilltop bifurcation and possible imperfections

We review a multiple Duffing oscillation, based on static bifurcation theory. We find that it is useful to consider the structural instability of a folding truss with possible imperfections as a theoretical model for a Duffing problem with multiple potential wells. Theoretical bifurcation analysis revealed that the equilibrium path on this model has a “hilltop bifurcation.” In addition, we have considered the elastic (in-)stability of several folding models with imperfections. The present model is very sensitive near a critical point, leading to strong geometrical nonlinearity. We found that there are both global and local dynamic behaviours that are related to bifurcation and imperfect influences, which correspond to the structure of the multiple homo- and heteroclinic orbits. We suggest a theoretical model for hilltop bifurcation, based on the static bifurcation problem and perturbation theory, to assist in the identification of the structural mechanisms of the global and local dynamics of different paths. Such models are very useful for investigating the essential and invariant nonlinear phenomena of the extended Duffing oscillator model.

A unifying model for non-adiabatic coupling at metallic surfaces beyond the local harmonic approximation: From vibrational relaxation to scanning tunneling microscopy

A model for treating excitation and relaxation of adsorbates at metallic surfaces induced by non-adiabatic coupling is developed. The derivation is based on the concept of resonant electron transfer, where the adsorbate serves as a molecular bridge for the inelastic transition between an electron source and a sink. In this picture, energy relaxation and scanning tunneling microscopy (STM) at metallic surfaces are treated on an equal footing as a quasi-thermal process. The model goes beyond the local harmonic approximation and allows for an unbiased description of floppy systems with multiple potential wells. Further, the limitation of the product ansatz for the vibronic wave function to include the position-dependence of the non-adiabatic couplings is avoided by explicitly enforcing detailed balance. The theory is applied to the excitation of hydrogen on palladium, which has multiple local potential minima connected by low energy barriers. The main aspects investigated are the lifetimes of adsorbate vibrations in different adsorption sites, as well as the dependence of the excitation, response, and transfer rates on an applied potential bias. The excitation and relaxation simulations reveal intricate population dynamics that depart significantly from the simplistic tunneling model in a truncated harmonic potential. In particular, the population decay from an initially occupied local minimum induced by the contact with an STM tip is found to be better described by a double exponential. The two rates are interpreted as a response to the system perturbation and a transfer rate following the perturbation. The transfer rate is found to obey a power law, as was the case in previous experimental and theoretical work.

Nanostructured Potential of Optical Trapping Using a Plasmonic Nanoblock Pair

We performed two-dimensional mapping of optical trapping potentials experienced by a 100 nm dielectric particle above a plasmon-resonant gold nanoblock pair with a gap of several nanometers. Our results demonstrate that the potentials have nanoscale spatial structures that reflect the near-field landscape of the nanoblock pair. When an incident polarization parallel to the pair axis is rotated by 90°, a single potential well turns into multiple potential wells separated by a distance smaller than the diffraction limit; this is associated with super-resolution optical trapping. In addition, we show that the trap stiffness can be enhanced by approximately 3 orders of magnitude compared to that with conventional far-field trapping.

Plasmonic optical trap having very large active volume realized with nano-ring structure

The feasibility of using gold nano-rings as plasmonic nano-optical tweezers is investigated. We found that at a resonant wavelength of λ=785 nm, the nano-ring produces a maximum trapping potential of ~32k(B)T on gold nanoparticles. The existence of multiple potential wells results in a very large active volume of ~10(6) nm(3) for trapping the target particles. The report nano-ring design provides an effective approach for manipulating nano-objects in very low concentration into the high-field region and is well suited for integration with microfluidics for lab-on-a-chip applications.