Multiple-well Potential Research Articles

A novel modified Modica–Mortola equation with a phase-dependent interfacial function

In this study, we propose a novel modified Modica–Mortola (MM) equation with a phase-dependent interfacial function. The classical MM functional has a multiple-well potential and the derived classical MM equation allows multiple local minima. The MM equation has a good property compared to its counterpart vector-valued Allen–Cahn (AC) system because the MM equation consists of a single equation. In the MM equation, one phase-field can represent multiple states. However, with a constant interfacial parameter, the interfacial transition layer is getting wider as the jump across two adjacent phases is higher. To overcome this unwanted phenomenon, we propose a phase-dependent interfacial function which has smaller value if gradient of phase-field is larger. We present several computational experiments to demonstrate the superior performance of the proposed modified MM equation over the conventional MM equation.

Nonequilibrium Transport and Phase Transitions in Driven Diffusion of Interacting Particles

Abstract Driven diffusive systems constitute paradigmatic models of nonequilibrium physics. Among them, a driven lattice gas known as the asymmetric simple exclusion process (ASEP) is the most prominent example for which many intriguing exact results have been obtained. After summarising key findings, including the mapping of the ASEP to quantum spin chains, we discuss the recently introduced Brownian ASEP (BASEP) as a related class of driven diffusive system with continuous space dynamics. In the BASEP, driven Brownian motion of hardcore-interacting particles through one-dimensional periodic potentials is considered. We study whether current–density relations of the BASEP can be considered as generic for arbitrary periodic potentials and whether repulsive particle interactions other than hardcore lead to similar results. Our findings suggest that shapes of current–density relations are generic for single-well periodic potentials and can always be attributed to the interplay of a barrier reduction, blocking, and exchange symmetry effect. This implies that in general up to five different phases of nonequilibrium steady states are possible for such potentials. The phases can occur in systems coupled to particle reservoirs, where the bulk density is the order parameter. For multiple-well periodic potentials, more complex current–density relations are possible, and more phases can appear. Taking a repulsive Yukawa potential as an example, we show that the effects of barrier reduction and blocking on the current are also present. The exchange symmetry effect requires hardcore interactions, and we demonstrate that it can still be identified when hardcore interactions are combined with weak Yukawa interactions. The robustness of the collective dynamics in the BASEP with respect to variations of model details can be a key feature for a successful observation of the predicted current–density relations in actual physical systems.

Nonlinear Dynamics and Performance Enhancement of Multi-stable Wideband Energy Harvesting: Theoretical Analysis

A new multi-stable wideband harvester is proposed to increase the power density of energy harvesting under wide-bandwidth and low-frequency operations. This harvester can possess with an arbitrary multi-stable state by combining the nonlinearities of stopper-engaged cantilevers and magnetic forces. The harvesting efficiency can be enhanced by interwell oscillations and the global amplitude of the harvester is restricted by the cantilever-stopper engagement. An electromechanical coupling model of the proposed harvester is established to describe its dynamic characteristics. To explore the influence of potential well configurations on a multi-stable energy harvesting, the proposed harvester with four typical potential well configurations is considered while its equivalent nonlinear restoring force can be experimentally identified. The dynamic characteristics of these four configurations at different levels of a harmonic excitation will be explored by both numerical and experimental methods. It is expected that a multiple-well potential with non-uniform depths and shallower outer wells is much more suitable for the proposed system when the external excitation exceeds a critical intensity.

Modeling of Semiconductor Electrostatic Qubits Realized Through Coupled Quantum Dots

Considering the enormous advances in nanometer-scale CMOS technology that now allows one to reliably fabricate billions of switching devices on a single silicon die, electrostatically controlled quantum dots (implemented as quantum wells) appear to be promising candidates for a massive implementation of quantum bits (qubits) and quantum logic circuits in order to facilitate high-volume production of quantum computers. In this paper, the case of finite two-well and multiple-well potentials arising from semiconductor charged-coupled structures are treated in a rigorous way by Schrodinger formalism. The modeling methodologies presented to allow one to describe the dynamics of quantum states in non-ideal geometries, account for some mechanisms of qubit decoherence and model electrostatic interaction between electrons that lead to entanglement. The presented methodology can be scaled up to circuits of greater complexity.

Energy splitting in a finite periodic multiple-well potential

The low-lying states for a one-dimensional potential consisting of N identical wells are considered, assuming that the wells are parabolic around the minima. The N localized approximate eigenfunctions, each of which matches an eigenfunction of the simple harmonic oscillator in one of the wells, are constructed, relying on the WKB approximation. Diagonalizing the Hamiltonian in the subspace spanned by the approximate eigenfunctions, a formula for the energy eigenvalues is obtained. The present work will be useful for introducing Bloch wave functions in a periodic potential to undergraduate and graduate students. It is shown that the formula for the eigenvalues can reproduce a known rigorous expression on the Mathieu equation.

On the motion law of fronts for scalar reaction-diffusion equations with equal depth multiple-well potentials

Slow motion for scalar Allen-Cahn type equation is a well-known phenomenon, precise motion law for the dynamics of fronts having been established first using the socalled geometric approach inspired from central manifold theory (see the results of Carr and Pego in 1989). In this paper, the authors present an alternate approach to recover the motion law, and extend it to the case of multiple wells. This method is based on the localized energy identity, and is therefore, at least conceptually, simpler to implement. It also allows to handle collisions and rough initial data.

Nonlinear Schrödinger equations with a multiple-well potential and a Stark-type perturbation

A Bose–Einstein condensate (BEC) confined in a one-dimensional lattice under the effect of an external homogeneous field is described by the Gross–Pitaevskii equation. Here we prove that such an equation can be reduced, in the semiclassical limit and in the case of a lattice with a finite number of wells, to a finite-dimensional discrete nonlinear Schrödinger equation. Then, by means of numerical experiments we show that the BEC’s center of mass exhibits an oscillating behavior with modulated amplitude; in particular, we show that the oscillating period actually depends on the shape of the initial wavefunction of the condensate as well as on the strength of the nonlinear term. This fact opens a question concerning the validity of a method proposed for the determination of the gravitational constant by means of the measurement of the oscillating period.

Toward accurate prediction of potential energy surfaces and the spectral density of hydrogen bonded systems

Despite the considerable progress made in quantum theory and computational methods, detailed descriptions of the potential energy surfaces of hydrogen-bonded systems have not yet been achieved. In addition, the hydrogen bond (H-bond) itself is still so poorly understood at the fundamental level that it remains unclear exactly what geometry constitutes a “real” H-bond. Therefore, in order to investigate features essential for hydrogen bonded complexes, a simple, efficient, and general method for calculating matrix elements of vibrational operators capable of describing the stretching modes and the H-bond bridges of hydrogen-bonded systems is proposed. The derived matrix elements are simple and computationally easy to evaluate, which makes the method suitable for vibrational studies of multiple-well potentials. The method is illustrated by obtaining potential energy surfaces for a number of two-dimensional systems with repulsive potentials chosen to be in Gaussian form for the stretching mode and of the Morse-type for the H-bond bridge dynamics. The forms of potential energy surfaces of weak and strong hydrogen bonds are analyzed by varying the asymmetry of the Gaussian potential. Moreover, the choice and applicability of the selected potential for the stretching mode and comparison with other potentials used in the area of hydrogen bond research are discussed.The approach for the determination of spectral density has been constructed in the framework of the linear response theory for which spectral density is obtained by Fourier transform of the autocorrelation function of the dipole moment operator of the fast mode. The approach involves anharmonic coupling between the high frequency stretching vibration (double well potential) and low-frequency donor-acceptor stretching mode (Morse potential) as well as the electrical anharmonicity of the dipole moment operator of the fast mode. A direct relaxation mechanism is incorporated through a time decaying exponential according to Rösch and Ratner theory.

Characterization of Surface Defect Sites on Bulk and Nanophase Anatase and Rutile TiO2 by Low-Temperature Specific Heat

The specific heats of highly hydrated 7 nm nanoparticles and lightly hydrated 200 nm particles of rutile and anatase show a small, broad maximum below 5 K. The excess heat capacity can be modeled with a two-level system having very small molar weighting factors (∼10–3 for nanoparticles, ∼10–5 for bulk) and energy separations of approximately (4.88 ± 0.06) K (7 nm particles) and (2.5 ± 0.3) K (200 nm particles). The weighting factors correspond to the number of very high-energy surface sites identified by adsorption calorimetry and to the number of isolated 3-coordinate Ti defects observed on models built to the nanoparticle morphologies. We attribute the features to water adsorbed at defect sites on the particle surfaces. The very low energies associated with the Schottky-type heat capacity maxima are consistent with tunnel splittings of the ground state of a multiple-well potential. A double-well potential model for the hydrogen bonding of an OH group associated with the defects is consistent with the maxima observed for both the 7 and 200 nm particles.

Nonlinear Schrödinger equations with multiple-well potential

We consider the stationary solutions for a class of Schrödinger equations with a N-well potential and a nonlinear perturbation. By means of semiclassical techniques we prove that the dominant term of the ground state solutions is described by a N-dimensional Hamiltonian system, where the coupling term among the coordinates is a tridiagonal Toeplitz matrix. In particular, in the limit of large focusing nonlinearity we prove that the ground state stationary solutions consist of N wavefunctions localized on a single well. Furthermore, we consider in detail the case of N=4 wells, where we show the occurrence of spontaneous symmetry-breaking bifurcation effect.

Interaction-induced excited-band condensate in a double-well optical lattice

We show theoretically that interaction effects in a double-well optical lattice can induce condensates in an excited band. For a symmetric double-well lattice, bosons condense into the bottom of the excited band at the edge of the Brillouin zone if the chemical potential is above a critical value. For an asymmetric lattice, a condensate with zero momentum is automatically induced in the excited band by the condensate in the lowest band. This is due to a combined effect of interaction and lattice potential, which reduces the band gap and breaks the inversion symmetry. Our work can be generalized to a superlattice composed of multiple-well potentials at each lattice site, where condensates can be induced in even higher bands.

Slow motion for gradient systems with equal depth multiple-well potentials

For scalar reaction–diffusion in one space dimension, it has been known for a long time that fronts move with an exponentially small speed for potentials with several distinct minimizers. The purpose of this paper is to provide a similar result in the case of systems. Our method relies on a careful study of the evolution of localized energy. This approach also has the advantage of relaxing the preparedness assumptions on the initial datum.

Transient-chaos induced directed transport in a spatially-open Hamiltonian system

We study the autonomous Hamiltonian dynamics of non-interacting particles trapped initially in one well of a symmetric multiple-well washboard potential. The particles interact locally with an anharmonic oscillator acting as an energy deposit. For a range of interaction strengths, the particles gain sufficient energy during a chaotic transient to escape from the well and afterwards settle onto regular (rotational) dynamics. Strikingly, microcanonical ensembles of initial conditions that are unbiased with respect to the washboard coordinate nevertheless give rise to net directed motion. We demonstrate that for unbiased spatially localized initial conditions, violation of parity prevents the existence of pairs of counter-propagating trajectories within the ensemble, despite the time-reversibility symmetry of the equations of motion, allowing for a nonzero current. Recent studies have shown that particle current may be induced in other systems with preserved spatial symmetry by an external periodic but asymmetric driving force, however, averaging over the phase of the latter yields zero current. The system we propose is novel in that averaging over the phase of the oscillator still yields nonzero current. Furthermore, no mixed phase space is required, and chaos is needed only in an initial stage of the dynamics to guide trajectories from the interior of separatrices onto sustained integrable rotational motion.

Strain of interatomic bonds and multiple-well potential in crystals with perovskite structure

A method for determining the strain characteristics of interatomic bonds in crystals of ABX3 compounds with perovskite structure is developed. The bond strain energy ΔU s is estimated. This energy is responsible for the formation of ordered lattice distortions related to the rotation of octahedra and/or the cooperative displacement of cations. It is shown that ΔU s correlates with the characteristics of ordered distortions and the temperature of the phase transition of compounds to the cubic structure. It is ascertained that the interatomicinteraction potential in crystals with strained bonds is a local multiple-well potential, the form of which depends on the nature of interacting atoms. It is shown that the occurrence of the ferroelectric state in the noted crystals may be due to the presence of bond strains and the anisotropy of the covalent component of the cation-anion interaction.

Role of substrate geometry in sliding friction

Via a Frenkel–Kontorova (FK) model approach, we study, in the context of solid frictionphenomena, the role of the substrate potential geometry on the underdamped dynamics ofa chain of interacting atoms. The choice of an on-site potential defined by thesum of two sinusoidal functions with different periodicity allows us to simulatemicroscopic sliding over simply periodic, multiple-well periodic and quasiperiodicsubstrates. We analyse the mobility–force characteristics and the related staticfriction, commenting on the nature of the particle dynamics in the vicinity of thepinning–depinning transition point and considering the role played by the coveragevariable on the depinning mechanism. For the quasiperiodic cases, we also examinethe hysteretic behaviour of the chain mobility as a function of the adiabaticallyvarying force at zero temperature. For multiple-well periodic potentials, we observethe possible formation of commensurate dynamical structures during sliding.

Commensurate sliding on a multiple-well substrate potential

In the context of microscopic solid friction phenomena, we study the underdamped dynamics of a Frenkel–Kontorova (FK) chain subject to an external dc force and a substrate potential defined by the sum of two sinusoidal functions with different periodicities. Particularly, focusing on mutually commensurate choices for the three inherent length scales of the system allows us to consider the formation of commensurate structures during sliding over the multiple-well on-site potential. These structures characterize the dynamical states observed during the chain motion at different strengths of the imposed driving. We comment on the nature of the particle dynamics in the vicinity of the pinning–depinning transition point and investigate the role played by the coverage variable on the depinning mechanism. The dependence of the minimal force required to initiate the chain motion ( static friction) on the ratio of the model interaction strengths is analyzed and compared to the well-known case of the Frenkel–Kontorova model.

Some further considerations on deriving matrix elements of non-periodic potentials in the basis of the non-degenerate harmonic oscillator wavefunctions

Using the properties of the generating function of the Hermite polynomials, we have calculated the matrix elements for the Gaussian-type potential V G (x)=A exp{−B(x−C) 2} and for the Morse-type potential V M( x)= D e[1−exp(− ax)] 2 in the basis of the non-degenerate harmonic oscillator wavefunctions. The final forms of these matrix elements are very simple to use and hence suitable for the numerical resolution of the Schrödinger equation for multiple-well potentials or anharmonic oscillators.

One-dimensional multiple-well oscillators: A time-dependent quantum mechanical approach

Time-dependent Schrodinger equation (TDSE) is solved numerically to calculate the ground- and first three excited-state energies, expectation values x 2 j j 1 2 6, and probability densities of quantum mechanical multiple-well oscillators. An imaginary-time evolution technique, coupled with the minimization of energy expectation value to reach a global minimum, subject to orthogonality constraint (for excited states) has been employed. Pseudodegeneracy in symmetric, deep multiple-well potentials, probability densities and the effect of an asymmetry parameter on pseudodegeneracy are discussed.

The harmonic oscillator tensor approach to the eigenspectra of multiple-well molecular potentials. I. Non-periodic potentials

Using an irreducible tensor technique, we have derived matrix elements for the non-periodical vibrational operators qk, qkeλη and qkeλq2 in the basis of the non-degenerate harmonic oscillator wave functions. The matrix elements are simple and computationally easy to evaluate which makes the method suitable for variational studies of multiple-well potentials.

Stationary layered solutions in ${\Bbb R}^2$ for an Allen-Cahn system with multiple well potential

We study entire solutions on \({\Bbb R}^2\) of the elliptic system \(-\Delta U + \nabla W(u)=0\) where \(W:{\Bbb R}^2\to{\Bbb R}^2\) is a multiple-well potential. We seek solutions \(U(x_1,x_2)\) which are “heteroclinic,” in two senses: for each fixed \(x_2\in{\Bbb R}\) they connect (at \(x_1=\pm\infty\)) a pair of constant global minima of \(W\), and they connect a pair of distinct one dimensional stationary wave solutions when \(x_2\to\pm \infty\). These solutions describe the local structure of solutions to a reaction-diffusion system near a smooth phase boundary curve. The existence of these heteroclinic solutions demonstrates an unexpected difference between the scalar and vector valued Allen–Cahn equations, namely that in the vectorial case the transition profiles may vary tangentially along the interface. We also consider entire stationary solutions with a “saddle” geometry, which describe the structure of solutions near a crossing point of smooth interfaces.