Multiwell Potential Research Articles

Local Structural Features Elucidate Crystallization of Complex Structures.

Complex crystal structures are composed of multiple local environments, and how this type of order emerges spontaneously during crystal growth has yet to be fully understood. We study crystal growth across various structures and along different crystallization pathways, using self-assembly simulations of identical particles that interact via multiwell isotropic pair potentials. We apply an unsupervised machine learning method to features from bond-orientational order metrics to identify different local motifs present during a given structure's crystallization process. In this manner, we distinguish different crystallographic sites in highly complex structures. Tailoring this order parameter to structures of varying complexity and coordination number, we study the emergence of local order along a multistep crystal growth pathway─from a low-density fluid to a high-density, supercooled amorphous liquid droplet and to a bulk crystal. We find a consistent under-coordination of the liquid relative to the average coordination number in the bulk crystal. We use our order parameter to analyze the geometrically frustrated growth of a Frank-Kasper phase and discover how structural defects compete with the formation of crystallographic sites that are more high-coordinated than the liquid environments. The method presented here for classifying order on a particle-by-particle level has broad applicability to future studies of structural self-assembly and crystal growth, and they can aid in the design of building blocks and for targeting pathways of formation of soft-matter structures.

Breaking down walls: Continuous potential models for internal motions in NMR spin relaxation

Simple physical models for restricted diffusion in a potential, which provide important insights for NMR spin relaxation, usually are based on free diffusion within rigid boundaries or diffusion in relatively simple continuous potential energy surfaces. The diffusion-in-a-cone model is an example of the former and diffusion in an N-fold cosine potential is an example of the latter. The present work models restricted diffusion for arbitrary potential energy functions on the surface of a cone or a sphere, by expanding the potentials in Fourier or spherical harmonic series, respectively. The results exhibit simple relationships between generalized order parameters and effective correlation times, critical for analysis of experimental spin relaxation data, and illustrate the transition from diffusive-like to jump-like behavior in multi-well potentials.

Heteroclinic traveling waves of two-dimensional parabolic Allen–Cahn systems

In this paper we show the existence of traveling waves w\colon [0,+\infty) \times \mathbb{R}^{2} \to \mathbb{R}^{k} ( k \geq 2 ) for the parabolic Allen–Cahn system \partial_{t} w - \Delta w = -\nabla_{u} V(w) in [0,+\infty) \times \mathbb{R}^{2} , satisfying some heteroclinic conditions at infinity. The potential V is a nonnegative and smooth multi-well potential, which means that its null set is finite and contains at least two elements. The traveling wave w propagates along the horizontal axis according to a speed c^{\star}>0 and a profile \mathfrak{U} . The profile \mathfrak{U} joins as x_{1} \to \pm \infty (in a suitable sense) two locally minimizing one-dimensional heteroclinics which have different energies, and the speed c^{\star} satisfies certain uniqueness properties. The proof is variational and, in particular, it requires the assumption of an upper bound, depending on V , on the difference between the energies of the one-dimensional heteroclinics.

Approaching certain central-field Schrödinger problems through the harmonic oscillator green's function

We propose a method to analytically solve the Schrödinger equation with central potentials. By means of Sturm–Liouville expansion, the Green's function (GF) of the Schrödinger differential operator can be represented in the GF of the isotropic harmonic oscillator. Applying this straightforward approach, we solve the Schrödinger problem for the Coulomb potential, for the Morse potential, and for the respective inverted potentials. Using the obtained results, we formulate a method to analytically solve Schrodinger equations for multi-well potentials, demonstrating that classically forbidden regions can be treated as inverted harmonic oscillators, whereas harmonic oscillators or Morse potentials can approximate the wells.

Quantitative convergence of the vectorial Allen–Cahn equation towards multiphase mean curvature flow

Phase-field models such as the Allen–Cahn equation may give rise to the formation and evolution of geometric shapes, a phenomenon that may be analyzed rigorously in suitable scaling regimes. In its sharp-interface limit, the vectorial Allen–Cahn equation with a potential with N\geq 3 distinct minima has been conjectured to describe the evolution of branched interfaces by multiphase mean curvature flow. In the present work, we give a rigorous proof for this statement in two and three ambient dimensions and for a suitable class of potentials: as long as a strong solution to multiphase mean curvature flow exists, solutions to the vectorial Allen–Cahn equation with well-prepared initial data converge towards multiphase mean curvature flow in the limit of vanishing interface width parameter \varepsilon\searrow 0 . We even establish the rate of convergence O(\varepsilon^{1/2}) . Our approach is based on the gradient-flow structure of the Allen–Cahn equation and its limiting motion: building on the recent concept of “gradient-flow calibrations” for multiphase mean curvature flow, we introduce a notion of relative entropy for the vectorial Allen–Cahn equation with multi-well potential. This enables us to overcome the limitations of other approaches, e.g. avoiding the need for a stability analysis of the Allen–Cahn operator or additional convergence hypotheses for the energy at positive times.

Multi-scroll chaotic attractors with multi-wing via oscillatory potential wells

The oscillatory potential well can cause the mass points in it to move chaotically, which can be considered as a physical mechanism of chaos generation. Based on this physical mechanism, the oscillatory multi-well potential with multi-concave is used to generate controllable multi-scroll chaotic attractors with multi-wing. These attractors have two kinds of topological units: scroll and wing. The topological structure of the attractor depends on the shape of the potential well, that is, the wells and the concaves correspond to the scrolls and wings of the attractor. By constructing potential wells with different shapes, we can obtain attractors with different topological structure. The number of scrolls and wings of the chaotic attractor can be controlled by the oscillation amplitude of the potential well and damping coefficient, that is, stronger oscillation or smaller damping allows the mass points to enter the higher potential wells or concaves, resulting in attractors with more scrolls or wings. As the number of scrolls or wings increases, the attractors become more complex, as expressed by Poincaré maps and quantified by the Kaplan-Yorke dimensions. Kaplan-Yorke dimensions of attractors can be adjusted continuously from 2 to 3 by the two control parameters. Finally, the results are confirmed by Multisim simulation circuit and actual analog circuit.

Transient response of energy harvesting systems with multi-well potential under Poisson white noise excitations

This paper studies the transient response of vibration-based energy harvesting (VEH) systems characterized by multi-well potential functions exposed to Poisson white noise excitations. With the recently developed radial basis function neural networks (RBFNN) approach, the generalized Fokker–Planck–Kolmogorov (GFPK) equation in three-dimensional state space for the transient probability density function (PDF) is solved. The trial solution is composed of a series of radial basis Gaussian functions, each multiplied by a time-varying weight coefficient. The method of Lagrange multiplier is introduced for the constraint of weight coefficients to normalized the trial solution of the PDF solution. As examples to validate the proposed solution approach, the mono-stable and tri-stable energy harvester systems are studied. Monte Carlo simulations (MCS) have been applied to check the analytical solution. Some theoretical trends of the harvested mean square voltage (MSV) as a function of system parameters are examined. The non-Gaussian excitation and time-varying nature of the system can significantly affect the MSV output. It is noted that this work provides researchers and engineers a tool to optimize energy harvesting systems operated in random environment.

Targeted Discovery of Low-Coordinated Crystal Structures via Tunable Particle Interactions

Particles interacting via isotropic, multiwell pair potentials have been shown to self-assemble into a range of crystal structures, yet how the characteristics of the underlying interaction potential give rise to the resultant structure remains largely unknown. We have thus developed a functional form for the interaction potential in which all features can be tuned independently. We perform continuous parameter space searches by systematically changing pairs of parameters, controlling the various features of the interaction potential. By enforcing a repulsive first well (controlling particle interactions of the first neighbor shell), we stimulate the formation of low-coordinated assemblies. We report the self-assembly of 20 previously unknown crystal structure types, 14 of which have low coordination numbers. Despite limiting the search to a small region of the vast parameter space of possible particle interactions, a wealth of complexity and symmetry is apparent within these crystal structures, which include clathrates with empty cages and low-symmetry structures. Our findings suggest that an unknown number of previously undiscovered crystal structure configurations are possible through self-assembly, which can serve as interesting design targets for soft condensed matter synthesis.

Spontaneous symmetry breaking, stability and adiabatic changes of 2D quantum droplets in amended Gross–Pitaevskii equation with multi-well potential

In this paper, we report the spontaneous symmetry breaking (SSB) of two-dimensional (2D) quantum droplets (QDs) for the amended Gross–Pitaevskii equation with Lee–Huang–Yang corrections and a Gaussian quadruple-well potential. The complete pitchfork symmetry breaking bifurcation diagrams are presented for the possible stationary states with four modes, which involve twelve different real solution branches and one complex solution branch, where the continuous asymmetric QDs branches bifurcate from the fundamental symmetric or antisymmetric one as the strength of nonlinearity exceeds a threshold value. Meanwhile the stability of the 2D QDs is explored by means of linear stability analysis and direct numerical simulations. Especially, for the asymmetric QDs, the SSB effect and the trend for getting the droplet asymmetry are discussed by changing the chemical potential. Finally, the adiabatic changes of stable stationary QDs are explored by modulating the position, depth, and width of the potential well as functions of time, and the strong robustness of QDs is found. Our results provide some theoretical basis for the SSB phenomena and related physical experiments on 2D asymmetric QDs.

A class of three dimensional Cahn-Hilliard equation with nonlinear diffusion

In this paper, we study the well-posedness and asymptotic behavior for a class of Cahn-Hilliard equation with nonlinear diffusion in R3. In order to overcome the difficulties caused by the derivatives of multi-well potential and the nonlinear terms, we “borrow” a linear principle part from the derivatives of multi-well potential, rewrite the equation as an equivalent equation with linear principle part. Then, on the basis of the higher order norm estimates of solutions and the mollifier technique, we obtain the local well-posedness of strong solutions. Moreover, by using pure energy method, standard continuity argument together with negative Sobolev norm estimates, one proves the global well-posedness and time decay estimates provided that the H4-norm of initial data is sufficiently small. In the end, based on parabolic interpolation inequality, bootstrap argument and some weighted estimates, we also establish the space-time decay estimates of strong solutions.

Orbital and equilibrium dynamics of a multiwell potential

We consider a Hamiltonian system associated with the two-well umbilical catastrophe potential D5, while the corresponding potential contains only one free parameter. We determine how this free parameter affects the equilibrium dynamics of the system by computing their coordinates on the configuration plane, along with their linear stability and type. Additionally, we discuss the influence of the same free parameter on the orbital dynamics of the system by performing a systematic and thorough orbit classification that allows us to reveal the bounded or escaping motion of the test particle.

Manifold dynamics and periodic orbits in a multiwell potential

In this article, we explore the dynamics as well as the geometry of the invariant manifolds that determine the escapes from a multiwell potential. We also present the network of both symmetric and asymmetric solutions of the system, while at the same time we extract valuable information about the periodic solutions, such as their locations, multiplicity, and linear stability.

Non-minimizing connecting orbits for multi-well systems

Given a nonnegative, smooth potential \(V: {{\mathbb {R}}}^k \rightarrow {{\mathbb {R}}}\) (\(k \ge 2\)) with multiple zeros, we say that a curve \({\mathfrak {q}}: {{\mathbb {R}}}\rightarrow {{\mathbb {R}}}^k\) is a connecting orbit if it solves the autonomous system of ordinary differential equations $$\begin{aligned} {\mathfrak {q}}''= \nabla _{\mathbf{u}} V({\mathfrak {q}}) , \quad \text{ in }\;\, {{\mathbb {R}}}\end{aligned}$$and tends to a zero of V at \(\pm \infty \). Broadly, our goal is to study the existence of connecting orbits for the problem above using variational methods. Despite the rich previous literature concerning the existence of connecting orbits for other types of second order systems, to our knowledge only connecting orbits which minimize the associated energy functional in a suitable function space were proven to exist for autonomous multi-well potentials. The contribution of this paper is to provide, for a class of such potentials, some existence results regarding non-minimizing connecting orbits. Our results are closely related to the ones in the same spirit obtained by J. Bisgard in his PhD thesis (University of Wisconsin-Madison, 2005), where non-autonomous periodic multi-well potentials (ultimately excluding autonomous potentials) are considered. Our approach is based on several refined versions of the classical Mountain Pass Lemma.

A Variational Approach to Morphogenesis : Recovering Spatial Phenotypic Features from Epigenetic Landscapes.

We model the process of cell fate determination of the flower Arabidopsis-thaliana employing a system of reaction-diffusion equations governed by a potential field. This potential field mimics the flower's epigenetic landscape as defined by Waddington. It is derived from the underlying genetic regulatory network (GRN), which is based on detailed experimental data obtained during cell fate determination in the early stages of development of the flower. The system of equations has a variational structure, and we use minimax techniques (in particular the Mountain Pass Lemma) to show that the minimal energy solution of our functional is, in fact, the one that traverses the epigenetic landscape (the potential field) in the spatial order that corresponds to the correct architecture of the flower, that is, following the observed geometrical features of the meristem. This approach can generally be applied to systems with similar structures to establish a genotype to phenotype correspondence. From a broader perspective, this problem is related to phase transition models with a multiwell vector potential, and the results and methods presented here can potentially be applied in this case.

Slow and fast relaxation times of quantum lattice model with local multi-well potentials: phenomenological dynamics for Sn2P2S6 ferroelectric crystals

As a continuation of the previously published work [Velychko O. V., Stasyuk I. V., Phase Transitions, 2019, 92, 420], a phenomenological framework for the relaxation dynamics of quantum lattice model with multi-well potentials is given in the case of deformed Sn2P2S6 ferroelectric lattice. The framework is based on the combination of statistical equilibrium theory and irreversible thermodynamics. In order to study these dynamics in a connected way we assume that the dipole ordering or polarization (η) and volume deformation (u) can be treated as fluxes and forces in the sense of Onsager theory. From the linear relations between the forces and fluxes, the rate equations are derived and characterized by two relaxation times (τS , τF) which describe the irreversible process near the equilibrium states. The behaviors of τS and τF in the vicinity of ferroelectric phase transitions are studied.

Orbital and Equilibrium Dynamics of a Multiwell Potential

Orbital and Equilibrium Dynamics of a Multiwell Potential

Hallmarks of tunneling dynamics with broken reflective symmetry

We study features of tunneling dynamics in an exactly-solvable model of N=4 supersymmetric quantum mechanics with a multi-well potential and with broken reflective symmetry. Quantum systems with a phenomenological potential of this type demonstrate the phenomenon of partial localization of under-barrier states, possibly resulting in the appearance of the so-called “resonant” tunneling, or the phenomenon of coherent tunneling destruction, referring to the complete localization. Taking the partial localization and the coherent tunneling destruction as basic examples, we indicate main advantages of using isospectral exactly-solvable Hamiltonians in studies quantum mechanical systems with two- and three-well potentials. They, in particular, are: having enough freedom of changing the potential shape in a wide range, that allows one to choose an exactly-solvable model close to characteristics of the phenomenological one; ability of changing the number of local minima and symmetry characteristics of the potential (symmetric or deformed) without changing the main part of the spectrum; engaging a smart basis of states, that dramatically decreases the dimensionality of matrices used in the diagonalization procedure of the corresponding spectral problem.

Probing the edge between integrability and quantum chaos in interacting few-atom systems

Interacting quantum systems in the chaotic domain are at the core of various ongoing studies of many-body physics, ranging from the scrambling of quantum information to the onset of thermalization. We propose a minimum model for chaos that can be experimentally realized with cold atoms trapped in one-dimensional multi-well potentials. We explore the emergence of chaos as the number of particles is increased, starting with as few as two, and as the number of wells is increased, ranging from a double well to a multi-well Kronig-Penney-like system. In this way, we illuminate the narrow boundary between integrability and chaos in a highly tunable few-body system. We show that the competition between the particle interactions and the periodic structure of the confining potential reveals subtle indications of quantum chaos for 3 particles, while for 4 particles stronger signatures are seen. The analysis is performed for bosonic particles and could also be extended to distinguishable fermions.

Moving beyond the constraints of chemistry via crystal structure discovery with isotropic multiwell pair potentials

The rigid constraints of chemistry-dictated by quantum mechanics and the discrete nature of the atom-limit the set of observable atomic crystal structures. What structures are possible in the absence of these constraints? Here, we systematically crystallize one-component systems of particles interacting with isotropic multiwell pair potentials. We investigate two tunable families of pairwise interaction potentials. Our simulations self-assemble a multitude of crystal structures ranging from basic lattices to complex networks. Sixteen of the structures have natural analogs spanning all coordination numbers found in inorganic chemistry. Fifteen more are hitherto unknown and occupy the space between covalent and metallic coordination environments. The discovered crystal structures constitute targets for self-assembly and expand our understanding of what a crystal structure can look like.

A hyper-block self-consistent approach to nonlinear Schrodinger equations: Breeding, metamorphosis, and killing of Hofstadter Butterflies

Nonlinear Schrödinger equations play essential roles in different physics and engineering fields. In this paper, a hyper-block finite-difference self-consistent method (HFDSCF) is employed to solve this stationary nonlinear eigenvalue equation and demonstrated its accuracy. By comparing the results with the Sinc self-consistent (SSCF) method and the exact available results, we show that the HFDSCF gives quantum states with high accuracy and can even solve the strongly nonlinear Schrodinger equations. Then, by applying our method to the Hofstadter butterfly problem, we describe the breeding, metamorphosis, and killing of these butterflies by using nonlinear interactions and two constant length multi-well and sinusoidal potentials.