Double-well Potential Research Articles

Universal calcium fluctuations in Hydra morphogenesis

Understanding the collective physical processes that drive robust morphological transitions in animal development necessitates the characterization of the relevant fields involved in morphogenesis. Calcium (Ca2+) is recognized as one such field. In this study, we demonstrate that the spatial fluctuations of Ca2+ during Hydra regeneration exhibit universal characteristics. To investigate this phenomenon, we employ two distinct controls, an external electric field and heptanol, a gap junction-blocking drug. Both lead to the modulation of the Ca2+ activity and a reversible halting of the regeneration process. The application of an electric field enhances Ca2+ activity in the Hydra’s tissue and increases its spatial correlations, while the administration of heptanol inhibits its activity and diminishes the spatial correlations. Remarkably, the statistical characteristics of Ca2+ spatial fluctuations, including the coefficient of variation and skewness, manifest universal shape distributions across tissue samples and conditions. We introduce a field-theoretic model, describing fluctuations in a tilted double-well potential, which successfully captures these universal properties. Moreover, our analysis reveals that the Ca2+ activity is spatially localized, and the Hydra’s tissue operates near the onset of bistability, where the local Ca2+ activity fluctuates between low and high excited states in distinct regions. These findings highlight the prominent role of the Ca2+ field in Hydra morphogenesis and provide insights into the underlying mechanisms governing robust morphological transitions.

Electrophysical Properties of Methylammonium Alum Single Crystals: The Role of the Hydrogen Bond Network in the Formation of Temperature Anomalies

Pyroelectric, dielectric properties, and thermodepolarization currents of methylammonium alum crystallohydrate single crystals were studied in temperature range 120–200 K. Temperature dependences of these quantities demonstrated peaks at ∼143, ∼160, and 176.4 K (Curie point). The behavior, which is typical to many crystals with a network of hydrogen bonds with an O–H...O distance of ∼2.7 ± 0.1 ˚A, is considered from the point of view of a quantum chemical model of a hydrogen bond network containing water clusters in addition to individual H2O molecules. The movement of bridging protons in molecular clusters in electric fields is coherent and correlated, and all bridging protons of the system are involved in it. These displacements are associated with oscillations of the oxygen frame of the compression-tension type. The discreteness of the temperature peaks of the electrophysical properties is associated with the discreteness of the energy levels of the double-well potential, caused by the splitting of the excited levels.

Quantum Information Entropy for Another Class of New Proposed Hyperbolic Potentials.

In this work, we investigate the Shannon entropy of four recently proposed hyperbolic potentials through studying position and momentum entropies. Our analysis reveals that the wave functions of the single-well potentials U0,3 exhibit greater localization compared to the double-well potentials U1,2. This difference in localization arises from the depths of the single- and double-well potentials. Specifically, we observe that the position entropy density shows higher localization for the single-well potentials, while their momentum probability density becomes more delocalized. Conversely, the double-well potentials demonstrate the opposite behavior, with position entropy density being less localized and momentum probability density showing increased localization. Notably, our study also involves examining the Bialynicki-Birula and Mycielski (BBM) inequality, where we find that the Shannon entropies still satisfy this inequality for varying depths u¯. An intriguing observation is that the sum of position and momentum entropies increases with the variable u¯ for potentials U1,2,3, while for U0, the sum decreases with u¯. Additionally, the sum of the cases U0 and U3 almost remains constant within the relative value 0.01 as u¯ increases. Our study provides valuable insights into the Shannon entropy behavior for these hyperbolic potentials, shedding light on their localization characteristics and their relation to the potential depths. Finally, we extend our analysis to the Fisher entropy F¯x and find that it increases with the depth u¯ of the potential wells but F¯p decreases with the depth.

Primordial black holes and inflation from double-well potentials

We investigate the formation of large peaks in the inflationary curvature power spectrum from double-well potentials. In such scenarios, the initial CMB spectrum is created at large field values. Subsequently, the inflaton will cross one of the minima and will decelerate rapidly as it reaches the local maximum at the origin, either falling back or crossing it. During this final phase, a significant peak in the curvature power spectrum can be generated. Our analysis reveals that this class of models produces more pronounced peaks than most quasi-inflection point scenarios with less tuning for the model parameters. Finally, we construct an explicit theoretically motivated inflationary scenario that is consistent with the latest CMB observations and capable of generating sufficiently large curvature perturbations for primordial black holes.

Tunneling Mechanism for Changing the Motion Direction of a Pulsating Ratchet. Temperature Effect

A pulsating ratchet with a spatially periodic double-well potential profile undergoing shift fluctuations for half a period is considered. The motion direction in such a ratchet is determined by the probability of overcoming which of the barriers surrounding the shallow potential well is greater. At relatively high temperatures, in accordance with the Arrhenius law, the probabilities of overcoming the barriers are determined by their heights, and at temperatures close to absolute zero, when the ratchet moves according to the tunnel mechanism, the barrier shapes are also important. Therefore, for narrow high and low wide barriers, the overcoming mechanism may turn out to be different and, moreover, dependent on temperature. As a result, a temperature-induced change in the direction of the ratchet motion is possible. A simple interpolation theory is presented to illustrate this effect. Simple criteria are formulated for the shape of the potential relief, using which one can experimentally observe motion reversal.

A solvable model for symmetry-breaking phase transitions

Analytically solvable models are benchmarks in studies of phase transitions and pattern-forming bifurcations. Such models are known for phase transitions of the second kind in uniform media, but not for localized states (solitons), as integrable equations which produce solitons do not admit intrinsic transitions in them. We introduce a solvable model for symmetry-breaking phase transitions of both the first and second kinds (alias sub- and supercritical bifurcations) for solitons pinned to a combined linear-nonlinear double-well potential, represented by a symmetric pair of delta-functions. Both self-focusing and defocusing signs of the nonlinearity are considered. In the former case, exact solutions are produced for symmetric and asymmetric solitons. The solutions explicitly demonstrate a switch between the symmetry-breaking transitions of the first and second kinds (i.e., sub- and supercritical bifurcations, respectively). In the self-defocusing model, the solution demonstrates the transition of the second kind which breaks antisymmetry of the first excited state.

Tunneling times of an electron in one-dimensional symmetric and asymmetric harmonic double-well potentials

Tunneling times of an electron in one-dimensional symmetric and asymmetric harmonic double-well potentials

Non-Markovian effects of conformational fluctuations on the global diffusivity in Langevin equation with fluctuating diffusivity.

Local diffusivity of a protein depends crucially on the conformation, and the conformational fluctuations are often non-Markovian. Here, we investigate the Langevin equation with non-Markovian fluctuating diffusivity, where the fluctuating diffusivity is modeled by a generalized Langevin equation under a double-well potential. We find that non-Markovian fluctuating diffusivity affects the global diffusivity, i.e., the diffusion coefficient obtained by the long-time trajectories when the memory kernel in the generalized Langevin equation is a power-law form. On the other hand, the diffusion coefficient does not change when the memory kernel is exponential. More precisely, the global diffusivity obtained by a trajectory whose length is longer than the longest relaxation time in the memory kernel is not affected by the non-Markovian fluctuating diffusivity. We show that these non-Markovian effects are the consequences of an everlasting effect of the initial condition on the stationary distribution in the generalized Langevin equation under a double-well potential due to long-term memory.

Logical and thermodynamical reversibility: Optimized experimental implementation of the not operation.

The not operation is a reversible transformation acting on a 1-bit logical state and should be achievable in a physically reversible manner at no energetic cost. We experimentally demonstrate a bit-flip protocol based on the momentum of an underdamped oscillator confined in a double-well potential. The protocol is designed to be reversible in the ideal dissipationless case, and the thermodynamic work required is inversely proportional to the quality factor of the system. Our implementation demonstrates an energy dissipation significantly lower than the minimal cost of information processing in logically irreversible operations. It is, moreover, performed at high speed: A fully equilibrated final state is reached in only half a period of the oscillator. The results are supported by an analytical model that takes into account the presence of irreversibility. This Research Letter concludes with a discussion of optimization strategies.

Application of Approximate Maximum-Entropy Moment Closure to the Wigner Equation

This paper investigates the potential of moment closures as a possible path to future quantum hydrodynamics models. Moment closures are known to produce accurate predictions of continuum and non-equilibrium classical gas flows while offering modeling and numerical advantages over other commonly used methods. The maximum-entropy hierarchy of moment closures holds the promise of robustly hyperbolic stable moment equations, however, the predicted distribution function in phase space is positive by design, which is in disagreement with the quasi-distribution function described by the Wigner equation. In this paper, predictions made by an interpolative one-dimensional five-moment system are compared to direct numerical solutions of the Wigner equation for a simple low-energy quantum state in unbounded double-well potentials. Numerical solutions for a particle in various potentials described by quartic polynomials are presented. The capacity of moment methods to provide accurate and efficient models for quantum systems is explored.

Effects of quantum fluctuations on macroscopic quantum tunneling and self-trapping of a BEC in a double-well trap

In this paper, we study the influence of quantum fluctuations (QFs) on the macroscopic quantum tunneling and self-trapping (ST) of a two-component Bose–Einstein condensate in a double-well trap. QFs are described by the Lee–Huang–Yang (LHY) term in the modified Gross–Pitaevskii (GP) equation. Employing the modified GP equation in a scalar approximation, we derive a dimer model using a two-mode approximation. The frequencies of Josephson oscillations and ST conditions under QFs are found analytically and proven by numerical simulations of the modified GP equation. The tunneling and localization phenomena are also investigated for the case of the LHY fluid loaded in a double-well potential.

Nonlinear dynamics of an artificial muscle with elastomer–electrode inertia: Modelling and analysis

Artificial muscles are dielectric elastomeric system that mimic the response of the natural muscle and are helpful in many bio-mimicking applications. The existing literature is based on the assumption that electrodes are compliant and are focused on materials with greater extensibility than biological matter. The present work deals with the modelling and nonlinear dynamic analysis of dielectric elastomers considering the elastomer–electrode inertia and damping effect for bio-mimicking. The distinguishing point of the proposed work is the appearance of terms related to the elastomer–electrode inertia effect in the governing equation. Further, the biological muscle’s nonlinear elastic behaviour is incorporated using the Fung material model. Equilibrium stretch variation with parameter exhibits cusp catastrophe. The effect of electrode inertia on the equilibria is examined. Single-well and double-well potential energy characteristics are obtained for constant voltage. The basin of attraction illustrates the sensitiveness of system dynamics on the initial state and damping. The system response, when subjected to time-varying voltage, is explored. Multi-frequency response exists due to harmonic, sub-harmonic, and super-harmonic resonance condition. The resonant frequency variation with geometry and inertia is presented. Time-varying voltage may lead to chaotic behaviour illustrated through the bifurcation diagram. The presence of a strange attractor and positive Lyapunov exponent in the chaotic region further justify the existence of chaos. Additionally, chaotic behaviour is obtained in the case of single-well and double-well potential. Also, the existence of similar attractors with same fractal dimension is shown. The inertia effect significantly changes the dynamic characteristic from chaotic to non-chaotic or vice-versa. The mechanical and electrical load may lead to chaotic or non-chaotic behaviour depending on the geometry of the elastomer. The chaotic region’s dependence on the damping value is also obtained. Our results show that electrode inertia significantly impacts the elastomer under quasi-static and dynamic conditions. The proposed work gives a foundation for precisely designing artificial muscle for practical applications like soft robotics, artificial arms, and different prosthetic implants.

Freestanding graphene heat engine analyzed using stochastic thermodynamics

We present an Ito-Langevin model for freestanding graphene connected to an electrical circuit. The graphene is treated as a Brownian particle in a double-well potential and is adjacent to a fixed electrode to form a variable capacitor. The capacitor is connected in series with a battery and a load resistor. The capacitor and resistor are given separate thermal reservoirs. We have solved the coupled Ito-Langevin equations for a broad range of temperature differences between the two reservoirs. Using ensemble averages, we report the rate of change in energy, heat, and work using stochastic thermodynamics. When the resistor is held at higher temperatures, the efficiency of the heat engine rises linearly with temperature. However, when the graphene is held at higher temperatures, the efficiency instantly rises and then plateaus. Also, twice as much entropy is produced when the resistor is hotter compared to when the graphene is hotter. Unexpectedly, the temperature of the capacitor is found to alter the dissipated power of the resistor.

Modeling of Double-Well Potentials for the Schrödinger Equation

Modeling of Double-Well Potentials for the Schrödinger Equation

Bayesian Estimation of Experimental Parameters in Stochastic Inertial Systems: Theory, Simulations, and Experiments with Objects Levitated in Vacuum

High-quality nanomechanical oscillators can sensitively probe force, mass, or displacement in experiments bridging the gap between the classical and quantum domain. Dynamics of these stochastic systems is inherently determined by the interplay between acting external forces, viscous dissipation, and random driving by the thermal environment. The relevance of inertia then dictates that both position and momentum must, in principle, be known to fully describe the system, which makes its quantitative experimental characterization rather challenging. We introduce a general method of Bayesian inference of the force field and environmental parameters in stochastic inertial systems that operates solely on the time series of recorded noisy positions of the system. The method is first validated on simulated trajectories of model stochastic harmonic and anharmonic oscillators with damping. Subsequently, the method is applied to experimental trajectories of particles levitating in tailored optical fields and used to characterize the dynamics of particle motion in a nonlinear Duffing potential, a static or time-dependent double-well potential, and a nonconservative force field. The presented inference procedure does not make any simplifying assumptions about the nature or symmetry of the acting force field and provides robust results with trajectories 2 orders of magnitude shorter than those typically required by alternative inference schemes. In addition to being a powerful tool for quantitative data analysis, it can also guide experimentalists in choosing appropriate sampling frequency (at least 20 measured points per single characteristic period) and length of the measured trajectories (at least 10 periods) to estimate the force field and environmental characteristics with a desired accuracy and precision.

Calculating Vibrational Excited State Absorptions with Excited State Constrained Minimized Energy Surfaces.

The modeling and interpretation of vibrational spectra are crucial for studying reaction dynamics using vibrational spectroscopy. Most prior theoretical developments focused on describing fundamental vibrational transitions while fewer developments focused on vibrational excited state absorptions. In this study, we present a new method that uses excited state constrained minimized energy surfaces (CMESs) to describe vibrational excited state absorptions. The excited state CMESs are obtained similarly to the previous ground state CMES development in our group but with additional wave function orthogonality constraints. Using a series of model systems, including the harmonic oscillator, Morse potential, double-well potential, quartic potential, and two-dimensional anharmonic potential, we demonstrate that this new procedure provides good estimations of the transition frequencies for vibrational excited state absorptions. These results are significantly better than those obtained from harmonic approximations using conventional potential energy surfaces, demonstrating the promise of excited state CMES-based methods for calculating vibrational excited state absorptions in real systems.

On the calculation of bound-state energies supported by hyperbolic double well potentials

We obtain eigenvalues and eigenfunctions of the Schrödinger equation with a hyperbolic double-well potential. We consider exact polynomial solutions for some particular values of the potential-strength parameter and also numerical energies for arbitrary values of this model parameter. We test the numerical method by means of a suitable exact asymptotic expression for the eigenvalues and also calculate critical values of the strength parameter that are related to the number of bound states supported by the potential.

Uphill inflation

Primordial black holes (PBH) may form from large cosmological perturbations, produced during inflation when the inflaton's velocity is sufficiently slowed down. This usually requires very flat regions in the inflationary potential. In this paper we investigate another possibility, namely that the inflaton climbs up its potential. When it turns back, its velocity crosses zero, which triggers a short phase of “uphill inflation” during which cosmological perturbations grow at a very fast rate. This naturally occurs in double-well potentials if the width of the well is close to the Planck scale. We include the effect of quantum diffusion in this scenario, which plays a crucial role, by means of the stochastic-δN formalism. We find that ultra-light black holes are produced with very high abundances, which do not depend on the energy scale at which uphill inflation occurs, and which suffer from substantially less fine tuning than in alternative PBH-production models. They are such that PBHs later drive a phase of PBH domination.

Quantifying protocol efficiency: A thermodynamic figure of merit for classical and quantum state-transfer protocols

Manipulating quantum systems undergoing non-Gaussian dynamics in a fast and accurate manner is becoming fundamental to many quantum applications. Here, we focus on classical and quantum protocols transferring a state across a double-well potential. The classical protocols are achieved by deforming the potential, while the quantum ones are assisted by a counterdiabatic driving. We show that quantum protocols perform more quickly and accurately. Finally, we design a figure of merit for the performance of the transfer protocols---namely, the protocol grading---that depends only on fundamental physical quantities, and which accounts for the quantum speed limit, the fidelity, and the thermodynamics of the process. We test the protocol grading with classical and quantum protocols, and show that quantum protocols have higher protocol grading than the classical ones.

Quantum behavior of the Duffing oscillator at the dissipative phase transition

The non-deterministic behavior of the Duffing oscillator is classically attributed to the coexistence of two steady states in a double-well potential. However, this interpretation fails in the quantum-mechanical perspective which predicts a single unique steady state. Here, we measure the non-equilibrium dynamics of a superconducting Duffing oscillator and experimentally reconcile the classical and quantum descriptions as indicated by the Liouvillian spectral theory. We demonstrate that the two classically regarded steady states are in fact quantum metastable states. They have a remarkably long lifetime but must eventually relax into the single unique steady state allowed by quantum mechanics. By engineering their lifetime, we observe a first-order dissipative phase transition and reveal the two distinct phases by quantum state tomography. Our results reveal a smooth quantum state evolution behind a sudden dissipative phase transition and form an essential step towards understanding the intriguing phenomena in driven-dissipative systems.