General Interaction Potential Research Articles

The collective dynamics of a stochastic Port-Hamiltonian self-driven agent model in one dimension

This paper studies the collective motion of self-driven agents in a one-dimensional space with periodic boundaries, using a stochastic Port-Hamiltonian system (PHS) with symmetric nearest-neighbor interactions and additive Brownian noise as an external input. In the case of a quadratic potential the PHS is an Ornstein-Uhlenbeck process for which we explicitly determine the distribution for any time t ≥ 0 and in the limit t → ∞. In particular, we characterize the collective motion by showing that the agents’ positions tend to build exactly one cluster. This is confirmed in simulations that show rapid and coordinated motion among agents, driven by noise, despite the absence of a preferred direction of motion in the model. Remarkably, the theoretical properties observed in the Ornstein-Uhlenbeck process also emerge in simulations of the nonlinear model incorporating a general interaction potential.

The influence of magnetic and Aharanov–Bohm fields on energy spectra of diatomic molecules in the framework of the Dirac equation with the generalized interaction potential

AbstractIn this study, we present the relativistic and non‐relativistic solutions of the Dirac equation with the spin symmetry for the generalized Cornell potential (GCP) using the wave function ansatz method in the existence of external magnetic and Aharanov–Bohm (AB) flux fields. The relativistic energy eigenvalues and the corresponding eigenfunctions are found for varied vibrational and magnetic quantum numbers. By adapting the GCP parameters, we derive the relativistic and non‐relativistic energy eigenvalues with and without external fields for a set of potential models including the Killingbeck, harmonic oscillator, pseudoharmonic, anharmonic, Cornell, Coulomb, Kratzer, and modified Kratzer potentials. Additionally, the non‐relativistic energy spectra of the Kratzer and modified Kratzer potentials are reported with and without external magnetic and AB flux fields for several diatomic molecules (DMs). We discovered that in the existence of external magnetic and AB flux fields, the non‐relativistic energy spectrum increases and degeneracy disappears. Furthermore, the AB flux field has a stronger impact on the energy spectrum than the magnetic field. To substantiate our findings, we calculate the energy levels of the Kratzer and modified Kratzer potentials for diverse DMs and find that they are perfectly consistent with previous studies.

Existence of exponentially and superexponentially spatially localized breather solutions for nonlinear klein–gordon lattices in ℤd, d ≥ 1

AbstractWe prove the existence of exponentially and superexponentially localized breather solutions for discrete nonlinear Klein–Gordon systems. Our approach considers $d$-dimensional infinite lattice models with general on-site potentials and interaction potentials being bounded by an arbitrary power law, as well as, systems with purely anharmonic forces, cases which are much less studied particularly in a higher-dimensional set-up. The existence problem is formulated in terms of a fixed-point equation considered in weighted sequence spaces, which is solved by means of Schauder's Fixed-Point Theorem. The proofs provide energy bounds for the solutions depending on the lattice parameters and its dimension under physically relevant non-resonance conditions.

Ground state of asymmetric tops with DMRG: Water in one dimension.

We propose an approach to compute the ground state properties of collections of interacting asymmetric top molecules based on the density matrix renormalization group method. Linear chains of rigid water molecules of varying sizes and density are used to illustrate the method. A primitive computational basis of asymmetric top eigenstates with nuclear spin symmetry is used, and the many-body wave function is represented as a matrix product state. We introduce a singular value decomposition approach in order to represent general interaction potentials as matrix product operators. The method can be used to describe linear chains containing up to 50 water molecules. Properties such as the ground state energy, the von-Neumann entanglement entropy, and orientational correlation functions are computed. The effect of basis set truncation on the convergence of ground state properties is assessed. It is shown that specific intermolecular distance regions can be grouped by their von-Neumann entanglement entropy, which in turn can be associated with electric dipole-dipole alignment and hydrogen bond formation. Additionally, by assuming conservation of local spin states, we present our approach to be capable of calculating chains with different arrangements of the para and ortho spin isomers of water and demonstrate that for the water dimer.

On the derivation of a Nonlinear Generalized Langevin Equation

We recast the Zwanzig’s derivation of a nonlinear generalized Langevin equation (GLE) for a heavy particle interacting with a heat bath in a more general framework. We show that it is necessary to readjust the Zwanzig’s definitions of the kernel matrix and noise vector in the GLE in order to recover the correct definition of fluctuation-dissipation theorem and to be able performing consistently the continuum limit. As shown by Zwanzig, the nonlinear feature of the resulting GLE is due to the nonlinear dependence of the equilibrium map by the heavy particle variables. Such an equilibrium map represents the global equilibrium configuration of the heat bath particles for a fixed (instantaneous) configuration of the system. Following the same derivation of the GLE, we show that a deeper investigation of the equilibrium map, considered in the Zwanzig’s Hamiltonian, is necessary. Moreover, we discuss how to get an equilibrium map given a general interaction potential. Finally, we provide a renormalization procedure which allows to divide the dependence of the equilibrium map by coupling coefficient from the dependence by the system variables yielding a more rigorous mathematical structure of the nonlinear GLE.

Dziobek Equilibrium Configurations on a Sphere

We investigate the n-body problem on a sphere with a general interaction potential that depends on the mutual distances. We focus on the equilibrium configurations, especially on the Dziobek equilibrium configurations, which is an analogy of Dziobek central configurations of the classical n-body problem. We obtain a criterion and then reduce it to two sets of equations. Then we apply these equations to the curved n-body problem in \({\mathbb {S}}^3\). In the end, we find the derivative of the Cayley-Menger determinant.

Bound states of a pair of particles on the half-line with a general interaction potential

In this paper we study an interacting two-particle system on the half-line. We focus on spectral properties of the Hamiltonian for a large class of two-particle potentials. We characterize the essential spectrum and prove, as a main result, the existence of eigenvalues below the bottom of it. We also prove that the discrete spectrum contains only finitely many eigenvalues.

A finite-volume method for fluctuating dynamical density functional theory

We introduce a finite-volume numerical scheme for solving stochastic gradient flow equations. Such equations are of crucial importance within the framework of fluctuating hydrodynamics and dynamic density functional theory. Our proposed scheme deals with general free-energy functionals, including, for instance, external fields or interaction potentials. This allows us to simulate a range of physical phenomena where thermal fluctuations play a crucial role, such as nucleation and other energy-barrier crossing transitions. A positivity-preserving algorithm for the density is derived based on a hybrid space discretization of the deterministic and the stochastic terms and different implicit and explicit time integrators. We show through numerous applications that not only our scheme is able to accurately reproduce the statistical properties (structure factor and correlations) of physical systems, but also allows us to simulate energy barrier crossing dynamics, which cannot be captured by mean-field approaches.

Fast evaluation of interaction integrals for confined systems with machine learning

The calculation of interaction integrals is a bottleneck for the treatment of many-body quantum systems due to its high numerical cost. We conduct configuration interaction calculations of the few-electron states confined in III-V semiconductor 2D structures using a shallow neural network to calculate the two-electron integrals, that can be used for general isotropic interaction potentials. This approach allows for a speed up of the evaluation of the energy levels and a controllable accuracy.

A general purpose acetonitrile interaction potential to describe its liquid, solid and gas phases

A flexible and polarizable force field to describe acetonitrile in its three states of matter has been developed on the basis of an ab initio potential energy surface at the MP2 level. The acetonitrile molecule is represented by a sophisticated twelve-site model: the six nuclei plus six mobile charges. Three thousands structures have been employed for the fitting including monomers, dimers and trimers. Gas phase behavior has been tested by analyzing aggregates from dimers to 27-mer which give accurate reproducibility of the experimental and quantum-mechanical results. Classical MD simulations have been performed for the liquid and its crystalline α- and β-forms. An analysis of liquid in a wide range of structural, dynamic, energetics and spectroscopic properties has been performed, as well as neutron diffraction data. Comparison with experimental information is satisfactory. Due to the first principles nature and the polarizable character of this force field properties of the three states of matter are properly reproduced.

Effect of the magnetic field on the energy spectra of a quantum dot system

Analytical solutions to the radial Schrodinger equation are obtained for a two-dimensional two-electron quantum dot system within the framework of a general interaction potential using the Taylor expansion method. The calculated results are compared with other theoretical works to check the efficacy of the present method. Effects of the Coulomb interaction, the anharmonic potential terms and the magnetic field on the energy spectra of quantum dot systems are also highlighted.

Universal scaling of the thermalization time in one-dimensional lattices.

We show that, in the thermodynamic limit, a one-dimensional (1D) nonlinear lattice can always be thermalized for arbitrarily small nonlinearity, thus proving the equipartition theorem for a class of systems. Particularly, we find that in the lattices with nearest-neighbor interaction potential V(x)=x^{2}/2+λx^{n}/n with n≥4, the thermalization time, T_{eq}, follows a universal scaling law; i.e., T_{eq}∝λ^{-2}ε^{-(n-2)}, where ε is the energy per particle. Numerical simulations confirm that it is accurate for an even n, while a certain degree of deviation occurs for an odd n, which is attributed to the extra vibration modes excited by the asymmetric interaction potential. This finding suggests that although the symmetry of interactions will not affect the system reaching equipartition eventually, it affects the process toward equipartition. Based on the scaling law found here, a unified formula for the thermalization time of a 1D general nonlinear lattice is obtained.

Transfer Free Energies of Test Proteins Into Crowded Protein Solutions Have Simple Dependence on Crowder Concentration.

The effects of macromolecular crowding on the thermodynamic properties of test proteins are determined by the latter's transfer free energies from a dilute solution to a crowded solution. The transfer free energies in turn are determined by effective protein-crowder interactions. When these interactions are modeled at the all-atom level, the transfer free energies may defy simple predictions. Here we investigated the dependence of the transfer free energy (Δμ) on crowder concentration. We represented both the test protein and the crowder proteins atomistically, and used a general interaction potential consisting of hard-core repulsion, non-polar attraction, and solvent-screened electrostatic terms. The chemical potential was rigorously calculated by FMAP (Qin and Zhou, 2014), which entails expressing the protein-crowder interaction terms as correlation functions and evaluating them via fast Fourier transform (FFT). To high accuracy, the transfer free energy can be decomposed into an excluded-volume component (Δμe−v), arising from the hard-core repulsion, and a soft-attraction component (Δμs−a), arising from non-polar and electrostatic interactions. The decomposition provides physical insight into crowding effects, in particular why such effects are very modest on protein folding stability. Further decomposition of Δμs−a into non-polar and electrostatic components does not work, because these two types of interactions are highly correlated in contributing to Δμs−a. We found that Δμe−v fits well to the generalized fundamental measure theory (Qin and Zhou, 2010), which accounts for atomic details of the test protein but approximates the crowder proteins as spherical particles. Most interestingly, Δμs−a has a nearly linear dependence on crowder concentration. The latter result can be understood within a perturbed virial expansion of Δμ (in powers of crowder concentration), with Δμe−v as reference. Whereas the second virial coefficient deviates strongly from that of the reference system, higher virial coefficients are close to their reference counterparts, thus leaving the linear term to make the dominant contribution to Δμs−a.

Local conservation laws in ultracold Fermi systems with time-dependent interaction potential.

In the context of an ultracold Fermi gas, we derive conservation laws for mass, energy and momentum based on a generalized nonlocal Boltzmann equation with gradient corrections in the collision integral. The corrections are expressed in terms of effective collision duration, particle displacement and changes of total momentum and energy. Their origin is in the in-medium T matrix. Using variations of the optical theorem, we show that in the collision integral the particle-hole symmetry can be recast into a form of collision symmetry amenable to semiclassical simulation. Pauli-blocked collisions are distinguished from Bose-stimulated nondissipative ones; the latter are not present in the absence of gradient corrections. Consolidating with the microscopic theory, we extract local conservation laws for a general time-dependent interaction potential, and demonstrate how both types of collisions affect densities and flows of conserving quantities. Comparison is made with the approach of Nozières and Schmitt-Rink in the limit of thermal equilibrium. Under approximations used for normal-state ultracold Fermi gases interacting via Feshbach resonances we demonstrate the effect of the collision delay on the shear viscosity.

Existence of ground states for aggregation-diffusion equations

We analyze free energy functionals for macroscopic models of multi-agent systems interacting via pairwise attractive forces and localized repulsion. The repulsion at the level of the continuous description is modeled by pressure-related terms in the functional making it energetically favorable to spread, while the attraction is modeled through nonlocal forces. We give conditions on general entropies and interaction potentials for which neither ground states nor local minimizers exist. We show that these results are sharp for homogeneous functionals with entropies leading to degenerate diffusions while they are not sharp for fast diffusions. The particular relevant case of linear diffusion is totally clarified giving a sharp condition on the interaction potential under which the corresponding free energy functional has ground states or not.

Exchange interactions and itinerant ferromagnetism in ultracold Fermi gases

Itinerant ferromagnetism, i.e. spontaneous polarization of non-localized particles, is expected to occur for strong repulsive interactions in a spin-1/2 Fermi system. However, this state has proven notoriously hard to find experimentally, both in ultracold gases and in solids. This raises questions about the stability of the itinerant ferromagnetic state itself. Here we develop a new approach to describe both the direct and exchange interactions for a general interaction potential in the path-integral formalism and we apply this method to itinerant ferromagnetism in three-dimensional ultracold Fermi gases. We show that the exchange interactions are lost in the Hubbard-Stratonovich transformation and we propose to explicitly include the exchange effects in a new modified interaction potential. In the saddle-point approximation, the effect of interactions can be taken into account using only three parameters. If the interactions become too strong, all saddle points become unstable to density fluctuations. This greatly restricts the area in the phase diagram where uniform itinerant ferromagnetism is expected to occur.

Bound state solutions to the Schrödinger equation for some diatomic molecules

The bound state solutions to the radial Schrodinger equation are obtained in three-dimensional space using the series expansion method within the framework of a general interaction potential. The energy eigenvalues of the pseudoharmonic and Kratzer potentials are given as special cases. The obtained analytical results are applied to several diatomic molecules, i.e. $$\mathrm {N}_2, \mathrm {CO}, \mathrm {NO}$$ and $$\mathrm {CH}$$ . In order to check the accuracy of the present method, a comparison is made with similar results obtained in the literature by using other techniques.

Mass Spectra of Heavy and Light Mesons Using Asymptotic Iteration Method

The non-relativistic radial Schrödinger equation is analytically solved using asymptotic iteration method within the framework of a general interaction potential whose special cases are the Cornell and Cornell plus harmonic potentials. The energy eigenvalues expression is derived in three dimensional space, which is further used to calculate the mass spectra of c̄c, b̄b, b̄c, cs̄, bs̄ and b̄q mesons. The obtained results of this work are in good agreement with experimental and other relativistic results and also improved in comparison with other non-relativistic recent studies.

Existence and convergence of solutions of the boundary value problem in atomistic and continuum nonlinear elasticity theory

We show existence of solutions for the equations of static atomistic nonlinear elasticity theory on a bounded domain with prescribed boundary values. We also show their convergence to the solutions of continuum nonlinear elasticity theory, with energy density given by the Cauchy–Born rule, as the interatomic distances tend to zero. These results hold for small data close to a stable lattice for general finite range interaction potentials. We also discuss the notion of stability in detail.

Consistency of multi-time Dirac equations with general interaction potentials

In 1932, Dirac proposed a formulation in terms of multi-time wave functions as candidate for relativistic many-particle quantum mechanics. A well-known consistency condition that is necessary for existence of solutions strongly restricts the possible interaction types between the particles. It was conjectured by Petrat and Tumulka that interactions described by multiplication operators are generally excluded by this condition, and they gave a proof of this claim for potentials without spin-coupling. Under suitable assumptions on the differentiability of possible solutions, we show that there are potentials which are admissible, give an explicit example, however, show that none of them fulfills the physically desirable Poincaré invariance. We conclude that in this sense, Dirac’s multi-time formalism does not allow to model interaction by multiplication operators, and briefly point out several promising approaches to interacting models one can instead pursue.