Metropolis–Hastings thermal state sampling for numerical simulations of Bose–Einstein condensates

We demonstrate the application of the Metropolis–Hastings algorithm to sampling of classical thermal states of one-dimensional Bose–Einstein quasicondensates in the classical fields approximation, both in untrapped and harmonically trapped case. The presented algorithm can be easily generalized to higher dimensions and arbitrary trap geometry. For truncated Wigner simulations the quantum noise can be added with conventional methods (half a quantum of energy in every mode). The advantage of the presented method over the usual analytical and stochastic ones lies in its ability to sample not only from canonical and grand canonical distributions, but also from the generalized Gibbs ensemble, which can help to shed new light on thermodynamics of integrable systems.     

Unified way for computing dynamics of Bose–Einstein condensates and degenerate Fermi gases

In this work we present a very simple and efficient numerical scheme which can be applied to study the dynamics of bosonic systems like, for instance, spinor Bose–Einstein condensates (BEC) with non-local interactions but equally well works for Fermi gases. The method we use is a modification of well known Split Operator Method (SOM). We carefully examine this algorithm in the case of F=1 spinor BEC without and with dipolar interactions and for strongly interacting two-component Fermi gas. Our extension of the SOM method has many advantages: it is fast, stable, and keeps constant all the physical constraints (constants of motion) at high level.     

Projection gradient method for energy functional minimization with a constraint and its application to computing the ground state of spin–orbit-coupled Bose–Einstein condensates

In this study, we propose a projection gradient method for energy functional minimization with a constraint, which we use to compute the ground state of spin–orbit-coupled Bose–Einstein condensates at extremely low temperatures. The method has the advantage that it maintains the constraint when evolving a gradient flow to find the energy functional minimization under a constraint. The original gradient projection method for energy functional minimization under a constraint only considers an energy functional with real functions as variables. Thus, we extend it to consider complex functions as independent variables. We apply the newly proposed method to study the ground state solution of spin–orbit-coupled pseudo-spin 1/2 Bose–Einstein condensates. Detailed numerical results demonstrate the effectiveness of our method. Using this method, we found various types of ground state structures of spin–orbit coupled Bose–Einstein condensates.     

Numerical exploration of vortex matter in Bose–Einstein condensates

We describe a finite element numerical approach to the full Hartree-Fock-Bogoliubov treatment of a vortex lattice in a rapidly rotating Bose–Einstein condensate. We study the system in the regime of high thermal or significant quantum fluctuations where we are presented with a very large nonlinear unsymmetric eigenvalue problem which is indefinite and which possesses low-lying excitations clustered arbitrarily close to zero, a problem that requires state-of-the-art numerical techniques.     

Excited Bose–Einstein condensates: Quadrupole oscillations and dark solitons

We study the dynamics of atomic Bose–Einstein condensates (BECs), when the quadrupole mode is excited. Within the Thomas–Fermi approximation, we derive an exact first-order system of differential equations that describes the parameters of the BEC wave function. Using perturbation theory arguments, we derive explicit analytical expressions for the phase, density and width of the condensate. Furthermore, it is found that the observed oscillatory dynamics of the BEC density can even reach a quasi-resonance state when the trap strength varies according to a time-periodic driving term. Finally, the dynamics of a dark soliton on top of a breathing BEC are also briefly discussed.     

OCTBEC—A Matlab toolbox for optimal quantum control of Bose–Einstein condensates

OCTBEC is a Matlab toolbox designed for optimal quantum control, within the framework of optimal control theory (OCT), of Bose–Einstein condensates (BEC). The systems we have in mind are ultracold atoms in confined geometries, where the dynamics takes place in one or two spatial dimensions, and the confinement potential can be controlled by some external parameters. Typical experimental realizations are atom chips, where the currents running through the wires produce magnetic fields that allow to trap and manipulate nearby atoms. The toolbox provides a variety of Matlab classes for simulations based on the Gross–Pitaevskii equation, the multi-configurational Hartree method for bosons, and on generic few-mode models, as well as optimization problems. These classes can be easily combined, which has the advantage that one can adapt the simulation programs flexibly for various applications.     

Multi-parameter continuation and collocation methods for rotating multi-component Bose–Einstein condensates

We describe multi-parameter continuation methods combined with spectral collocation methods for computing numerical solutions of rotating two-component Bose–Einstein condensates (BECs), which are governed by the Gross–Pitaevskii equations (GPEs). Various types of orthogonal polynomials are used as the basis functions for the trial function space. A novel multi-parameter/multiscale continuation algorithm is proposed for computing the solutions of the governing GPEs, where the chemical potential of each component and angular velocity are treated as the continuation parameters simultaneously. The proposed algorithm can effectively compute numerical solutions with abundant physical phenomena. Numerical results on rotating two-component BECs are reported.     

A splitting compact finite difference method for computing the dynamics of dipolar Bose–Einstein condensate

We numerically study the nonlocal Gross–Pitaevskii equation (NGPE) which describes the dynamics of Bose–Einstein condensates (BEC) with dipole–dipole interaction at extremely low temperature. In preparation for the numerics, first we reformulate the dimensionless NGPE into a Schrödinger–Poisson system. Then, we discretize the three-dimensional Schrödinger–Poisson system in space by a sixth-order compact finite difference method and in time by a splitting technique. By means of three-dimensional discrete fast Sine transform, we develop a fast solver for the resulting discretized system. Finally, we present numerical examples in three dimensions to demonstrate the power of the numerical methods and to discuss some physics of dipolar BEC. The merits of the proposed method for the NGPE are that it is fast and unconditionally stable. Moreover, the method is of spectral-like accuracy in space, and conserves the particle number and the energy of the system in the discretized level.     

Efficient spectral collocation and continuation methods for symmetry-breaking solutions of rotating Bose–Einstein condensates

We describe an efficient spectral collocation method (SCM) for symmetry-breaking solutions of rotating Bose–Einstein condensates (BECs) which is governed by the Gross–Pitaevskii equation (GPE). The Lagrange interpolants using the Legendre–Gauss–Lobatto points are used as the basis functions for the trial function space. Some formulas for the derivatives of the basis functions are given so that the GPE can be efficiently computed. The SCMs are incorporated in the context of a predictor–corrector continuation algorithm for tracing primary and secondary solution branches of the GPE. Symmetry-breaking solutions are numerically presented for both rotating BECs, BECs in optical lattices, and two-component BECs in optical lattices. Our numerical results show that the numerical algorithm we propose in this paper outperforms the classical orthogonal Legendre polynomials.     

A numerical study of the Navier–Stokes-αβ model

We present a numerical study of the NS-αβ model, which is a recently proposed multiscale variation of the NS-α model that attempts to recapture scales lost through over-regularization by separately modeling dissipation-range scales. We develop a similarity theory for the new model which shows that it is better equipped than the NS-α model to capture smaller-scale behavior. Next, we propose and study an unconditionally stable, optimally accurate, and efficient finite-element implementation for the NS-αβ model; rigorous proofs for stability and convergence are provided. Finally, we present results from two numerical experiments that demonstrate the advantages of the NS-αβ model over the NS-α model.     

Efficient continuation methods for spin-1 Bose–Einstein condensates in a magnetic field

We study linear stability analysis for spin-1 Bose–Einstein condensates (BEC). We show that all bounded solutions of this physical system are neutrally stable. In particular, all steady-state solutions of the physical system, and the associated discrete steady-state solutions are neutrally stable. Next, we consider the physical system without the affect of magnetic field. By exploiting the physical properties of both ferromagnetic and antiferromagnetic cases, we develop efficient multi-level pseudo-arclength continuation algorithms combined with a spectral collocation method for these two cases, respectively. When the magnetic field is imposed on the physical system, an additional multi-level continuation algorithm is described for the ferromagnetic case. Extensive numerical results for spin-1 BEC in a magnetic field, and in optical lattices are reported.     

Determination of the protonation state for the catalytic dyad in β-secretase when bound to hydroxyethylamine transition state analogue inhibitors: A molecular dynamics simulation study

BACE1 is an aspartyl protease of pharmacological interest for its direct participation in Alzheimer’s disease (AD) through β-amyloid peptide production. Two aspartic acid residues are present in the BACE1 catalytic region which can adopt multiple protonation states depending on the chemical nature of its inhibitors, i.e., monoprotonated, diprotonated and di-deprotonated states. In the present study a series of protein-ligand molecular dynamics (MD) simulations was carried out to identify the most feasible protonation state adopted by the catalytic dyad in the presence of hydroxyethylamine transition state analogue inhibitors. The MD trajectories revealed that the di-deprotonated state is most prefered in the presence of hydroxyethilamine (HEA) family inhibitors. This appears as a result after evaluating, for all 9 protonation state configurations during the simulation time, the deviations of a set of distances and dihedral angles measured on the ligand, protein and protein-ligand complex with reference to an X-ray experimental BACE1/HEA crystallographic structure. These results will help to clarify the phenomena related to the HEAs inhibitory pathway, and improve HEAs databases’ virtual screening and ligand design processes targeting β-secretase protein.     

A heterogeneous parallel Red–Black SOR technique and the numerical study on SIMPLE

The Journal of Supercomputing - A basic heterogeneous parallel Red–Black successive over-relaxation (SOR) implement, the mono-color floating-point scheme, was developed on graphics processing...     

Thermodynamic assessment of the Co–Cr–Fe system and atomic mobility study of its fcc phase

In the present work, the Co–Cr–Fe system was thermodynamically assessed by using the CALPHAD approach coupled with first-principles calculations and experimental data from the literature. Subsequently, the fcc Co–Cr–Fe ternary diffusion couples annealed at 1273 and 1473K were scanned by using electron probe microanalysis (EPMA) to obtain the composition profiles, from which the interdiffusion coefficients were extracted by the Whittle-Green method. Based on these interdiffusion coefficients and present thermodynamic parameters, the atomic mobilities of Co, Cr, and Fe in the fcc Co–Cr–Fe alloys were assessed by means of DICTRA software. The calculated interdiffusion coefficients, composition profiles and diffusion paths of the fcc Co–Cr–Fe alloys were consistent with the experimental data, which verifies the accuracy of the assessed atomic mobilities.     

System identification and control of the ground operationmode of a hybrid aerial–ground robot

This paper presents an in-depth analytical and empirical assessment of the performance of DoubleBee, a novel hybrid aerial–ground robot. Particularly, the dynamic model of the robot with ground contact is analyzed, and the unknown parameters inthe model are identified. We apply an unscented Kalman filter-based approach and a least square-based approach to estimatethe parameters with given measurements and inputs at every time step. Real data are collected and used to estimate theparameters; test data verify that the values obtained are able to model the rotation of the robot accurately. A gain-scheduledfeedback controller is proposed, which leverages the identified model to generate accurate control inputs to drive the systemto the desired states. The system is proven to track a constant-velocity reference signal with bounded error. Simulations andreal-world experiments using the proposed controller show improved performance than the PID-based controller in trackingstep commands and maintaining attitude under robot movement.     

A small dispersion limit to the Camassa–Holm equation: A numerical study

In this paper we take up the question of a small dispersion limit for the Camassa–Holm equation. The particular limit we study involves a modification of the Camassa–Holm equation, seen in the recent theoretical developments by Himonas and Misio?ek, as well as the first author, where well-posedness is proved in weak Sobolev spaces. This work led naturally to the question of how solutions actually behave in these modified equations as time evolves. While the dispersive limit studied here is inspired by the work of Lax and Levermore on the zero dispersion limit of the Korteweg–de Vries equation to the inviscid Burgers’ equation, here there is no known Inverse Scattering theory. Consequently, we resort to a sophisticated numerical simulation to study two representative (one smooth and one peakon), but by no means exhaustive, initial conditions in the modified Camassa–Holm equation. In both cases there appears to be a strong limit of the modified Camassa–Holm equation to the Camassa–Holm equation as the dispersive parameter tends to zero, provided that solutions have not evolved for too long (time sufficiently small). For the smooth initial condition considered, this time must be chosen before the solution approaches steepening; beyond this time the computed solution becomes increasingly complicated as the dispersive term tends towards zero, and there does not appear to be a limit. By contrast, for the peakon initial condition this limit does appear to exist for all times considered. While in many cases the computations required few discretization points, there were some very challenging cases (particularly for the small dispersion computations) where an enormous number of unknowns were required to properly resolve the solution.     

Numerical study on gas–liquid nano-flows with pseudo-particle modeling and soft-particle molecular dynamics simulation

We couple pseudo-particle modeling (PPM, Ge and Li in Chem Eng Sci 58(8):1565–1585, 2003), a variant of hard-particle molecular dynamics, with standard soft-particle molecular dynamics (MD) to study an idealized gas–liquid flow in nano-channels. The coupling helps to keep sharp contrast between gas and liquid behaviors and the simulations conducted provide a reference frame for exploring more complex and realistic gas–liquid nano-flows. The qualitative nature and general flow patterns of the flow under such extreme conditions are found to be consistent with its macro-scale counterpart.

Comparison study on the different dynamics between the Allen–Cahn and the Cahn–Hilliard equations

We perform a comparison study on the different dynamics between the Allen–Cahn (AC) and the Cahn–Hilliard (CH) equations. The AC equation describes the evolution of a non-conserved order field during anti-phase domain coarsening. The CH equation describes the process of phase separation of a conserved order field. The AC and the CH equations are second-order and fourth-order nonlinear parabolic partial differential equations, respectively. Linear stability analysis shows that growing and decaying modes for both the equations are the same. While the growth rates are monotonically decreasing with respect to the modes for the AC equation, the growth rates for the CH equation are increasing and then decreasing with respect to the modes. We perform various numerical tests using the Fourier spectral method to highlight the different evolutionary dynamics between the AC and the CH equations.     

Generalized perturbed complex Toda chain for Manakov system and exact solutions of Bose–Einstein mixtures

We analyze, both analytically and numerically, the dynamical behavior of the N-soliton train in the Manakov system with perturbations due to an external potential. The perturbed complex Toda chain model has been employed to describe adiabatic interactions within the N Manakov-soliton train. Simulations performed demonstrate that external potentials can play a stabilizing role to provide bound state regime of the train propagation. We also present new stationary and travelling wave solutions to equations describing Bose–Fermi mixtures in an elliptic external potential. Precise conditions for the existence of every class of solutions are derived. There are indications that such waves and localized objects may be observed in experiments with cold quantum degenerate gases.