Finite Number Of Particles Research Articles

Integration of A* Search and Classic Optimal Control for Safe Planning of Continuum Deformation of a Multiquadcopter System

This article offers an algorithmic approach to plan continuum deformation of a multiquadcopter system (MQS) in an obstacle-laden environment. We treat the MQS as a finite number of particles of a deformable body coordinating under a homogeneous transformation. We define the MQS homogeneous deformation coordination as a decentralized leader–follower problem and integrate the principles of continuum mechanics, A* search method, and optimal control approach to safety and optimally plan MQS continuum deformation coordination. In particular, we apply the principles of continuum mechanics to obtain the safety constraints, use the A* search method to assign the intermediate configurations of the leaders by minimizing the travel distance of the MQS, and determine the leaders’ optimal trajectories by solving a constrained optimal control problem. The optimal planning of the continuum deformation coordination is acquired by the quadcopter team in a decentralized fashion through local communication.

Exact solution of an integrable non-equilibrium particle system

We consider the integrable family of symmetric boundary-driven interacting particle systems that arise from the non-compact XXX Heisenberg model in one dimension with open boundaries. In contrast to the well-known symmetric exclusion process, the number of particles at each site is unbounded. We show that a finite chain of N sites connected at its ends to two reservoirs can be solved exactly, i.e., the factorial moments of the non-equilibrium steady-state can be written in the closed form for each N. The solution relies on probabilistic arguments and techniques inspired by integrable systems. It is obtained in two steps: (i) the introduction of a dual absorbing process reducing the problem to a finite number of particles and (ii) the solution of the dual dynamics exploiting a symmetry obtained from the quantum inverse scattering method. Long-range correlations are computed in the finite-volume system. The exact solution allows us to prove by a direct computation that, in the thermodynamic limit, the system approaches local equilibrium. A by-product of the solution is the algebraic construction of a direct mapping between the non-equilibrium steady state and the equilibrium reversible measure.

Limiting Current Distribution for a Two Species Asymmetric Exclusion Process

We study current fluctuations of a two-species asymmetric exclusion process, known as the Arndt-Heinzel-Rittenberg model. For a step-Bernoulli initial condition with finite number of particles, we provide an explicit multiple integral expression for a certain joint current probability distribution. By performing an asymptotic analysis we prove that the joint current distribution is given by a product of a Gaussian and a GUE Tracy-Widom distribution in the long time limit, as predicted by non-linear fluctuating hydrodynamics.

On the Features of Ideal Bose-Gas Thermodynamic Prop-erties at a Finite Particle Number

The paper is devoted to the theory of an ideal Bose-gas with a finite number N of particles. The exact expressions for the partition functions and occupation numbers of the model in the grand canonical, canonical, and microcanonical ensembles are found. From the calculations, it is followed that, oppositely to the accepted opinion that the chemical potential μ of an ideal Bose-gas is only negative, it can take values in the range −∞ < μ < ∞. The asymptotic expressions (in the case N ≫ 1) for the partition functions and occupation numbers for all above-mentioned thermodynamic ensembles are also evaluated.

Unruh Effect and Information Entropy Approach

The Unruh effect can be considered a source of particle production. The idea has been widely employed in order to explain multiparticle production in hadronic and heavy-ion collisions at ultrarelativistic energies. The attractive feature of the application of the Unruh effect as a possible mechanism of the multiparticle production is the thermalized spectra of newly produced particles. In the present paper, the total entropy generated by the Unruh effect is calculated within the framework of information theory. In contrast to previous studies, here the calculations are conducted for the finite time of existence of the non-inertial reference frame. In this case, only a finite number of particles are produced. The dependence on the mass of the emitted particles is taken into account. Analytic expression for the entropy of radiated boson and fermion spectra is derived. We study also its asymptotics corresponding to low- and high-acceleration limiting cases. The obtained results can be further generalized to other intrinsic degrees of freedom of the emitted particles, such as spin and electric charge.

Asynchronous finite differences in most probable distribution with finite numbers of particles

For a discrete function fx on a discrete set, the finite difference can be either forward and backward. If fx is a sum of two such functions fx=f1x+f2x, the first order difference of Δfx can be grouped into four possible combinations, in which two are the usual synchronous ones Δff1x+Δff2x and Δbf1x+Δbf2x, and other two are asynchronous ones Δff1x+Δbf2x and Δbf1x+Δff2x, where Δf and Δb denote the forward and backward difference respectively. Thus, the first order variation equation δfx=0 for this function fx gives at most four different solutions which contain both true and false one. A formalism of the discrete calculus of variations is developed to single out the true one by means of comparison of the second order variations, in which the largest value in magnitude indicates the true solution, yielding the exact form of the distributions for Boltzmann, Bose and Fermi system without requiring the numbers of particle to be infinitely large. When there is only one particle in the system, all distributions reduce to be the Boltzmann one.

Motion modeling of cable-driven continuum robots using vector form intrinsic finite element method

ABSTRACT This paper presents the motion modeling of a cable-driven, single backbone continuum robot via the vector form intrinsic finite element (VFIFE) method. The VFIFE method is a solution framework based on vector mechanics. The algorithm describes a continuous body by using a finite number of particles instead of a mathematical function. We first approached the model by defining path elements and allocating particles (mass nodes) for the structure. We then established constitutive conditions as the generalized forces derived from both the external environment and internal deformed structural elements. We found that the dynamic equation of particles can be governed by Newton’s law of motion and solved by a general explicit time integration technique. Furthermore, we performed static modeling of the system by employing a dynamic relaxation algorithm with kinetic damping in the dynamic equations. Finally, from experiments, we validated the simulated static model of a single backbone continuum robot. It is shown that this method is capable of describing a continuum robot’s motion by solving for frame structures that undergo large deformation.

Path-Integral Monte Carlo Worm Algorithm for Bose Systems with Periodic Boundary Conditions

We provide a detailed description of the path-integral Monte Carlo worm algorithm used to exactly calculate the thermodynamics of Bose systems in the canonical ensemble. The algorithm is fully consistent with periodic boundary conditions, which are applied to simulate homogeneous phases of bulk systems, and it does not require any limitation in the length of the Monte Carlo moves realizing the sampling of the probability distribution function in the space of path configurations. The result is achieved by adopting a representation of the path coordinates where only the initial point of each path is inside the simulation box, the remaining ones being free to span the entire space. Detailed balance can thereby be ensured for any update of the path configurations without the ambiguity of the selection of the periodic image of the particles involved. We benchmark the algorithm using the non-interacting Bose gas model for which exact results for the partition function at finite number of particles can be derived. Convergence issues and the approach to the thermodynamic limit are also addressed for interacting systems of hard spheres in the regime of high density.

Quantum heat engines with Carnot efficiency at maximum power

Heat engines constitute the major building blocks of modern technologies. However, conventional heat engines with higher power yield lesser efficiency and vice versa and respect various power-efficiency trade-off relations. This is also assumed to be true for the engines operating in the quantum regime. Here we show that these relations are not fundamental. We introduce quantum heat engines that deliver maximum power with Carnot efficiency in the one-shot finite-size regime. These engines are composed of working systems with a finite number of quantum particles and are restricted to one-shot measurements. The engines operate in a one-step cycle by letting the working system simultaneously interact with hot and cold baths via semi-local thermal operations. By allowing quantum entanglement between its constituents and, thereby, a coherent transfer of heat from hot to cold baths, the engine implements the fastest possible reversible state transformation in each cycle, resulting in maximum power and Carnot efficiency. Finally, we propose a physically realizable engine using quantum optical systems.

Skyrme-based extrapolation for the static response of neutron matter

The study of inhomogeneous neutron matter can provide insights into the structure of neutron stars as well as their dynamics in neutron-star mergers. In this work we tackle pure neutron matter in the presence of a periodic external field by considering a finite (but potentially large) number of particles placed in periodic boundary conditions. We start with the simpler setting of a noninteracting gas and then switch to a Skyrme-Hartree-Fock approach, showing static-response results for five distinct Skyrme parametrizations. We explain both the technical details of our computational approach, as well as the significance of these results as a general finite-size extrapolation scheme that may be used by ab initio practitioners to approach the static-response problem of neutron matter in the thermodynamic limit.

Transition between vacuum and finite-density states in the infinite-dimensional Bose–Hubbard model with spatially inhomogeneous dissipation

Abstract We analyze the dynamics of the infinite-dimensional Bose–Hubbard model with spatially inhomogeneous dissipation in the hardcore boson limit by solving the Lindblad master equation with use of the Gutzwiller variational method. We consider dissipation processes that correspond to inelastic light scattering in the case of Bose gases in optical lattices. We assume that the dissipation is applied to half of the lattice sites in a spatially alternating manner. We focus on steady states at which the system arrives after long time evolution. We find that when the average particle density is varied, the steady state exhibits a transition between a state in which the sites without dissipation are vacuum and one containing a finite number of particles at those sites. We associate the transition with the notion that the sites with dissipation tend towards a local state at infinite temperature.

Spatial birth-and-death processes with a finite number of particles

The aim of this work is to establish essential properties of spatial birth-and-death processes with general birth and death rates on ${\mathbb{R}^{\mathrm{d}}}$. Spatial birth-and-death processes with time dependent rates are obtained as solutions to certain stochastic equations. The existence, uniqueness, uniqueness in law and the strong Markov property of unique solutions are proven when the integral of the birth rate over ${\mathbb{R}^{\mathrm{d}}}$ grows not faster than linearly with the number of particles of the system. Martingale properties of the constructed process provide a rigorous connection to the heuristic generator. The pathwise behavior of an aggregation model is also studied. The probability of extinction and the growth rate of the number of particles under condition of nonextinction are estimated.

On number of particles in coalescing-fragmentating Wasserstein dynamics

We consider the system of sticky-reflected Brownian particles on the real line proposed in [4]. The model is a modification of the Howitt-Warren flow but now the diffusion rate of particles is inversely proportional to the mass which they transfer. It is known that the system consists of a finite number of distinct particles for almost all times. In this paper, we show that the system also admits an infinite number of distinct particles on a dense subset of the time interval if and only if the function responsible for the splitting of particles takes an infinite number of values.

Thermodynamic characteristics of ideal quantum gases in harmonic potentials within exact and semiclassical approaches

We theoretically examine equilibrium properties of the harmonically trapped ideal Bose and Fermi gases in the quantum degeneracy regime. We analyze thermodynamic characteristics of gases with a finite number of atoms by means of the known semiclassical approach and perform comparison with exact numerical results. For a Fermi gas, we demonstrate deviations in the Fermi energy values originating from a discrete level structure and show that these are observable only for a small number of particles. For a Bose gas, we observe characteristic softening of phase transition features, which contrasts to the semiclassical predictions and related approximations. We provide a more accurate methodology of determining corrections to the critical temperature due to finite number of particles.

Major merging of galaxies in multicomponent numerical models: mass loss and exchange

The process of collision of two multicomponent galaxies is considered in detail based on numerical simulations of the dynamics of gravitating gas, stars and dark mass. To solve the equations of motion of the gas component, we use the Smoothed Particle Hydrodynamics method. Modeling of collisionless components is based on the N-body model. The computations of gravitational forces are carried out using both the approximate hierarchical TreeCode algorithm and the direct method of summing the gravitational contribution from all particles, which provides an accurate solution. This approach allows testing various models and evaluating the resulting errors associated with the calculation of gravitational forces and a finite number of particles in each of the components. Both methods for calculating gravity are software implemented as parallel codes for Nvidia Tesla GPUs. The estimates of the lost mass and the efficiency of matter exchange between galaxies are discussed depending on the model parameters.

Bose-Einstein statistics for a finite number of particles

This article presents a study of the grand canonical Bose-Einstein (BE) statistics for a finite number of particles in an arbitrary quantum system. The thermodynamical quantities that identify BE condensation -- namely, the fraction of particles in the ground state and the specific heat -- are calculated here exactly in terms of temperature and fugacity. These calculations are complemented by a numerical calculation of fugacity in terms of the number of particles, without taking the thermodynamic limit. The main advantage of this approach is that it does not rely on approximations made in the vicinity of the usually defined critical temperature, rather it makes calculations with arbitrary precision possible, irrespective of temperature. Graphs for the calculated thermodynamical quantities are presented in comparison to the results previously obtained in the thermodynamic limit. In particular, it is observed that for the gas trapped in a 3-dimensional box the derivative of specific heat reaches smaller values than what was expected in the thermodynamic limit -- here, this result is also verified with analytical calculations. This is an important result for understanding the role of the thermodynamic limit in phase transitions and makes possible to further study BE statistics without relying neither on the thermodynamic limit nor on approximations near critical temperature.

Three-dimensional vortex-induced vibration analysis of catenary-type risers under flow with different incident angles

This paper proposes a three-dimensional (3D) numerical model for the analysis of vortex-induced vibration (VIV) on catenary-type risers (CTRs). Fully considering the geometrical nonlinearities and large deformation behavior of CTR, this model is established based on the vector form intrinsic finite element (VFIFE) theory and the independence principle (i.e., consider only the flow component perpendicular to the riser axis). The riser is discretized into a finite number of particles, the motion of these particles only has to satisfy Newton's second law and the structure is in perpetual dynamic equilibrium under the combined excitation of internal and external forces on the particles. Moreover, the unsteady hydrodynamic force associated with cross-flow (CF) vibration is modeled as distributed Van der Pol wake oscillators. The model is used to predict the VIV response of the CTRs under various flow conditions (i.e., in-plane uniform flow, shear flow, and diverse angled flow), and the results show well agreement with the corresponding experimental results. Then, the CTR's VIV characteristics under diverse angled flow are further analyzed. The obtained results highlight that the incident angle of the flow has a great influence on the mode transition, frequency locking position, energy transfer, and internal forces.

Finite-size effects and thermodynamic limit in one-dimensional Janus fluids

The equilibrium properties of a Janus fluid made of two-face particles confined to a one-dimensional channel are revisited. The exact Gibbs free energy for a finite number of particles N is exactly derived for both quenched and annealed realizations. It is proved that the results for both classes of systems tend in the thermodynamic limit (N → ∞) to a common expression recently derived (Maestre and Santos 2020 J. Stat. Mech. 063217). The theoretical finite-size results are particularized to the Kern–Frenkel model and confirmed by Monte Carlo simulations for quenched and (both biased and unbiased) annealed systems.

Hierarchy of many-body invariants and quantized magnetization in anomalous Floquet insulators

We uncover a new family of few-body topological phases in periodically driven fermionic systems in two dimensions. These phases, which we term correlation-induced anomalous Floquet insulators (CIAFIs), are characterized by quantized contributions to the bulk magnetization from multi-particle correlations, and are classified by a family of integer-valued topological invariants. The CIAFI phases do not require many-body localization, but arise in the generic situation of k-particle localization, where the system is localized (due to disorder) for any finite number of particles up to a maximum number, k. We moreover show that, when fully many-body localized, periodically driven systems of interacting fermions in two dimensions are characterized by a quantized magnetization in the bulk, thus confirming the quantization of magnetization of the anomalous Floquet insulator. We demonstrate our results with numerical simulations.

A particle flow filter for high-dimensional system applications.

A novel particle filter proposed recently, the particle flow filter (PFF), avoids the long‐existing weight degeneracy problem in particle filters and, therefore, has great potential to be applied in high‐dimensional systems. The PFF adopts the idea of a particle flow, which sequentially pushes the particles from the prior to the posterior distribution, without changing the weight of each particle. The essence of the PFF is that it assumes the particle flow is embedded in a reproducing kernel Hilbert space, so that a practical solution for the particle flow is obtained. The particle flow is independent of the choice of kernel in the limit of an infinite number of particles. Given a finite number of particles, we have found that a scalar kernel fails in high‐dimensional and sparsely observed settings. A new matrix‐valued kernel is proposed that prevents the collapse of the marginal distribution of observed variables in a high‐dimensional system. The performance of the PFF is tested and compared with a well‐tuned local ensemble transform Kalman filter (LETKF) using the 1,000‐dimensional Lorenz 96 model. It is shown that the PFF is comparable to the LETKF for linear observations, except that explicit covariance inflation is not necessary for the PFF. For nonlinear observations, the PFF outperforms LETKF and is able to capture the multimodal likelihood behavior, demonstrating that the PFF is a viable path to fully nonlinear geophysical data assimilation.