Scalar Lattice Research Articles

Equivalence of lattice operators and graph matrices

Abstract We explore the relationship between lattice field theory and graph theory, placing special emphasis on the interplay between Dirac and scalar lattice operators and matrices within the realm of spectral graph theory. Beyond delving into fundamental concepts of spectral graph theory, such as adjacency and Laplacian matrices, we introduce a novel matrix called an “antisymmetrized adjacency matrix”, specifically tailored for cycle digraphs (T1 lattice) and simple directed paths (B1 lattice). The nontrivial relationship between graph theory matrices and lattice operators shows that the graph Laplacian matrix mirrors the lattice scalar operator and the Wilson term in lattice fermions, while the antisymmetrized adjacency matrix, along with its extensions to higher dimensions, is equivalent to naive lattice Dirac operators. Building upon these connections, we provide rigorous proofs for two key assertions: (i) The count of zero-modes in a free lattice scalar operator coincides with the zeroth Betti number of the underlying graph (lattice). (ii) The maximum count of Dirac zero-modes in a free lattice fermion operator is equivalent to the cumulative sum of all Betti numbers when the D-dimensional graph results from a Cartesian product of cycle digraphs (T1 lattice) and simple directed paths (B1 lattice).

Combining neutral scalar and isothermal Shan-Chen lattice Boltzmann method to simulate droplet placement on a wall in isothermal and non-isothermal states

BackgroundIn the current article, two-phase thermal fluxes are created by combining the thermal model of the neutral scalar model with the two-phase Shan-Chen model of the lattice Boltzmann method (LBM). MethodsThe different intermolecular powers for the isothermal Shan-Chen model show how a droplet would be placed on a wall. By raising the droplet intermolecular power parameter, the surface area increases and becomes wet. Next, the isothermal Shan-Chen method and the neutral scalar method are combined to investigate multi-phase thermal problems. The droplet placement on the hot wall is therefore done at relatively high Rayleigh numbers. By raising the Rayleigh number, the isothermal lines within the droplet's interior gradually become less ascending and less descending until they eventually achieve a uniform state when it is placed against a hot wall. Additionally, the channel's Rayleigh-Bénard convective heat transfer is enhanced by increasing the Rayleigh number. Significant findingsNatural convection in the enclosures can be used in solar collectors. As the Rayleigh number increases, the average Nusselt number (Nuavg) rises as would be expected. The results demonstrate that LBM is a practical method for simulating multi-phase thermal flows.

Achieving the Dirty Paper Channel Capacity With Scalar Lattices and Probabilistic Shaping

Achieving the Dirty Paper Channel Capacity With Scalar Lattices and Probabilistic Shaping

Side-jump scattering enhanced spin Hall effect in SrTiO3-implanted Pt

A spin Hall effect (SHE) enables the electrical generation and detection of spin currents for promising applications in spintronics, but heavy metals with low spin Hall angle θSH limit the development of SHE devices. In this work, we have introduced dielectric oxide material SrTiO3 into Pt by magnetron sputtering and measured the θSH on the NiCo/Pt1–x(STO)x heterostructure through spin-torque ferromagnetic resonance. Our results demonstrate that the maximum spin Hall angle in Pt0.98(STO)0.02 is 0.121 ± 0.003, which is approximately twice that of pure Pt (0.064 ± 0.003). Moreover, theoretical analysis has revealed that the spin Hall angle arises from a complementary interplay between intrinsic and extrinsic mechanisms, namely, the strong spin–orbit coupling in Pt for the intrinsic mechanism and side-jump scattering caused by scalar potential and lattice expansion at dielectric impurities for the extrinsic mechanism. This interplay significantly contributes to enhancing the spin Hall angle. This work demonstrates an effective strategy for fabricating high-performance spin Hall materials with low resistivity, large spin Hall angle, and excellent compatibility with semiconductor processes in low-power spin-torque devices.

Motion of localized disturbances in scalar harmonic lattices.

We present analytical and numerical investigations of energy propagation in systems of massive particles that interact via harmonic (linear) forces. The particle motion is described by a scalar displacement, and the particles are arranged in a simple crystal lattice. For the systems under consideration we prove the conservation of the total energy flux analytically. Then, using a sample case of a square lattice, we confirm the analytical results numerically. We create disturbances of a special kind which can move with a predefined velocity with a minor change in their shape. We show that a clot of energy, associated with each disturbance, moves similarly to a free body of matter in classical mechanics. We also numerically study a simultaneous propagation of a number of energy clots as an analogy to the motion of point masses in the conventional mechanics of particles. The obtained results demonstrate that an energy flow in lattices can be described in terms of numerous separated energy bodies, making a step towards a linkage between lattice dynamics and the kinetic theory of heat transfer in solids.

Discrete space and the scalar lattice 3.0

Discrete space and the scalar lattice 3.0

Discrete space and the scalar lattice

Discrete space and the scalar lattice

Discrete space and the scalar lattice

Discrete space and the scalar lattice

Scaling Limits of Lattice Quantum Fields by Wavelets

We present a rigorous renormalization group scheme for lattice quantum field theories in terms of operator algebras. The renormalization group is considered as an inductive system of scaling maps between lattice field algebras. We construct scaling maps for scalar lattice fields using Daubechies’ wavelets, and show that the inductive limit of free lattice ground states exists and the limit state extends to the familiar massive continuum free field, with the continuum action of spacetime translations. In particular, lattice fields are identified with the continuum field smeared with Daubechies’ scaling functions. We compare our scaling maps with other renormalization schemes and their features, such as the momentum shell method or block-spin transformations.

Asymptotic Description of Fast Thermal Processes in Scalar Harmonic Lattices

Thermal vibrations in a d-dimensional (d = 1, 2) scalar harmonic lattice of a simple structure are under consideration. Redistribution of the averaged kinetic and potential energies of particles after instantaneous thermal excitation (fast process) is described. It is established that the difference between the kinetic and potential energies undergoes power-law damped oscillations at times much longer than the characteristic atomic-vibration period. A typical exponent is –d/2. The oscillation frequencies are determined from the dispersion relation for the lattice. The algorithm of proving allows generalization to scalar and vector lattices with a complex structure and different dimensions. Thermal vibrations in a two-dimensional hexagonal lattice (graphene lattice) are considered as an example of this generalization.

Асимптотическое описание быстрых тепловых процессов в скалярных гармонических решетках

Thermal processes in d-dimensional (d = 1,2) scalar harmonic lattices with simple structure are considered. The redistribution between the averaged kinetic and potential energies of particles after instantaneous thermal excitation is described. At times much longer than the characteristic period of atomic oscillations, the difference between the kinetic and potential energies oscillates with amplitude decreasing by a power law (typically, the exponent is -d/2). The frequencies of the oscillations are determined from the dispersion relation for the lattice. The above results can be generalized for scalar and vector lattices of complex structure and various dimensions. As an example, the oscillations of the kinetic tmperature in the two-dimensional hexagonal lattice (graphene lattice) are investigated.

Solution formulas for the two-dimensional Toda lattice and particle-like solutions with unexpected asymptotic behaviour

The first main aim of this article is to derive an explicit solution formula for the scalar two-dimensional Toda lattice depending on three independent operator parameters, ameliorating work in [31]. This is achieved by studying a noncommutative version of the 2d-Toda lattice, generalizing its soliton solution to the noncommutative setting.The purpose of the applications part is to show that the family of solutions obtained from matrix data exhibits a rich variety of asymptotic behaviour. The first indicator is that web structures, studied extensively in the literature, see [4] and references therein, are a subfamily. Then three further classes of solutions (with increasingly unusual behaviour) are constructed, and their asymptotics are derived.

Steady-state kinetic temperature distribution in a two-dimensional square harmonic scalar lattice lying in a viscous environment and subjected to a point heat source

We consider heat transfer in an infinite two-dimensional square harmonic scalar lattice lying in a viscous environment and subjected to a heat source. The basic equations for the particles of the lattice are stated in the form of a system of stochastic ordinary differential equations. We perform a continualization procedure and derive an infinite system of linear partial differential equations for covariance variables. The most important results of the paper are the deterministic differential–difference equation describing non-stationary heat propagation in the lattice and the analytical formula in the integral form for its steady-state solution describing kinetic temperature distribution caused by a point heat source of a constant intensity. The comparison between numerical solution of stochastic equations and obtained analytical solution demonstrates a very good agreement everywhere except for the main diagonals of the lattice (with respect to the point source position), where the analytical solution is singular.

Integrable Seven-Point Discrete Equations and Second-Order Evolution Chains

We consider differential-difference equations that determine the continuous symmetries of discrete equations on the triangular lattice. It is shown that a certain combination of continuous flows can be represented as a scalar evolution lattice equation of order 2. The general scheme is illustrated by a number of examples, including an analog of the elliptic Yamilov lattice equation.

Laser cooling of rubidium atoms in a 2D optical lattice

Lossless polarization gradient cooling of atoms in a far-detuned 2D optical lattice is demonstrated. Temperatures down to K and phase space densities as high as 1 / 1000 are achieved in a total duty cycle of . It is shown that utilizing the vector component of the optical lattice allows lower temperatures to be achieved when compared with pure scalar lattices.

Stability limits of the single relaxation-time advection–diffusion lattice Boltzmann scheme

In many cases, multi-species and/or thermal flows involve large discrepancies between the different diffusion coefficients involved — momentum, heat and species diffusion. In the context of classical passive scalar lattice Boltzmann (LB) simulations, the scheme is quite sensitive to such discrepancies, as relaxation coefficients of the flow and passive scalar fields are tied together through their common lattice spacing and time-step size. This in turn leads to at least one relaxation coefficient, [Formula: see text] being either very close to 0.5 or much larger than unity which, in the case of the former (small relaxation coefficient), has been shown to cause instability. The present work first establishes the stability boundaries of the passive scalar LB method in the sense of von Neumann and as a result shows that the scheme is unconditionally stable, even for [Formula: see text], provided that the nondimensional velocity does not exceed a certain threshold. Effects of different parameters such as the distribution function and lattice speed of sound on the stability area are also investigated. It is found that the simulations diverge for small relaxation coefficients regardless of the nondimensional velocity. Numerical applications and a study of the dispersion–dissipation relations show that this behavior is due to numerical noise appearing at high wave numbers and caused by the inconsistent behavior of the dispersion relation along with low dissipation. This numerical noise, known as Gibbs oscillations, can be controlled using spatial filters. Considering that noise is limited to high wave numbers, local filters can be used to control it. In order to stabilize the scheme with minimal impact on the solution even for cases involving high wave number components, a local Total Variation Diminishing (TVD) filter is implemented as an additional step in the classical LB algorithm. Finally, numerical applications show that this filter eliminates the unwanted oscillations while closely reproducing the reference solution.

Fast and slow thermal processes in harmonic scalar lattices

An approach for analytical description of thermal processes in harmonic lattices is presented. We cover longitudinal and transverse vibrations of chains and out-of-plane vibrations of two-dimensional lattices with interactions of an arbitrary number of neighbors. The motion of each particle is governed by a single scalar equation and therefore the notion ‘scalar lattice’ is used. The evolution of initial temperature field in an infinite lattice is investigated. An exact equation describing the evolution is derived. Continualization of this equation with respect to spatial coordinates is carried out. The resulting continuum equation is solved analytically. The solution shows that the kinetic temperature is represented as the sum of two terms, one describing short time behavior, the other large time behavior. At short times, the temperature performs high-frequency oscillations caused by redistribution of energy among kinetic and potential forms (fast process). Characteristic time of this process is of the order of ten periods of atomic vibrations. At large times, changes of the temperature are caused by ballistic heat transfer (slow process). The temperature field is represented as a superposition of waves having the shape of initial temperature distribution and propagating with group velocities dependent on the wave vector. Expressions describing fast and slow processes are invariant with respect to substitution t by . However, examples considered in the paper demonstrate that these processes are irreversible. Numerical simulations show that presented theory describes the evolution of temperature field at short and large time scales with high accuracy.

Entire solutions for bistable lattice differential equations with obstacles

We consider scalar lattice dierential equations posed on square lattices in two space dimensions. Under certain natural conditions we show that wave-like solutions exist when obstacles (characterized by \holes) are present in the lattice. Our work generalizes to the discrete spatial setting the results obtained in [9] for the propagation of waves around obstacles in continuous spatial domains. The analysis hinges upon the development of sub and super-solutions for a class of discrete bistable reaction-diusion problems and on a generalization of a classical result due to Aronson and Weinberger that concerns the spreading of localized disturbances.

Normal heat conductivity in two-dimensional scalar lattices

The paper revisits recent counterintuitive results on the divergence of the heat conduction coefficient in two-dimensional lattices. It was reported that in certain lattices with on-site potential, for which a one-dimensional chain has convergent conductivity, the latter diverges in the 2D counterpart. We demonstrate that this conclusion is an artifact caused by the insufficient size of the simulated system. To overcome computational restrictions, a ribbon of relatively small width is simulated instead of a more traditional square specimen. It is further demonstrated that the heat conduction coefficient in the “long” direction of the ribbon ceases to depend on the width, as the latter achieves only 10 to 20 chains. So, one can consider the dynamics of much longer systems, than in the traditional setting, and still can gain reliable information regarding the 2D lattice. It turns out that for all considered models, for which the conductivity is convergent in the 1D case, it is convergent also in the 2D case. At the same time, however, the length of the system, necessary to reveal the convergence in the 2D case, may be much bigger than in its 1D counterpart.

Investigation of vapor condensation on a flat plate and horizontal cryogenic tube using lattice Boltzmann method

In this paper simulation of vapor condensation, liquid film formation, and dew drop sprinkling from cryogenic horizontal tube base on two-dimensional lattice Boltzmann method is presented. Lee's multiphase model is used which is applicable to high density and viscosity ratios. The passive scalar thermal lattice Boltzmann framework and proper source term due to phase change are combined with the multiphase model to simulate the film condensation. Lee's model is based on convective Cahn-Hilliard equation and the divergence-free condition of the velocity field. However, as the phase change occurs at the interface, the divergence-free condition is no longer satisfied. The simple passive scalar approach is employed for the evolution of the temperature field in the computational domain and the flow field is affected by temperature under the hypothesis of Boussinesq. The D2Q9 lattice structure is used for the case of density ratio of 25), and the effects of gravitational superheating, contact angle and the temperature of sub-cooled tube on vapor condensation are illustrated. Finally flow fields and three-phase contact line movement of film condensation are analyzed.