Normal modes of oscillations of lattices

Polynomial equations are obtained for the solutions of the vibrational frequencies of a simple cubic, primitive orthorhombic and tetragonal Bravais lattices of finite size with particles connected to their nearest neighbours through both central and non-centrarl forces, but with arbitrary forces connecting the surface atoms to the rigid walls. The exact expressions of the different normal modes of oscillations and the amplitudes of vibration of different particles in various modes are obtained by solving three decoupled partial difference equations.     

Reliability Polynomials and Their Asymptotic Limits for Families of Graphs

We present exact calculations of reliability polynomials R(G,p) for lattice strips G of fixed widths L _y 4 and arbitrarily great length L _x with various boundary conditions. We introduce the notion of a reliability per vertex, r({G},p)=lim_|V|R(G,p)^1/|V| where |V| denotes the number of vertices in G and {G} denotes the formal limit lim_|V|G. We calculate this exactly for various families of graphs. We also study the zeros of R(G,p) in the complex p plane and determine exactly the asymptotic accumulation set of these zeros , across which r({G}) is nonanalytic.     

Asymptotic Behavior of Inflated Lattice Polygons

We study the inflated phase of two dimensional lattice polygons with fixed perimeter N and variable area, associating a weight exp [pA−Jb] to a polygon with area A and b bends. For convex and column-convex polygons, we calculate the average area for positive values of the pressure. For large pressures, the area has the asymptotic behaviour , where , and ρ<1. The constant K(J) is found to be the same for both types of polygons. We argue that self-avoiding polygons should exhibit the same asymptotic behavior. For self-avoiding polygons, our predictions are in good agreement with exact enumeration data for J=0 and Monte Carlo simulations for J≠0. We also study polygons where self-intersections are allowed, verifying numerically that the asymptotic behavior described above continues to hold.     

Flow Polynomials and Their Asymptotic Limits for Lattice Strip Graphs

We present exact calculations of flow polynomials F(G,q) for lattice strips of various fixed widths L _y 4 and arbitrarily great lengths L _x , with several different boundary conditions. Square, honeycomb, and triangular lattice strips are considered. We introduce the notion of flows per face fl in the infinite-length limit. We study the zeros of F(G,q) in the complex q plane and determine exactly the asymptotic accumulation sets of these zeros in the infinite-length limit for the various families of strips. The function fl is nonanalytic on this locus. The loci are found to be noncompact for many strip graphs with periodic (or twisted periodic) longitudinal boundary conditions, and compact for strips with free longitudinal boundary conditions. We also find the interesting feature that, aside from the trivial case L _y =1, the maximal point, q _cf , where crosses the real axis, is universal on cyclic and Möbius strips of the square lattice for all widths for which we have calculated it and is equal to the asymptotic value q _cf =3 for the infinite square lattice.     

Coulomb energy of cubic lattices

The Coulomb energies of simple cubic, face-centered cubic and body-centered cubic point lattices with uniform neutralizing background are given for crystals with up to 16 atoms per unit cell.     

Photoresistivity and optical switching of graphene with DNA lattices

We present the photon induced conductivity of 2D DNA lattices with and without graphene and demonstrate the switching current responses controlled by light irradiation. The conductivity in the DNA lattices with protein streptavidin controlled by blue and white lights shows significant enhancement with the addition of graphene. An optical pulse response of a graphene immobilized DNA lattice is encouraging and may lead to various bio-sensing applications such as immunological assays, DNA forensics, and toxin detection.     

Surface defect solitons at interfaces between dual-frequency and simple lattices

We reveal theoretically the existence and stability of surface defect solitons (SDSs) at interfaces between dual-frequency and simple lattices with focusing saturable nonlinearity. Solitons with some unique properties exist in such composite structures with the change of defect intensity. For zero defect or positive defect, the surface solitons exist at the semi-infinite gap and cannot exist in the first gap, and solitons are stable at lower power but unstable at high power. For the case of negative defect, the surface solitons exist not only in the semi-infinite gap, but also in the first gap. With increasing the defect depth, the stable region of surface solitons becomes narrower in the semi-infinite gap, these solitons are stable within a moderate power region in the first gap within unstable solitons in the entire semi-infinite gap.     

Calculation of electrostatic energy of planar lattices

The electrostatic potential of planar lattices semiinfinite crystals and films are calculated by the Fourier transformation of Coulomb potential in one direction only. Some examples, of application of calculation of electrostatic energy are presented.     

Asymptotic results on occupancy times of random walks

In this note we derive, using Wald's theorem asymptotic results on mean occupancy time of an interval for random walks with arbitrary transition probabilities. We show that our results are consistent with those obtained (by Weiss, Ref. 2) via the master equation approach, by demonstrating that the resulting infinite series can be summed exactly.     

Survival probabilities for random walks on lattices with randomly distributed traps

I present here a numerical procedure to compute survival probabilities for random walks on lattices with randomly distributed traps. The procedure has some advantages over existing methods, and its performance is evaluated for the 1D simple random walk, for which some exact results are known. Thereafter, I apply the procedure to 1D random walks with variable step length and to 3D simple random walks.     

Quench dynamics of ultracold atoms in one-dimensional optical lattices with artificial gauge fields

We study the quench dynamics of noninteracting ultracold atoms loaded in one-dimensional(1D) optical lattices with artificial gauge fields, which are modeled by lattices with complex hopping coefficients. After suddenly changing the hopping coefficient, time evolutions of the density distribution, momentum distribution, and mass current at the center are studied for both finite uniform systems and trapped systems. Effects of filling factor, system size, statistics, harmonic trap, and phase difference in hopping are identified, and some interesting phenomena show up. For example, for a finite uniform fermionic system shock and rarefaction wave plateaus are formed at two ends, whose wave fronts move linearly with speed equaling to the maximal absolute group velocity. While for a finite uniform bosonic system the whole density distribution moves linearly at the group velocity. Only in a finite uniform fermionic system there can be a constant quasisteady-state current, whose amplitude is decided by the phase difference and filling factor. The quench dynamics can be tested in ultracold atoms with minimal modifications of available experimental techniques, and it is a very interesting and fundamental example of the transport phenomena and the nonequilibrium dynamics.     

The triangular kagomé lattices revisited

The dimer problem, Ising spins and bond percolation on the triangular kagomé lattice have been studied extensively by physicists. In this paper, based on the fact the triangular kagomé lattice with toroidal boundary condition can be regarded as the line graph of 3.12.12 lattice with toroidal boundary condition, we derive the formulae of the number of spanning trees, the energy, and the Kirchhoff index of the triangular kagomé lattice with toroidal boundary condition.     

Asymptotic behavior of energy band associated with a negative energy level

The mathematical foundation of the tight binding approximation is given. If ₀ is a negative energy level of a real potentialq, then there exists an energy band for a one-dimensional chain with period 2T of the same atoms which lies near ₀. We study this band whenT tends to infinity.On leave of absence from the Department of Physics, Leningrad State University, Leningrad, USSR.     

Investigation of the characteristic energy losses of platinum (111)

With the help of a combined LEED- and Auger-investigation, the surface of a platinum (111) crystal was cleaned first. Then, the spectrum of the characteristic energy losses for both contaminated and cleaned surfaces is studied. On the cleaned surface the following losses were found: ΔE ₁=7.4 eV, ΔE ₂=13.5 eV, ΔE ₃=24.8 eV, ΔE ₄=31.8 eV, ΔE ₅=45.1 eV, ΔE ₆=54.1 eV, ΔE ₇=71.2 eV. The present results are compared with the measurement of other investigators. In particular, in good agreement with optical measurements we identify ΔE ₁ and ΔE ₂ as interband transitions, and ΔE ₃ and ΔE ₄ as surface and volume plasma loss, respectively.     

Analysis of thermal energy harvesting using ferromagnetic materials

This Letter aims at giving a preliminary investigation of the thermal energy harvesting capabilities of a technique using the temperature-dependent permeability of ferromagnetic materials. The principles lie in the modification of the magnetic field caused by the variation of the permeability due to the temperature change, hence generating a voltage across a coil surrounding the circuit. The technique can be made truly passive by the use of magnets for applying bias magnetic field. Theoretical results, validated by experimental measurements, show a voltage output of 1.2 mV at optimal load of 2 Ω under 60 K temperature variation in 5 s (with a maximum slope of 25 K s⁻¹). Further improvements, such as the use of low resistivity coil and magnet with high remnant magnetic field, indicate that it is possible to convert up to 7.35 μJ cm⁻³ K⁻² cycle⁻¹.     

The travelling salesman problem on randomly diluted lattices: Results for small-size systems

If one places N cities randomly on a lattice of size L, we find that and vary with the city concentration p=N/L ², where is the average optimal travel distance per city in the Euclidean metric and is the same in the Manhattan metric. We have studied such optimum tours for visiting all the cities using a branch and bound algorithm, giving the exact optimized tours for small system sizes () and near-optimal tours for bigger system sizes (). Extrapolating the results for , we find that for p=1, and and with as . Although the problem is trivial for p=1, for it certainly reduces to the standard travelling salesman problem on continuum which is NP-hard. We did not observe any irregular behaviour at any intermediate point. The crossover from the triviality to the NP-hard problem presumably occurs at p=1. Received 15 April 2000     

Asymptotic analysis of the lattice Boltzmann equation

In this article we analyze the lattice Boltzmann equation (LBE) by using the asymptotic expansion technique. We first relate the LBE to the finite discrete-velocity model (FDVM) of the Boltzmann equation with the diffusive scaling. The analysis of this model directly leads to the incompressible Navier–Stokes equations, as opposed to the compressible Navier–Stokes equations obtained by the Chapman–Enskog analysis with convective scaling. We also apply the asymptotic analysis directly to the fully discrete LBE, as opposed to the usual practice of analyzing a continuous equation obtained through the Taylor-expansion of the LBE. This leads to a consistency analysis which provides order-by-order information about the numerical solution of the LBE. The asymptotic technique enables us to analyze the structure of the leading order errors and the accuracy of numerically derived quantities, such as vorticity. It also justifies the use of Richardson’s extrapolation method. As an example, a two-dimensional Taylor-vortex flow is used to validate our analysis. The numerical results agree very well with our analytic predictions.