Quantum dynamics of charge transfer on the one-dimensional lattice:Wave packet spreading and recurrence

The wave function temporal evolution on the one-dimensional(1D) lattice is considered in the tight-binding approximation. The lattice consists of N equal sites and one impurity site(donor). The donor differs from other lattice sites by the on-site electron energy E and the intersite coupling C. The moving wave packet is formed from the wave function initially localized on the donor. The exact solution for the wave packet velocity and the shape is derived at different values E and C. The velocity has the maximal possible group velocity v = 2. The wave packet width grows with time ～ t1/3and its amplitude decreases ～ t-1/3. The wave packet reflects multiply from the lattice ends. Analytical expressions for the wave packet front propagation and recurrence are in good agreement with numeric simulations.     

Complex scaling and residual flavour symmetry in the neutrino mass matrix

We present an instance from nonequilibrium statistical mechanics which combines increase in entropy and finite Poincaré recurrence time. The model we consider is a variation of the well-known Kac’s ring where we consider balls of four colours. As is known, Kac introduced this model where balls arranged between lattice sites, in each time step, move one step clockwise. The colour of the balls change as they cross marked sites. This very simple example rationalize the increase in entropy and recurrence. In our variation, the interesting quantity which counts the difference in the number of balls of different colours is shown to reduce to a set of linear equations if the probability of change of colour is symmetric among a pair of colours. The transfer matrix turns out to be non-Hermitian with real eigenvalues, leading to all colours being equally likely for long times, and a monotonically varying entropy. The new features appearing due to four colours is very instructive.     

Fermi–Pasta–Ulam recurrence and modulation instability

We give a qualitative conceptual explanation of the Fermi–Pasta–Ulam (FPU) like recurrence in the onedimensional focusing nonlinear Schrodinger equation (NLSE). The recurrence can be considered as a result of the nonlinear development of the modulation instability. All known exact localized solitary wave solutions describing propagation on the background of the modulationally unstable condensate show the recurrence to the condensate state after its interaction with solitons. The condensate state locally recovers its original form with the same amplitude but a different phase after soliton leave its initial region. Based on the integrability of the NLSE, we demonstrate that the FPU recurrence takes place not only for condensate, but also for a more general solution in the form of the cnoidal wave. This solution is periodic in space and can be represented as a solitonic lattice. That lattice reduces to isolated soliton solution in the limit of large distance between solitons. The lattice transforms into the condensate in the opposite limit of dense soliton packing. The cnoidal wave is also modulationally unstable due to soliton overlapping. The recurrence happens at the nonlinear stage of the modulation instability. Due to generic nature of the underlying mathematical model, the proposed concept can be applied across disciplines and nonlinear systems, ranging from optical communications to hydrodynamics.     

Thermodynamics of a model wih interacting annealed bond impurities on the Bethe lattice

A magnetic model is considered consisting of annealed, mutually repelling ferromagnetic bond impurities in an antiferromagnetic host lattice. Using recurrence relation techniques, the grand-canonical version of this model is solved on the three-coordinated Bethe lattice. A generic phase diagram is obtained containing, apart from the usual ferro- and antiferromagnetic regimes, two distinct incommensurate phases as well as a period-four modulated phase. Evidence is obtained that in one of the two incommensurate phases impurity pairing occurs.     

Robust dynamical recurrences based on Floquet spectrum

Robust recurrence behavior of wave packets in periodically driven systems and coupled higher dimensional systems is analyzed, which takes place in the realm of higher coupling/modulation strength. We analyze the wave packet dynamics close to nonlinear resonances developed in the systems and provide the analytical understanding of recurrence times. We apply these analytical results to investigate the recurrence times of matter waves in optical lattice in the presence of external periodic forcing. The obtained analytical results can experimentally be observed using currently available experimental setups.     

On the concentration dependence of wings of spectra of spin correlation functions of diluted Heisenberg paramagnets

Singular points of the autocorrelation function on the imaginary time axis that is averaged over the location of spins in the magnetically dilute spin lattice with isotropic spin–spin interaction at a high temperature have been studied. For the autocorrelation function in the approximation of the self-consistent fluctuating local field, nonlinear integral equations have been proposed which reflect the separation of the inhomogeneous spin systems into close spins and other spins. The coordinates of the nearest singular points have been determined in terms of the radius of convergence of the expansion in powers of time, the coefficients of which have been calculated from recurrence equations. It has been shown that the coordinates of singular points and, consequently, the wings of the autocorrelation function spectrum at strong magnetic dilution are determined by the modulation of the local field by the nearest pairs of spins leading to its logarithmic concentration dependence.     

Corrected MK Transformation of ZN Gauge Theory

The corrected Migdal-Kadanoff recurrence relations of lattice gauge systems is obtained and the phase diagram of Z₄ system is calculated.It is found that the fixed poind and the phase diagram obtained in this paper are numerically closer to that given by Monte Carlo simulation.Therefore,the corrected recurrence relation is more significant than MK relation.     

Double Dirac cones at k = 0 in pinned platonic crystals

In this paper, we compute the band structure for a pinned elastic plate which is constrained at the points of a hexagonal lattice. Existing work on platonic crystals has been restricted to square and rectangular array geometries, and an examination of other Bravais lattice geometries for platonic crystals has yet to be made. Such hexagonal arrays have been shown to support Dirac cone dispersion at the center of the Brillouin zone for phononic crystals, and we demonstrate the existence of double Dirac cones for the first time in platonic crystals here. In the vicinity of these Dirac points, there are several complex dispersion phenomena, including a multiple interference phenomenon between families of waves which correspond to free space transport and those which interact with the pins. An examination of the reflectance and transmittance for large finite gratings arranged in a hexagonal fashion is also made, where these effects can be visualized using plane waves. This is achieved via a recurrence relation approach for the reflection and transmission matrices, which is computationally stable compared to transfer matrix approaches.     

Memory in the occurrence of earthquakes

We study the statistics of the recurrence times tau between earthquakes above a certain magnitude M in six (one global and five regional) earthquake catalogs. We find that the distribution of the recurrence times strongly depends on the previous recurrence time tau0, such that small and large recurrence times tend to cluster in time. This dependence on the past is reflected in both the conditional mean recurrence time and the conditional mean residual time until the next earthquake, which increase monotonically with tau0. As a consequence, the risk of encountering the next event within a certain time span after the last event depends significantly on the past, an effect that has to be taken into account in any effective earthquake prognosis.     

On the relationship between continuous time random walks and the non-equilibrium Ornstein-Zernike equation

An exact non-equilibrium Ornstein-Zernike (OZ) equation is derived for lattice fluid systems whose time development is given by a generalized master equation. The derivation is based on a generalization of the Montroll-Weiss continuous-time random walk on a lattice, and on their relationship with master equation solutions. Time dependent direct and total correlation functions are defined in terms of the generating functions for the probability densities of the random walker, such that, in the infinite time limit the equilibrium OZ equation is recovered. A perturbative analysis of the time dependent OZ equation is shown to be formally analogous to the perturbation of the Bloch equation in quantum field theory. Analytic results are obtained, under the mean spherical approximation, for the time dependent total correlation function for a one-dimensional lattice fluid with exponential attraction.     

Macroscopic Diffusion from a Hamilton-like Dynamics

We introduce and analyze a model for the transport of particles or energy in extended lattice systems. The dynamics of the model acts on a discrete phase space at discrete times but has nonetheless some of the characteristic properties of Hamiltonian dynamics in a confined phase space: it is deterministic, periodic, reversible and conservative. Randomness enters the model as a way to model ignorance about initial conditions and interactions between the components of the system. The orbits of the particles are non-intersecting random loops. We prove, by a weak law of large number, the validity of a diffusion equation for the macroscopic observables of interest for times that are arbitrary large, but small compared to the minimal recurrence time of the dynamics.     

Lattice Models Solvable Through the Full Interval Method on Links

A two state model on a one dimensional lattice is considered, where the evolution of the state of each site is determined by the states of that site and its neighboring sites. Corresponding to this original lattice, a derived lattice is introduced the sites of which are the links of the original lattice. It is shown that there is only one reaction on the original lattice, which results in the derived lattice being solvable through the full interval method. And that reaction corresponds to the one dimensional stochastic non-consensus opinion model. A one dimensional non-consensus opinion model with deterministic evolution has already been introduced. Here this is extended to be a model which has a stochastic evolution. Discrete time evolution of such a model is investigated, including the two limiting cases of small probabilities for evolution (resulting to an effectively continuous-time evolution), and deterministic evolution. The formal solution to the evolution equation is obtained and the large time behavior of the system is investigated. Some special cases are studied in more detail.     

A kinetic model for the premelting of a crystalline structure

An analytical kinetic approach to examine the premelting phenomenon is suggested by using a first passage time analysis. Premelting is considered to occur when the time of formation of a Frenkel type defect in the surface monolayer becomes sufficiently small. The mean time of defect formation on the surface lattice, i.e., the mean time necessary for a selected (surface-located) molecule to leave its lattice site and form a Frenkel defect, is calculated by using a first passage time analysis. The model is illustrated by numerical calculations for a crystalline structure composed of molecules interacting via the Lennard-Jones (LJ) potential. The lattice vectors in the plane parallel to the free surface of the crystal were assumed to be equal (to the lattice parameter) and the angle between them was varied. The model predictions of the Tammann temperature (of premelting) are very sensitive to the parameters of the LJ potential. In all the cases considered, the temperature dependence of the mean first passage time has two clearly distinct regimes: at low temperatures the dependence is sharp and at high temperatures it is weak.     

Diffusion on the Scaling Limit of the Critical Percolation Cluster in the Diamond Hierarchical Lattice

We construct critical percolation clusters on the diamond hierarchical lattice and show that the scaling limit is a graph directed random recursive fractal. A Dirichlet form can be constructed on the limit set and we consider the properties of the associated Laplace operator and diffusion process. In particular we contrast and compare the behaviour of the high frequency asymptotics of the spectrum and the short time behaviour of the on-diagonal heat kernel for the percolation clusters and for the underlying lattice. In this setting a number of features of the lattice are inherited by the critical cluster.     

Two-Particle Lattice Green Functions on a Two-Dimensional Rectangular Lattice with Next-Nearest Neighbor Hopping

The paper shows a way to find two-particle lattice Green functions (LGF) on any site of a two-dimensional, rectangular lattice with hopping to nearest- and next-nearest neighbors. Exact, analytical formulas using elliptic integrals for the on-site—G _0,0, nearest neighbor—G _0,1 and G _1,0 and next-nearest neighbor—G _1,1 LGF’s are shown. Difference equation for general G _m,n is given together with five kind of recurrence relations among LGF’s and their derivatives. A way of assembling recurrence relations into closed sets of equations, enabling to find G _n,m on any lattice site (requiring the knowledge of five parameters) is described. The differences between one- and two-particle LGF’s are shown.     

近共振三维光学晶格中铯原子温度的参数依赖

介绍了四光束三维近共振光学晶格的方案,在铯原子磁光阱和光学粘团的基础上搭建了近共振光学晶格的光路,实现了光学晶格中冷原子的装载.利用短程飞行时间吸收法测量了近共振光学晶格中冷原子的温度,通过改变晶格的光强和失谐等条件,对近共振光学晶格中铯原子的亚多普勒冷却的参数依赖关系作了实验研究,并与光学粘团作了比较.     

Numerical method based on the lattice Boltzmann model for the Fisher equation

In this paper, a lattice Boltzmann model for the Fisher equation is proposed. First, the Chapman-Enskog expansion and the multiscale time expansion are used to describe higher-order moment of equilibrium distribution functions and a series of partial differential equations in different time scales. Second, the modified partial differential equation of the Fisher equation with the higher-order truncation error is obtained. Third, comparison between numerical results of the lattice Boltzmann models and exact solution is given. The numerical results agree well with the classical ones.     

Asymptotics of Selberg-like integrals by lattice path counting

We obtain explicit expressions for positive integer moments of the probability density of eigenvalues of the Jacobi and Laguerre random matrix ensembles, in the asymptotic regime of large dimension. These densities are closely related to the Selberg and Selberg-like multidimensional integrals. Our method of solution is combinatorial: it consists in the enumeration of certain classes of lattice paths associated to the solution of recurrence relations.     

Experimental evidence of the spin lattice character of surface relaxation in CESR

By measuring spin lattice relaxation time independently of any linewidth determination, we show the spin lattice character of surface relaxation in CESR. The method is based on amplitude modulation of the microwave field and detection of the longitudinal magnetization.     

Diffusion in a heterogeneous medium for low concentrations

The equations describing diffusion on a heterogeneous lattice for low concentrations are considered taking into account lattice site blocking. It is shown that lattice site blocking cannot be disregarded in the case of a strongly heterogeneous lattice even for low concentrations. It is established that the equation with a fractional time derivative holds only in a bounded time interval. Anomalous diffusion, which is described by the equation with a fractional time derivative at the initial stage, must be described over long time periods by an ordinary diffusion equation with a concentration-dependent diffusion coefficient.