Hydrodynamics beyond Navier-Stokes: exact solution to the lattice Boltzmann hierarchy

The exact solution to the hierarchy of nonlinear lattice Boltzmann (LB) kinetic equations in the stationary planar Couette flow is found at nonvanishing Knudsen numbers. A new method of solving LB kinetic equations which combines the method of moments with boundary conditions for populations enables us to derive closed-form solutions for all higher-order moments. A convergence of results suggests that the LB hierarchy with larger velocity sets is the novel way to approximate kinetic theory.     

HCIZ integral and 2D Toda lattice hierarchy

The expression of the large N Harish Chandra–Itzykson–Zuber (HCIZ) integral in terms of the moments of the two matrices is investigated using an auxiliary unitary two-matrix model, the associated biorthogonal polynomials and integrable hierarchy. We find that the large N HCIZ integral is governed by the dispersionless Toda lattice hierarchy and derive its string equation. We use this to obtain various exact results on its expansion in powers of the moments.     

Single molecule electron transfer dynamics in complex environments

We propose a new theoretical approach to study the kinetics of the electron transfer (ET) under the dynamical influence of the complex environments with the first passage times (FPT) of the reaction events. By measuring the mean and high order moments of FPT and their ratios, the full kinetics of ET, especially the dynamical transitions across different temperature zones, is revealed. The potential applications of the current results to single molecule electron transfer are discussed.     

First passage time for a class of one-dimensional stochastic systems

We consider the time evolution of a class of stochastic systems of finite size with polynomial nearest neighbor transition rates. We obtain analytical expressions for the first passage time (FPT) and its moments. We show that the mean FPT, averaged over a uniform initial distribution, shows a simple asymptotoc behavior with the system size and the parameters of the transition rates.     

Some new aspects of dendrimer applications

Dendrimers are characterized by special features that make them promising candidates for many applications. Here we focus on two such applications: dendrimers as light harvesting antennae, and dendrimers as molecular amplifiers, which may serve as novel platforms for drug delivery. Both applications stem from the unique structure of dendrimers. We present a theoretical framework based on the master equation within which we describe these applications. The quantities of interest are the first passage time (FPT), probability density function (PDF) and its moments. We examine how the FPT PDF and its characteristics depend on the geometric and energetic structures of the dendrimeric system. In particular, we investigate the dependence of the FPT properties on the number of generations (dendrimer size) and the system bias. We present analytical expressions for the FPT PDF for very efficient dendrimeric antennae and for dendrimeric amplifiers. For these cases the mean FPT scales linearly with the system length, and fluctuations around the mean FPT are negligible for large systems. Relationships of the FPT to light harvesting process for other types of system-bias are discussed.     

New integrable lattice hierarchies

In this Letter we give a new integrable four-field lattice hierarchy, associated to a new discrete spectral problem. We obtain our hierarchy as the compatibility condition of this spectral problem and an associated equation, constructed herein, for the time-evolution of eigenfunctions. We consider reductions of our hierarchy, which also of course admit discrete zero curvature representations, in detail. We find that our hierarchy includes many well-known integrable hierarchies as special cases, including the Toda lattice hierarchy, the modified Toda lattice hierarchy, the relativistic Toda lattice hierarchy, and the Volterra lattice hierarchy. We also obtain here a new integrable two-field lattice hierarchy, to which we give the name of Suris lattice hierarchy, since the first equation of this hierarchy has previously been given by Suris. The Hamiltonian structure of the Suris lattice hierarchy is obtained by means of a trace identity formula.     

Two new integrable lattice hierarchies associated with a discrete Schrödinger nonisospectral problem and their infinitely many conservation laws

In this Letter, by means of using discrete zero curvature representation and constructing opportune time evolution problems, two new discrete integrable lattice hierarchies with n-dependent coefficients are proposed, which relate to a new discrete Schrödinger nonisospectral operator equation. The relation of the two new lattice hierarchies with the Volterra hierarchy is discussed. It has been shown that one lattice hierarchy is equivalent to the positive Volterra hierarchy with n-dependent coefficients and another lattice hierarchy with isospectral problem is equivalent to the negative Volterra hierarchy. We demonstrate the existence of infinitely many conservation laws for the two lattice hierarchies and give the corresponding conserved densities and the associated fluxes formulaically. Thus their integrability is confirmed.     

The influence of magnetic impurities on thermodynamic characteristics of antiferromagnets

The thermodynamic behaviour of an impurity spin interacting with both one and another sublattice of an antiferromagnet with the “easy axis” anisotropy type is considered in the framework of phenomenological theory. The direct interaction between impurity magnetic moments is assumed to be negligible. The ground (pure Neel) state of the host lattice is shown to become unstable in the presence of an impurity. At this condition a local magnetic distortion being “frozen” in the vicinity of an impurity takes place and gives rise to appearance of a local ferromagnetism vector perpendicular to the anisotropy axis. Such phenomenon may be described as a local phase transition within the host lattice.     

Toda lattice hierarchy and conservation laws

Time evolutions of the Toda lattice hierarchies of Ueno and Takasaki are induced by Hamiltonians which are conservation laws for the original (one and two dimensional) Toda lattice obtained by Olive and Turok. Moreover these Hamiltonians for two dimensional Toda lattice hierarchy are also conserved quantities of the two component KP hierarchy in which that system is embedded. The one dimensional Toda lattice hierarchy is characterized by the bilinear relations, and a new version of the one dimensional Toda lattice hierarchy is constructed. Generalized Toda lattice hierarchies associated to all affine Lie algebras are presented.     

A neutron diffraction and magnetization study of Heusler alloys containing Co and Zr,Hf, V or Nb

Neutron diffraction, X-ray diffraction and saturation magnetization measurements have been made on a series of ferromagnetic alloys at the composition Co₂YZ, where Y is a group IVA or VA element and Z is a group IIIB or IVB element. The alloys are mainly ordered in the Heusler L2₁ type chemical structure, but some show the presence of a small amount of additional phase, or some preferential disorder.The magnetic results form two distinct groups in which the moments are confined to the Co sites but their magnitude depends on the electron concentration determined by the Y and Z atoms. Although the magnetic moments and Curie temperatures of the alloys in the same group are similar, their lattice parameters differ according to whether Z is a group IIIB or IVB element. Such behaviour indicates that, as in other alloy series with the Heusler structure, a change in lattice parameter has little effect on the magnetic properties in comparison with the effect of a change in electron concentration.     

Soliton solutions by Darboux transformation for a Hamiltonian lattice system

Starting from a new discrete iso-spectral problem, we derive a hierarchy of Hamiltonian lattice equations. A Darboux transformation is established for the lattice soliton hierarchy. As applications, the soliton solutions of resulted lattice hierarchy are given.     

A hierarchy of Liouville integrable discrete Hamiltonian equations

Based on a discrete four-by-four matrix spectral problem, a hierarchy of Lax integrable lattice equations with two potentials is derived. Two Hamiltonian forms are constructed for each lattice equation in the resulting hierarchy by means of the discrete variational identity. A strong symmetry operator of the resulting hierarchy is given. Finally, it is shown that the resulting lattice equations are all Liouville integrable discrete Hamiltonian systems.     

An integrable coupling system of lattice hierarchy and its continuous limits

In [E.G. Fan, Phys. Lett. A 372 (2008) 6368], Fan present a lattice hierarchy and its continuous limits. In this Letter, we extend this method, by introducing a complex discrete spectral problem, a coupling lattice hierarchy is derived. It is shown that a new sequence of combinations of complex lattice spectral problem converges to the integrable coupling couplings of soliton equation hierarchy, which has the integrable coupling system of AKNS hierarchy as a continuous limit.     

Continuous limits for an integrable coupling system of Toda equation hierarchy

In this Letter, we present an integrable coupling system of lattice hierarchy and its continuous limits by using of Lie algebra sl(4). By introducing a complex discrete spectral problem, the integrable coupling system of Toda lattice hierarchy is derived. It is shown that a new complex lattice spectral problem converges to the integrable couplings of discrete soliton equation hierarchy, which has the integrable coupling system of C-KdV hierarchy as a new kind of continuous limit.     

Two hierarchies of integrable lattice equations associated with a discrete matrix spectral problem

Two hierarchies of nonlinear integrable positive and negative lattice models are derived from a discrete spectral problem. The two lattice hierarchies are proved to have discrete zero curvature representations associated with a discrete spectral problem, which also shows that the positive and negative hierarchies correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. Moreover, the integrable lattice models in the positive hierarchy are of polynomial type, and the integrable lattice models in the negative hierarchy are of rational type. Further, we construct infinite conservation laws of the positive hierarchy, then, the integrable coupling systems of the positive hierarchy are derived from enlarging Lax pair.     

Toda lattice realization of integrable hierarchies

We present a new realization of scalar integrable hierarchies in terms of the Toda lattice hierarchy. In other words, we show on a large number of examples that an integrable hierarchy, defined by a pseudo-differential Lax operator, can be embedded in the Toda lattice hierarchy. Such a realization in terms the Toda lattice hierarchy seems to be as general as the Drinfeld-Sokolov realization.     

Advanced linked cluster expansion scalar fields at finite temperature

Linked cluster expansions provide a useful tool for both analytical and numerical investigations of lattice field theories. The expansion parameter(s) being the interaction strength(s) fields at neighboured lattice sites are coupled, they result into convergent hopping parameter-like series for free energies, correlation functions and in particular susceptibilities. We consider scalar fields with O(N)-symmetric nearest-neighbour interactions on hypercubic lattices with possibly finite extension in some directions, thus including field theories at finite temperature T. We improve known and develop new techniques and algorithms to increase the order n. The expansions can be computed too in such a way that detailed information on critical behaviour can be extracted from the susceptibility series. This concerns both simple moments as well as higher correlations such as 4- and 6-point functions used to define renormalized coupling constants. Particular emphasis is done on finite-temperature field theory. In order to be able to measure finite-temperature critical behaviour, the order of explicit computation n has to be sufficiently large compared to T⁻¹ in lattice units. 2- and 4-point susceptibility series are computed up to and including the 18th order and beyond.     

Two new discrete integrable systems

In this paper,we focus on the construction of new(1+1)-dimensional discrete integrable systems according to a subalgebra of loop algebra A 1.By designing two new(1+1)-dimensional discrete spectral problems,two new discrete integrable systems are obtained,namely,a 2-field lattice hierarchy and a 3-field lattice hierarchy.When deriving the two new discrete integrable systems,we find the generalized relativistic Toda lattice hierarchy and the generalized modified Toda lattice hierarchy.Moreover,we also obtain the Hamiltonian structures of the two lattice hierarchies by means of the discrete trace identity.