Long-range effects on superdiffusive algebraic solitons in anharmonic chains

Studies on thermal diffusion of lattice solitons in Fermi-Pasta-Ulam (FPU)-like lattices were recently generalized to the case of dispersive long-range interactions (LRI) of the Kac-Baker form. The variance of the soliton position shows a stronger than linear time-dependence (superdiffusion) as found earlier for lattice solitons on FPU chains with nearest-neighbour interactions (NNI). Since the superdiffusion seems to be generic for nontopological solitons, we want to illuminate the role of the soliton shape on the superdiffusive mechanism. Therefore, we concentrate on an FPU-like lattice with a certain class of power-law long-range interactions where the solitons have algebraic tails instead of the exponential tails in the case of FPU-type interactions (with or without Kac-Baker LRI).Despite of structurally similar Langevin equations which hold for the soliton position and width of the two soliton types, the algebraic solitons reach the superdiffusive long-time limit with a characteristic t^3/2 time-dependence much faster than exponential solitons. The soliton shape determines the diffusion constant in the long-time limit that is approximately a factor of π smaller for algebraic solitons. Our results appear to be generic for nonlinear excitaitons in FPU-chains, because the same superdiffusive time-dependence was also observed in simulations with discrete breathers.     

New results for diffusion in Lorentz lattice gas cellular automata

New calculations to over ten million time steps have revealed a more complex diffusive behavior than previously reported of a point particle on a square and triangular lattice randomly occupied by mirror or rotator scatterers. For the square lattice fully occupied by mirrors where extended closed particle orbits occur, anomalous diffusion was still found. However, for a not fully occupied lattice the superdiffusion, first noticed by Owczarek and Prellberg for a particular concentration, obtains for all concentrations. For the square lattice occupied by rotators and the triangular lattice occupied by mirrors or rotators, an absence of diffusion (trapping) was found for all concentrations, except on critical lines, where anomalous diffusion (extended closed orbits) occurs and hyperscaling holds for all closed orbits withuniversal exponentsd _f =7/4 and =15/7. Only one point on these critical lines can be related to a corresponding percolation problem. The questions arise therefore whether the other critical points can be mapped onto a new percolation-like problem and of the dynamical significance of hyperscaling.     

Some fractal properties of the percolating backbone in two dimensions

A new algorithm is presented, based on elements of artificial intelligence theory, to determine the fractal properties of the backbone of the incipient infinite cluster. It is found that the fractal dimensionality of the backbone isd _f ^BB =1.61±0.01, the chemical dimensionality isd _t=1.40±0.01, and the fractal dimension of the minimum pathd _min=1.15 ± 0.02 for the two-dimensional triangular lattice.     

Calculation of superdiffusion for the Chirikov-Taylor model

It is widely known that the paradigmatic Chirikov-Taylor model presents enhanced diffusion for specific intervals of its stochasticity parameter due to islands of stability, which are elliptic orbits surrounding accelerator mode fixed points. In contrast with normal diffusion, its effect has never been analytically calculated. Here, we introduce a differential form for the Perron-Frobenius evolution operator in which normal diffusion and superdiffusion are treated separately through phases formed by angular wave numbers. The superdiffusion coefficient is then calculated analytically resulting in a Schloemilch series with an exponent beta=3/2 for the divergences. Numerical simulations support our results.     

On the ground state of ferromagnetic Hamiltonians

It is generally believed that the ground state of the ferromagnetic Heisenberg-Dirac-Van Vleck Hamiltonians acting ons=1/2 spins of a lattice withN sites has the maximum possible value of the total spinS=N/2 and isN+1 times degenerate. We present a rigorous proof of this statement, independent of the lattice dimension and topology. A member of the Doppler Institute of Mathematical Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University, Prague.     

Diffusion of adsorbates on random alloy surfaces

In this work the diffusion of non-interacting adsorbates on a random AB alloy surface is considered. For this purpose a simple cubic (sc), body-centered cubic (bcc) or face-centered cubic (fcc) auxiliary metal lattice is introduced. The auxiliary lattice is truncated parallel to its (100) plane in such a way that the fourfold hollow positions of the metal surface form a regular net of adsorption sites with square symmetry. The adsorption energy of each adsorption site is determined by its own environment, i.e. by the numbers of direct A or B neighbors. The Monte-Carlo method has been utilized to simulate surface diffusion of adsorbates on such energetically heterogeneous alloy surfaces and to calculate the tracer, jump and chemical diffusion coefficients. The chemical diffusion coefficient was calculated via two different approaches: the fluctuation and the Kubo-Green method. The influence of energetical heterogeneities on the surface diffusion is largely pronounced at low temperatures and low surface coverages, where most of the adatoms are trapped by deep adsorption sites. It was found that at low temperatures the sequential occupation of the different types of adsorption sites can be observed. Received: 24 October 1997 / Accepted: 17 December 1997     

Reply to criticisms of theB (3) field

The confusion and self-contradiction among recent critics of theB ⁽³⁾ (Evans-Vigier) field are analysed. Barron [17] and Buckingham [18] assert that the field is zero by symmetry. Grimes [21] asserts that the field isnon-zero butfortuitous. Lakhtakia in one paper [19] asserts thatB ⁽³⁾ isnon-zero butnot fundamental, and in a second paper that it isunknowlable and therefore may as well be zero. A rebuttal is given of each the individual papers, and it is shown that the Evans-Vigier field is the fundamental magnetizing field of electromagnetic radiation.     

The diffusion of self-avoiding random walk in high dimensions

We use the Brydges-Spencer lace expansion to prove that the mean square displacement of aT step strictly self-avoiding random walk in thed dimensional square lattice is asymptotically of the formDT asT approaches infinity, ifd is sufficiently large. The diffusion constantD is greater than one.     

Boundaries forSU(3) c xU(1)el lattice gauge theory with a chemical potential

It is shown forSU(N) andU(1) gauge groups that periodic spatial boundary conditions, as commonly used in lattice simulations, are not possible in the charged sectors of a local gauge theory. For charge-conjugate (C-)periodic boundary conditions the effective gauge action of fermions is derived. For nonzero chemical potential, the breakdown of translational invariance induced by the breakdown ofC symmetry is discussed. If translational invariance is abandoned, (anti)periodic spatial b.c. for fermions and for theSU(3) gauge field andC-periodic b.c. for theU(1) gauge field can be used.     

Clustering for algebraicK-systems

We prove that for a von Neumann algebra that is an algebraicK system with respect to some automorphism, the invariant state isK-clustering andr-clustering. Further, we study by using examples how far the von Neumann algebra inherits theK property from the underlyingC ^* algebra.     

A class of stochastic evolutions that scale to the porous medium equation

A class of reversible Markov jump processes on a periodic lattice is described and a result about their scaling behavior stated: Under diffusion scaling, the empirical measure converges to a solution of the porous medium equation on thed-dimensional torus. The process can be viewed as a randomly interacting configuration of sticks that evolves through exchanges of stick pieces between nearest neighbors through a zero-range pressure mechanism, with conservation of total stick length.     

A generalized treatment of the chemical diffusion coefficient in binary oxides as a function of the oxygen activity

The dependence of the chemical diffusion coefficient and of the self-diffusion coefficient of the metal on the oxide composition (a₀) in a binary oxide which is predominantly an electronic semiconductor and contains defects only in the cation sublattice has been examined theoretically. For the case where there is only one type dominant lattice defect following an ideal solution behavior the chemical diffusion coefficient is independent of a₀ When more than one type of defect is present instead, the chemical diffusion coefficient depends on a₀. Furthermore, it is shown that values of the chemical diffusion coefficient obtained by thermogravimetric analysis may differ from those obtained using changes in electrical conductivity. Results are compared with available experimental data for NiO and CoO at 1000°C and reasonable agreement is found.     

Superdiffusion and stable laws

The superdiffusion equation with a fractional Laplacian Δ ^α/2 in _N-dimensional space describes the asymptotic (t→∞) behavior of a generalized Poisson process with the range (discontinuity) distribution density ∼|x|−α−1. The solutions of this equation belong to a class of spherically symmetric stable distributions. The main properties of these solutions are given together with their representations in the form of integrals and series and the results of numerical calculations. It is shown that allowance for the finite velocity of free particle motion for α>1 merely amounts to a reduction in the diffusion coefficient with the form of the distribution remaining stable. For α<1 the situation changes radically: the expansion velocity of the diffusion packet exceeds the velocity of free particle motion and the superdiffusion equation becomes physically meaningless. Zh. éksp. Teor. Fiz. 115, 1411–1425 (April 1999)     

Diffusion in Lorentz lattice gas cellular automata: The honeycomb and quasi-lattices compared with the square and triangular lattices

We study numerically the nature of the diffusion process on a honeycomb and a quasi-lattice, where a point particle, moving along the bonds of the lattice, scatters from randomly placed scatterers on the lattice sites according to strictly deterministic rules. For the honeycomb lattice fully occupied by fixed rotators two (symmetric) isolated critical points appear to be present, with the same hyperscaling relation as for the square and the triangular lattices. No such points appear to exist for the quasi-lattice. A comprehensive comparison is made with the behavior on the previously studied square and triangular lattices. A great variety of diffusive behavior is found, ranging from propagation, superdiffusion, normal, quasi-normal, and anomalous, to absence of diffusion. The influence of the scattering rules as well as of the lattice structure on the diffusive behavior of a point particle moving on the all lattices studied so far is summarized.     

A reexamination of theI.S. results for the cluster compound Au55(PPh3)12Cl6

The observed lowI.S. values [1] for the four distinct gold sites in the cluster compound Au₅₅(PPh₃)₁₂Cl₆ have been explained in terms of a decrease of the 6s electron density at the nucleus, due either to a lattice expansion or to a delocalization of 6s electrons over the ligand shell surrounding the gold core. Recent EXAFS measurements [2] indicate a single average distance between the gold atoms, about 3.5% less than for bulk gold. This not only excludes the lattice expansion hypothesis, it effectively increases the density of 6s electrons, making an explanation for the observedI.S. even more difficult. For an understanding of theI.S. values it is necessary to reconsider the probable occupation of the 5d orbitals within the framework provided by the XPS surface atom core level shifts developed by Citrin and Wertheim [3]. Partial confirmation has been found in preliminary XPS results [5]. The consequences for thef-factor and specific heat results [1] will also be examined.     

Finite difference approximations for a fractional advection diffusion problem

The use of the conventional advection diffusion equation in many physical situations has been questioned by many investigators in recent years and alternative diffusion models have been proposed. Fractional space derivatives are used to model anomalous diffusion or dispersion, where a particle plume spreads at a rate inconsistent with the classical Brownian motion model. When a fractional derivative replaces the second derivative in a diffusion or dispersion model, it leads to enhanced diffusion, also called superdiffusion. We consider a one-dimensional advection–diffusion model, where the usual second-order derivative gives place to a fractional derivative of order α$\alpha$, with 1<α?2$1<\alpha ?2$. We derive explicit finite difference schemes which can be seen as generalizations of already existing schemes in the literature for the advection–diffusion equation. We present the order of accuracy of the schemes and in order to show its convergence we prove they are stable under certain conditions. In the end we present a test problem.     

Diffusion of hydrogen in metals

The present status of our understanding of the diffusion of hydrogen in metals, both experimental and theoretical, is reviewed. Discussions are focused on the mechanism of diffusion of hydrogen isotopes H, D and T in f.c.c. and b.c.c. metals; the positive muon (μ ⁺) is referred to where appropriate. An up-to-date compilation of diffusion data as a function of temperature and isotope mass has been made, and a clear distinction in general diffusion behaviour in f.c.c. and b.c.c. metals is noted. Subsequently, the results obtained from the Gorsky effect, nuclear magnetic resonance and quasi-elastic neutron scattering that provide information on elementary jump processes are discussed.

A conceptual framework of the quantum diffusion of light interstitials in metals is given, including the recent Kondo theory that emphasizes the crucial importance of particle-conduction electron interactions in the diffusion process, especially at low temperatures. It is shown with the help of recent estimates of the tunneling matrix element that the overall feature of diffusion of hydrogen isotopes in b.c.c. metals as well as μ ⁺ in f.c.c. metals can be explained consistently within the frame presented here.

Finally, recent advances in the diffusion studies on hydrogen in b.c.c. metals are described. They include a re-analysis of quench-recovery experiments that manifested nearly athermal diffusion of H, D and T in Ta at low temperatures, and an enormous enhancement of the diffusivity under stress (superdiffusion) observed for H and D in V.     

Finite-element lattice Hamiltonian matrix elements: Anharmonic oscillators

The finite-element approach to lattice field theory is both highly accurate (relative errors 1/N ², whereN is the number of lattice points) and exactly unitary (in the sense that canonical commutation relations are exactly preserved at the lattice sites). In this Letter, we construct matrix elements for the time evolution operator for the anharmonic oscillator, for which the continuum Hamiltonian isH=p ²/2+q ^2k /2k. Construction of such matrix elements does not require solving the implicit equations of motion. Low-order approximations turn out to be quite accurate. For example, the matrix element of the time evolution operator in the harmonic oscillator groundstate gives a result for thek=2 anharmonic oscillator groundstate energy accurate to better than 1% while a two-state approximation reduces the error to less than 0.1%. Accurate wavefunctions are also extracted. Analogous results may be obtained in the continuum, but there the computation is more difficult, and not generalizable to field theories in more dimensions.     

Specific heat of amorphous and polycrystalline lead-bismuth films

The specific heat of Pb₇₀Bi₃₀-films produced by quench condensation has been measured in the amorphous and the polycrystalline phase at temperatures between 0.5 K and 2 K. No evidence could be found for theT-proportional term which has been observed on many amorphous dielectrics.These results completely agree with the observations on amorphous Bi-films and confirm once more the assumption that the existence of covalent bonds in addition to the structural disorder is essential for the occurrence of a linear term in the temperature dependence of the lattice heat.     

μ + SR studies of antiferromagnetic chromiumSR studies of antiferromagnetic chromium

μ ⁺ SR measurements have been performed on Cr single crystals at temperatures 60 mK≤T≤295 K in applied magnetic fields 0≤B _appl≤1.5 T. The temperature dependence of the observed precession frequencies and transverse relaxation rates can be explained by the assumption that theμ ⁺ are hopping between adjacent tetrahedral interstices. At temperaturesT≤11 K evidence for an interaction between theμ ⁺ and the spin-density waves in Cr has been found. The directions and magnitudes of the lattice magnetic moments are unaffected by the applied magnetic fields.