0000000000931165
AUTHOR
W. Von Niessen
A Lotka-type model for oscillations in surface reactions
In this paper we introduce a reaction model on a lattice which leads to oscillations. The model consists of two monomolecular and one bimolecular reaction step and is related to the Lotka model. Despite the simple evolution rules, the model shows a complex behaviour (i.e. the appearance of oscillations). This offers us the opportunity to test different types of stochastic approximations and compare them with the results of a Monte Carlo simulation. The simulation is performed on a large lattice (L = 1024) in order to take long-range correlations into account. Comparing the results of this simulation with the stochastic approaches shows that only advanced numerical approximations are able to…
Kinetic model for surface reconstruction
Institut fu ¨r Physikalische und Theoretische Chemie, Technische Universitat Braunschweig, Hans-Sommer-Strase 10,38106 Braunschweig, Germany~Received 7 December 2001; published 25 July 2002!A microscopic kinetic model for the ab @e.g., hex131 for Pt~100! and 132131 for Pt~110!#surface reconstruction is investigated by means of the mean field approximation and Monte Carlo simulations.It considers homogeneous phase nucleation that induces small surface phase defects. These defects can grow ordecline via phase border propagation in dependence on the chemical coverage by an adsorbate A ~CO!.Anasymmetry in the adsorbate surface diffusion from one surface phase to the other gives rise to two criti…
The A + B → 0 reaction on a disordered lattice
Abstract In this paper a stochastic model for the A + B → 0 reaction with creation of particles on a disordered surface is studied for d = 2 and d = 3 spatial dimensions. Densities and correlations of the particles are examined in detail. We find that the stationary state which exists for d = 3 in case of an ordered lattice vanishes in the case of a disordered lattice. A stationary state for d = 2 never exists.
Internal Spatiotemporal Stochastic Resonance in a Microscopic Surface Reaction Model
We show the existence of internal stochastic resonance in a microscopic stochastic model for the oscillating CO oxidation on single crystal surfaces. This stochastic resonance arises directly from the elementary reaction steps of the system without any external input. The lattice gas model is investigated by means of Monte Carlo simulations. It shows oscillation phenomena and mesoscopic pattern formation. Stochastic resonance arises once homogeneous nucleation in the individual surface phases (reconstructed and non-reconstructed) is added. This nucleation is modelled as a noise process. As a result, synchronization of the kinetic oscillations is obtained. Internal stochastic resonance may t…
Global Synchronization via Homogeneous Nucleation in Oscillating Surface Reactions
The mechanism leading to globally synchronized oscillations in the $\mathrm{CO}+{\mathrm{O}}_{2}/\mathrm{Pt}\left(110\right)$ reaction system is investigated by means of Monte Carlo simulations. The model considers the reconstruction of the surface via phase border propagation and spontaneous phase nucleation. The reason for global oscillations turns out to be the spontaneous phase nucleation. This nucleation, which is modeled as a weak noise process, results in a random creation of dynamic defects and leads to global synchronization via stochastic resonance. The mechanism of global coupling via the gas phase, as it is proposed to date, does not occur.
Exact analytic solution of the multi-dimensional Anderson localization
The method proposed by the present authors to deal analytically with the problem of Anderson localization via disorder [J.Phys.: Condens. Matter {\bf 14} (2002) 13777] is generalized for higher spatial dimensions D. In this way the generalized Lyapunov exponents for diagonal correlators of the wave function, $$, can be calculated analytically and exactly. This permits to determine the phase diagram of the system. For all dimensions $D > 2$ one finds intervals in the energy and the disorder where extended and localized states coexist: the metal-insulator transition should thus be interpreted as a first-order transition. The qualitative differences permit to group the systems into two classes…
Reply to Comment on "Exact analytic solution for the generalized Lyapunov exponent of the 2-dimensional Anderson localization"
We reply to comments by P.Marko$\breve{s}$, L.Schweitzer and M.Weyrauch [preceding paper] on our recent paper [J. Phys.: Condens. Matter 63, 13777 (2002)]. We demonstrate that our quite different viewpoints stem for the different physical assumptions made prior to the choice of the mathematical formalism. The authors of the Comment expect \emph{a priori} to see a single thermodynamic phase while our approach is capable of detecting co-existence of distinct pure phases. The limitations of the transfer matrix techniques for the multi-dimensional Anderson localization problem are discussed.
The phase diagram of the multi-dimensional Anderson localization via analytic determination of Lyapunov exponents
The method proposed by the present authors to deal analytically with the problem of Anderson localization via disorder [J.Phys.: Condens. Matter {\bf 14} (2002) 13777] is generalized for higher spatial dimensions D. In this way the generalized Lyapunov exponents for diagonal correlators of the wave function, $$, can be calculated analytically and exactly. This permits to determine the phase diagram of the system. For all dimensions $D > 2$ one finds intervals in the energy and the disorder where extended and localized states coexist: the metal-insulator transition should thus be interpreted as a first-order transition. The qualitative differences permit to group the systems into two classes…
Discrete-lattice theory for Frenkel-defect aggregation in irradiated ionic solids
Institut fu ¨r Physikalische und Theoretische Chemie, Technische Universitat Braunschweig, D-38106 Braunschweig, Germany~Received 11 September 1997; revised manuscript received 6 April 1998!A microscopic theory of diffusion-controlled aggregation of radiation Frenkel defects—called in ionic solidsH and F centers—is presented. This is based on a discrete-lattice formalism for the single defect densities~concentrations! and the coupled joint densities of similar and dissimilar defects treated in terms of a modifiedKirkwood superposition approximation. The kinetics of defect aggregation is studied in detail; the cooperativecharacter of this process for both types of complementary defects is sho…
A new approach to the analytic solution of the Anderson localization problem for arbitrary dimensions
Subsequent to the ideas presented in our previous papers [J.Phys.: Condens. Matter {\bf 14} (2002) 13777 and Eur. Phys. J. B {\bf 42} (2004) 529], we discuss here in detail a new analytical approach to calculating the phase-diagram for the Anderson localization in arbitrary spatial dimensions. The transition from delocalized to localized states is treated as a generalized diffusion which manifests itself in the divergence of averages of wavefunctions (correlators). This divergence is controlled by the Lyapunov exponent $\gamma$, which is the inverse of the localization length, $\xi=1/\gamma$. The appearance of the generalized diffusion arises due to the instability of a fundamental mode cor…
Stochastic model for complex surface-reaction systems with application toNH3formation
A stochastic model is introduced that is appropriate to describe surface-reaction systems. These reaction systems are well suited for the description via master equations using their Markovian behavior. In this representation an infinite chain of master equations for the distribution functions of the state of the surface, of pairs of surface sites, etc., arises. This hierarchy is truncated by a superposition approximation. The resulting lattice equations are solved in a small region which contains all of the structure-sensitive aspects and can be connected to continuous functions which represent the behavior of the system for large distances from a reference point. In the present paper, we …
Oscillation Phenomena Leading to Chaos in a Stochastic Surface Reaction Model
A microscopic lattice gas model for the $\mathrm{CO}+\mathrm{NO}$ reaction on Pt(100) is studied by means of Monte Carlo simulations. It shows different kinetical phenomena such as steady state reaction, damped, regular, and irregular oscillations, as well as a transition into chaotical behavior via the Feigenbaum route. Because of its small number of parameters, each with a specific physical meaning, it enables the investigation of the whole parameter regime leading to a deeper insight to the mechanisms which create the oscillations and chaotical behavior.
Comment on "surface restructuring, kinetic oscillations, and chaos in heterogeneous catalytic reactions".
In a recent article Zhdanov studied the oscillating $\mathrm{NO}+{\mathrm{H}}_{2}$ reaction on the Pt(100) single-crystal surface [V. P. Zhdanov, Phys. Rev. E 59, 6292 (1999)]. We have scrutinized his model and found fundamental errors in the chemical modeling, in the modeling of the surface reconstruction and in the simulation procedure itself.
Exact analytic solution for the generalized Lyapunov exponent of the 2-dimensional Anderson localization
The Anderson localization problem in one and two dimensions is solved analytically via the calculation of the generalized Lyapunov exponents. This is achieved by making use of signal theory. The phase diagram can be analyzed in this way. In the one dimensional case all states are localized for arbitrarily small disorder in agreement with existing theories. In the two dimensional case for larger energies and large disorder all states are localized but for certain energies and small disorder extended and localized states coexist. The phase of delocalized states is marginally stable. We demonstrate that the metal-insulator transition should be interpreted as a first-order phase transition. Con…
Forced oscillations in a self-oscillating surface reaction model
A microscopic lattice gas model for the catalytic CO + O2 reaction on Pt(110) subject to external periodic forcing is studied by means of cellular automaton simulations. Harmonic resonance, subharmonic and superharmonic entrainment, quasiperiodic as well as chaotic behavior are among the observed phenomena in this model when the gas phase concentration of CO as an external control parameter is periodically varied and interacts with the self-oscillating reaction system.
Effect of reactant spatial distribution in theA+B→0reaction kinetics in one dimension with Coulomb interaction
The effect of nonequilibrium charge screening in the kinetics of the one-dimensional, diffusion-controlled $A+B\ensuremath{\rightarrow}0$ reaction between charged reactants in solids and liquids is studied. The incorrectness of the static, Debye-H\"uckel theory is shown. Our microscopic formalism is based on the Kirkwood superposition approximation for three-particle densities and the self-consistent treatment of the electrostatic interactions defined by the nonuniform spatial distribution of similar and dissimilar reactants treated in terms of the relevant joint correlation functions. Special attention is paid to the pattern formation due to a reaction-induced non-Poissonian fluctuation sp…
Random walk approach to the analytic solution of random systems with multiplicative noise—The Anderson localization problem
We discuss here in detail a new analytical random walk approach to calculating the phase-diagram for spatially extended systems with multiplicative noise. We use the Anderson localization problem as an example. The transition from delocalized to localized states is treated as a generalized diffusion with a noise-induced first-order phase transition. The generalized diffusion manifests itself in the divergence of averages of wavefunctions (correlators). This divergence is controlled by the Lyapunov exponent $\gamma$, which is the inverse of the localization length, $\xi=1/\gamma$. The appearance of the generalized diffusion arises due to the instability of a fundamental mode corresponding to…
Anderson localization problem: An exact solution for 2-D anisotropic systems
Our previous results [J.Phys.: Condens. Matter 14 (2002) 13777] dealing with the analytical solution of the two-dimensional (2-D) Anderson localization problem due to disorder is generalized for anisotropic systems (two different hopping matrix elements in transverse directions). We discuss the mathematical nature of the metal-insulator phase transition which occurs in the 2-D case, in contrast to the 1-D case, where such a phase transition does not occur. In anisotropic systems two localization lengths arise instead of one length only.