J. L. Vega

Kramers’ turnover theory for diffusion of Na atoms on a Cu(001) surface measured by He scattering

The diffusion of adatoms and molecules on a surface at low coverage can be measured by helium scattering. The experimental observable is the dynamic structure factor. In this article, we show how Kramers’ turnover theory can be used to infer physical properties of the diffusing particle from the experiment. Previously, Chudley and Elliot showed, under reasonable assumptions, that the dynamic structure factor is determined by the hopping distribution of the adsorbed particle. Kramers’ theory determines the hopping distribution in terms of two parameters only. These are an effective frequency and the energy loss of the particle to the bath as it traverses from one barrier to the next. Kramers’ theory, including finite barrier corrections, is tested successfully against numerical Langevin equation simulations, using both separable and nonseparable interaction potentials. Kramers’ approach, which really is a steepest descent estimate for the rate, based on the Langevin equation, involves closed analytical expressions and so is relatively easy to implement. Diffusion of Na atoms on a Cu(001) surface has been chosen as an example to illustrate the application of Kramers’ theory.

Collisional line shapes for low frequency vibrations of adsorbates on a metal surface

The dynamics of atoms or molecules adsorbed on a metal surface, and excited by collisions with an atomic beam, are treated within a theory that includes energy dissipation into lattice vibrations by means of a frequency and temperature dependent friction function. The theory provides dynamic structure factors for energy transfer derived from collisional time correlation functions. It describes the relaxation of a vibrationally excited atom or molecule within a model of a damped quantum harmonic oscillator bilinearly coupled to a bath of lattice oscillators. The collisional time correlation function is generalized to include friction effects and is applied to the vibrational relaxation of the frustrated translation mode of Na adsorbed on a Cu(001) surface, CO on Cu(001), and CO on Pt(111), following excitation by collisions with He atoms. Results for the frequency shift and width of line shapes versus surface temperature are in very good agreement with experimental measurements of inelastic He atom scattering. Our interpretation of the experimental results provides insight on the relative role of phonon versus electron-hole relaxation.

Damage spreading in the Bak-Sneppen model

We explain the results recently obtained for the damage-spreading behaviour in the Bak-Sneppen (BS) model (Tamarit et al 1998 Eur. Phys. J. B 1 545). We do this by relating the BS model to a much simpler one, which includes many features of the BS model and provides a clear explanation for the occurrence of power-law growth of the distance.

Quasielastic and low vibrational lineshapes in atom-surface diffusion

The diffusion and low frequency vibrational motions of atoms and molecules on surfaces can be measured by means of quasielastic helium atom scattering. In this paper, we discuss and investigate different analytical approximations, based on the theory of stochastic processes, to the dynamic structure factor, considering the two motions on an equal footing. Special emphasis is put on the nature of the corresponding lineshapes, explained in terms of the motional narrowing effect. We also discuss the influence of the diffusional and vibrational coupling and several ways of experimentally separating the two types of contributions to the dynamic structure factor. In particular, we propose that the so-called inelastic focusing singularity from atom–surface scattering controls the lineshapes of quasielastic and vibrational peaks.

Surface diffusion and low vibrational motion with interacting adsorbates: a shot noise description

Here, an approach in terms of shot noise is proposed to study and characterize surface diffusion and low vibrational motion when having interacting adsorbates on surfaces. In what we call statistical limit, that is, at long times and high number of collisions, one expects that diffusing particles display an essential Markovian behavior. Accordingly, the action of the pairwise potentials accounting for particle-particle collisions is equivalent to considering a shot noise acting on a single particle. We call this approach the interacting single adsorbate approximation, which gathers three important advantages: (i) the dynamics underlying surface diffusion and low vibrational motion can be easily understood in terms of relatively simple stochastic processes; (ii) from our model, appropriate (and well justified) working formulas are easily obtained, which explain the results arising from more complicated (but commonly used) molecular dynamics simulations within the Langevin formulation; and (iii), at the same time, it is less demanding computationally than the latter type of calculations. In order to illustrate the application of this model, numerical results are presented. Specially, our model reproduces the experimental observation regarding the broadening of the quasielastic peak ruling surface diffusion.

Generalized Chudley-Elliott vibration-jump model in activated atom surface diffusion

Here the authors provide a generalized Chudley-Elliott expression for the activated atom surface diffusion which takes into account the coupling between both low-frequency vibrational motion (namely, the frustrated translational modes) and diffusion. This expression is derived within the Gaussian approximation framework for the intermediate scattering function at low coverage. Moreover, inelastic contributions (arising from creation and annihilation processes) to the full width at half maximum of the quasielastic peak are also obtained.

Self-organized criticality driven by deterministic rules

Complex extended systems showing critical behavior, a lack of scale in their features, appear to be widespread in nature, being as diverse as earthquakes @1#, creep phenomena @2#, material fracturing @3‐5#, fluid displacement in porous media @6,7#, interface growth @8,9#, river networks @10‐12#, and biological evolution @13‐16#. At variance with equilibrium statistical mechanics, these systems do not need any fine tuning of a parameter to be in a critical state. To explain this behavior, Bak, Tang, and Wiesenfeld introduced the concept of self-organized criticality ~SOC! through the simple sandpile model @17,18#. In recent years, several models with extremal dynamics have been shown to exhibit SOC when noise is present @19#. In this Brief Report we show that for this class of systems, noise can be replaced by either a chaotic or a quasiperiodic signal without destroying criticality. To illustrate this point we consider, as an example, the model proposed by Bak and Sneppen to describe the coevolution of natural species @19#. The result that different, even deterministic, microscopic rules can induce SOC in the collective behavior of a population points to a greater relevance of SOC in nature. In the Bak-Sneppen ~BS! model an ecosystem is described by a one-dimensional lattice, every site of which is occupied by a species. Species with stronger mutual interactions in the ecosystem are arranged on nearest-neighbor sites ~the lattice can be interpreted as a food chain or as a food web in more than one dimension!. Each species is characterized by its fitness, describing the average number of offsprings an individual of that species can have in the given environment. This definition of the fitness also accounts for the greater resistance to mutations of fitter species since mutations must propagate over a greater number of individuals to become a genetic trait of the species. Thus the species with the lowest fitness is the one that feels the strongest evolutionary pressure @14‐16#. Its fate is to either evolve or get extinct, and its place will be taken by some newcomer species in the same ecological niche. Therefore, the fitness of the species occupying that site is the most likely to change in a short time. The nearest-neighbor species will find a different environment and their fitnesses will result changed too. As a result of such a simple dynamical rule, the system exhibits sequences of causally connected evolutionary events called avalanches @19#. The number of avalanches N follows a power-law distribution N~ s!;s 2t , ~1! where s is the size of the avalanche and t;1.07 @20,21# is the avalanche critical exponent. This kind of behavior, which is the essence of self-organized criticality, has actually been observed in paleontological data @13#, suggesting that evolution and extinction may be episodic at all scales ~a feature that goes under the name of punctuated equilibrium !@ 14‐ 16#. In nature, the evolution of the least-fit species is due to genetic mutation. In the BS model, this mutation is realized by giving to the corresponding species a random fitness. As shown in @19#, each lattice site j is assigned a fitness, namely, a random number between 0 and 1. At each time step in the simulation the smallest fitness is found. Then the fitnesses of the minimum and of the two nearest neighbors are updated according to the rule

Hamiltonian theory for vibrational line shapes of atoms adsorbed on surfaces

The vibrational motions of atomic adsorbates on surfaces can be probed by helium atom scattering. The experimental observable is the dynamic structure factor, which shows an inelastic peak around the vibrational frequency of the isolated adsorbates known as the frustrated translational or T-mode peak. In this paper we develop a theory for the line shape of this peak, as well as for its temperature-dependent shift and broadening, based on a Hamiltonian equivalent of the generalized Langevin equation. The theory can be used to infer physical parameters of the adatom-surface interaction, such as the friction coefficient, the barrier height to diffusion, and the anharmonicity parameter. Numerical simulations are used to ascertain the range of validity of the theory, which is also generalized to describe multidimensional systems and to include quantum corrections. We compare the theoretical predictions for the shift and broadening with experimental results for the Na/Cu(001) system, showing quantitative agreement within experimental resolution.

Self-organized criticality in deterministic systems with disorder

Using the Bak-Sneppen model of biological evolution as our paradigm, we investigate in which cases noise can be substituted with a deterministic signal without destroying Self-Organized Criticality (SOC). If the deterministic signal is chaotic the universality class is preserved; some non-universal features, such as the threshold, depend on the time correlation of the signal. We also show that, if the signal introduced is periodic, SOC is preserved but in a different universality class, as long as the spectrum of frequencies is broad enough.

Stochastic Description of Activated Surface Diffusion with Interacting Adsorbates

Activated surface diffusion on metal surfaces is receiving much attention both experimentally and theoretically. One of the main theoretical problems in this field is to explain the line-shape broadening observed when the surface coverage is increased. Recently, we have proposed a fully stochastic model, the interacting single adsorbate (ISA) model, aimed at explaining and understanding this type of experiments, which essentially consists of considering the classical Langevin formulation with two types of noise forces: (i) a Gaussian white noise accounting for the substrate friction, and (ii) a shot noise simulating the interacting adsorbates at different coverages. No interaction potential between adsorbates is included because any trace of microscopic interaction seems to be wiped out in a Markovian regime. This model describes in a good approximation, and at a very low computational cost, the line-shape broadening observed experimentally. Furthermore, its mathematical simplicity also allows to derive some analytical expressions which are of much help in the interpretation of the physics underlying surface diffusion processes.