arXiv subjects

O. Biham

Publications and source records attributed to O. Biham.

At least 19 recordsLinked to original sources

arXiv2020

Analysis of the convergence of the degree distribution of contracting random networks towards a Poisson distribution using the relative entropy

We present analytical results for the structural evolution of random networks undergoing contraction processes via generic node deletion scenarios, namely, random deletion, preferential deletion and propagating deletion. Focusing on configuration model networks, which exhibit a given degree distribution $P_0(k)$ and no correlations, we show using a rigorous argument that upon contraction the degree distributions of these networks converge towards a Poisson distribution. To this end, we use the relative entropy $S_t=S[P_t(k) || \pi(k|\langle K \rangle_t)]$ of the degree distribution $P_t(k)$ of the contracting network at time $t$ with respect to the corresponding Poisson distribution $\pi(k|\langle K \rangle_t)$ with the same mean degree $\langle K \rangle_t$ as a distance measure between $P_t(k)$ and Poisson. The relative entropy is suitable as a distance measure since it satisfies $S_t \ge 0$ for any degree distribution $P_t(k)$, while equality is obtained only for $P_t(k) = \pi(k|\langle K \rangle_t)$. We derive an equation for the time derivative $dS_t/dt$ during network contraction and show that the relative entropy decreases monotonically to zero during the contraction process. We thus conclude that the degree distributions of contracting configuration model networks converge towards a Poisson distribution. Since the contracting networks remain uncorrelated, this means that their structures converge towards an Erd{\H o}s-R\'enyi (ER) graph structure, substantiating earlier results obtained using direct integration of the master equation and computer simulations [I. Tishby, O. Biham and E. Katzav, {\it Phys. Rev. E} {\bf 100}, 032314 (2019)]. We demonstrate the convergence for configuration model networks with degenerate degree distributions (random regular graphs), exponential degree distributions and power-law degree distributions (scale-free networks).

physics.soc-ph↗

arXiv2011

Diffusion-limited reactions on disordered surfaces with continuous distributions of binding energies

We study the steady state of a stochastic particle system on a two-dimensional lattice, with particle influx, diffusion and desorption, and the formation of a dimer when particles meet. Surface processes are thermally activated, with (quenched) binding energies drawn from a \emph{continuous} distribution. We show that sites in this model provide either coverage or mobility, depending on their energy. We use this to analytically map the system to an effective \emph{binary} model in a temperature-dependent way. The behavior of the effective model is well-understood and accurately describes key quantities of the system: Compared with discrete distributions, the temperature window of efficient reaction is broadened, and the efficiency decays more slowly at its ends. The mapping also explains in what parameter regimes the system exhibits realization dependence.

cond-mat.stat-mech↗

arXiv2010

Diffusion-limited reactions on a two-dimensional lattice with binary disorder

Reaction-diffusion systems where transition rates exhibit quenched disorder are common in physical and chemical systems. We study pair reactions on a periodic two-dimensional lattice, including continuous deposition and spontaneous desorption of particles. Hopping and desorption are taken to be thermally activated processes. The activation energies are drawn from a binary distribution of well depths, corresponding to `shallow' and `deep' sites. This is the simplest non-trivial distribution, which we use to examine and explain fundamental features of the system. We simulate the system using kinetic Monte Carlo methods and provide a thorough understanding of our findings. We show that the combination of shallow and deep sites broadens the temperature window in which the reaction is efficient, compared to either homogeneous system. We also examine the role of spatial correlations, including systems where one type of site is arranged in a cluster or a sublattice. Finally, we show that a simple rate equation model reproduces simulation results with very good accuracy.

cond-mat.dis-nn↗

arXiv2009

Incorporation of stochastic chemistry on dust grains in the PDR code using moment equations

Unlike gas-phase reactions, chemical reactions taking place on interstellar dust grain surfaces cannot always be modeled by rate equations. Due to the small grain sizes and low flux,these reactions may exhibit large fluctuations and thus require stochastic methods such as the moment equations. We evaluate the formation rates of H2, HD and D2 molecules on dust grain surfaces and their abundances in the gas phase under interstellar conditions. We incorporate the moment equations into the Meudon PDR code and compare the results with those obtained from the rate equations. We find that within the experimental constraints on the energy barriers for diffusion and desorption and for the density of adsorption sites on the grain surface, H2, HD and D2 molecules can be formed efficiently on dust grains. Under a broad range of conditions, the moment equation results coincide with those obtained from the rate equations. However, in a range of relatively high grain temperatures, there are significant deviations. In this range, the rate equations fail while the moment equations provide accurate results. The incorporation of the moment equations into the PDR code can be extended to other reactions taking place on grain surfaces.

astro-ph.GA↗

arXiv2008

Molecular Hydrogen Formation on Low Temperature Surfaces in Temperature Programmed Desorption Experiments

The study of the formation of molecular hydrogen on low temperature surfaces is of interest both because it allows to explore elementary steps in the heterogeneous catalysis of a simple molecule and because of the applications in astrochemistry. Here we report results of experiments of molecular hydrogen formation on amorphous silicate surfaces using temperature-programmed desorption (TPD). In these experiments beams of H and D atoms are irradiated on the surface of an amorphous silicate sample. The desorption rate of HD molecules is monitored using a mass spectrometer during a subsequent TPD run. The results are analyzed using rate equations and the activation energies of the processes leading to molecular hydrogen formation are obtained from the TPD data. We show that a model based on a single isotope provides the correct results for the activation energies for diffusion and desorption of H atoms. These results can thus be used to evaluate the formation rate of H_2 on dust grains under the actual conditions present in interstellar clouds.

physics.chem-ph↗

arXiv2007

Efficient Stochastic Simulations of Complex Reaction Networks on Surfaces

Surfaces serve as highly efficient catalysts for a vast variety of chemical reactions. Typically, such surface reactions involve billions of molecules which diffuse and react over macroscopic areas. Therefore, stochastic fluctuations are negligible and the reaction rates can be evaluated using rate equations, which are based on the mean-field approximation. However, in case that the surface is partitioned into a large number of disconnected microscopic domains, the number of reactants in each domain becomes small and it strongly fluctuates. This is, in fact, the situation in the interstellar medium, where some crucial reactions take place on the surfaces of microscopic dust grains. In this case rate equations fail and the simulation of surface reactions requires stochastic methods such as the master equation. However, in the case of complex reaction networks, the master equation becomes infeasible because the number of equations proliferates exponentially. To solve this problem, we introduce a stochastic method based on moment equations. In this method the number of equations is dramatically reduced to just one equation for each reactive species and one equation for each reaction. Moreover, the equations can be easily constructed using a diagrammatic approach. We demonstrate the method for a set of astrophysically relevant networks of increasing complexity. It is expected to be applicable in many other contexts in which problems that exhibit analogous structure appear, such as surface catalysis in nanoscale systems, aerosol chemistry in stratospheric clouds and genetic networks in cells.

cond-mat.stat-mech↗

arXiv2007

Efficient Simulations of Interstellar Gas-Grain Chemistry Using Moment Equations

Networks of reactions on dust grain surfaces play a crucial role in the chemistry of interstellar clouds, leading to the formation of molecular hydrogen in diffuse clouds as well as various organic molecules in dense molecular clouds. Due to the sub-micron size of the grains and the low flux, the population of reactive species per grain may be very small and strongly fluctuating. Under these conditions rate equations fail and the simulation of surface-reaction networks requires stochastic methods such as the master equation. However, the master equation becomes infeasible for complex networks because the number of equations proliferates exponentially. Here we introduce a method based on moment equations for the simulation of reaction networks on small grains. The number of equations is reduced to just one equation per reactive specie and one equation per reaction. Nevertheless, the method provides accurate results, which are in excellent agreement with the master equation. The method is demonstrated for the methanol network which has been recently shown to be of crucial importance.

astro-ph↗

arXiv2005

Algebraic analysis of quantum search with pure and mixed states

An algebraic analysis of Grover's quantum search algorithm is presented for the case in which the initial state is an arbitrary pure quantum state of n qubits. This approach reveals the geometrical structure of the quantum search process, which turns out to be confined to a four-dimensional subspace of the Hilbert space. This work unifies and generalizes earlier results on the time evolution of the amplitudes during the quantum search, the optimal number of iterations and the success probability. Furthermore, it enables a direct generalization to the case in which the initial state is a mixed state, providing an exact formula for the success probability.

quant-ph↗

arXiv2003

The effect of unitary noise on Grover's quantum search algorithm

The effect of unitary noise on the performance of Grover's quantum search algorithm is studied. This type of noise may result from tiny fluctuations and drift in the parameters of the (quantum) components performing the computation. The resulting operations are still unitary, but not precisely those assumed in the design of the algorithm. Here we focus on the effect of such noise in the Hadamard gate $W$, which is an essential component in each iteration of the quantum search process. To this end $W$ is replaced by a noisy Hadamard gate $U$. The parameters of $U$ at each iteration are taken from an arbitrary probability distribution (e.g. Gaussian distribution) and are characterized by their statistical moments around the parameters of $W$. For simplicity we assume that the noise is unbiased and isotropic, namely all noise variables in the parametrization we use have zero average and the same standard deviation $ε$. The noise terms at different calls to $U$ are assumed to be uncorrelated. For a search space of size $N=2^n$ (where $n$ is the number of qubits used to span this space) it is found that as long as $ε< O(n^{-{1/2}} N^{-{1/4}})$, the algorithm maintains significant efficiency, while above this noise level its operation is hampered completely. It is also found that below this noise threshold, when the search fails, it is likely to provide a state that differs from the marked state by only a few bits. This feature can be used to search for the marked state by a classical post-processing, even if the quantum search has failed, thus improving the success rate of the search process.

quant-ph↗

arXiv2003

Analysis of Grover's quantum search algorithm as a dynamical system

Grover's quantum search algorithm is analyzed for the case in which the initial state is an arbitrary pure quantum state $|ϕ>$ of $n$ qubits. It is shown that the optimal time to perform the measurement is independent of $| ϕ>$, namely, it is identical to the optimal time in the original algorithm in which $| ϕ> = | 0>$, with the same number of marked states, $r$. The probability of success $P_{\rm s}$ is obtained, in terms of the amplitudes of the state $| ϕ>$, and is shown to be independent of $r$. A class of states, which includes fixed points and cycles of the Grover iteration operator is identified. The relevance of these results in the context of using the success probability as an entanglement measure is discussed. In particular, the Groverian entanglement measure, previously limited to a single marked state, is generalized to the case of several marked states.

quant-ph↗

arXiv2000

Glassy properties and localization of interacting electrons in two-dimensional systems

We present a computer simulation study of a disordered two-dimensional system of localized interacting electrons at thermal equilibrium. It is shown that the configuration of occupied sites within the Coulomb gap persistently changes at temperatures much less than the gap width. This is accompanied by large time dependent fluctuations of the site energies. The observed thermal equilibration at low temperatures suggests a possible glass transition only at T=0. We interpret the strong fluctuations in the occupation numbers and site energies in terms of the drift of the system between multiple energy minima. The results also imply that interacting electrons may be effectively delocalized within the Coulomb gap. Insulating properties, such as hopping conduction, appear as a result of long equilibration times associated with glassy dynamics. This may shine new light on the relation between the metal-insulator transition and glassy behavior.

cond-mat.dis-nn↗

arXiv1999

Molecular Hydrogen Formation on Astrophysically Relevant Surfaces

Recent experimental results about the formation of molecular hydrogen on astrophysically relevant surfaces under conditions close to those encountered in the interstellar medium are analyzed using rate equations. The parameters of the rate equation model are fitted to temperature-programmed desorption curves obtained in the laboratory. These parameters are the activation energy barriers for atomic hydrogen diffusion and desorption, the barrier for molecular hydrogen desorption, and the probability of spontaneous desorption of a hydrogen molecule upon recombination. The model is a generalization of the Polanyi-Wigner equation and provides a description of both first and second order kinetic processes within a single model. Using the values of the parameters that fit best the experimental results, the efficiency of hydrogen recombination on olivine and amorphous carbon surfaces is obtained for a range of hydrogen flux and surface temperature pertinent to a wide range of interstellar conditions.

astro-ph↗

arXiv1999

Models for adatom diffusion on fcc(001) metal surfaces

We present a class of models that describe self diffusion on several fcc(001) metal substrates within a common framework. The models are found to apply well for Cu(001), Ag(001), Au(001), Ni(001) and Pd(001).For each of these metals the models can be used to estimate the activation energy of any diffusion process using a few basic parameters which may be obtained from experiments, ab-initio or semi-empirical calculations. To demonstrate the approach, the parameters of the models are optimized to describe self-diffusion on the (001) surface, by comparing the energy barriers to a full set of barriers obtained from semi-empirical potentials via the embedded atom method (EAM). It is found that these models with at most four parameters,provide a good description of the full landscape of hopping energy barriers on FCC(001) surfaces. The main features of the diffusion processes revealed by EAM calculations are quantitatively reproducible by the models.

cond-mat↗

arXiv1998

Systematic Computation of the Least Unstable Periodic Orbits in Chaotic Attractors

We show that a recently proposed numerical technique for the calculation of unstable periodic orbits in chaotic attractors is capable of finding the least unstable periodic orbits of any given order. This is achieved by introducing a modified dynamical system which has the same set of periodic orbits as the original chaotic system, but with a tuning parameter which is used to stabilize the orbits selectively. This technique is central for calculations using the stability criterion for the truncation of cycle expansions, which provide highly improved convergence of calculations of dynamical averages in generic chaotic attractors. The approach is demonstrated for the Henon attractor.

chao-dyn↗

arXiv1998

H2 Formation on Interstellar Grains in Different Physical Regimes

An analysis of the kinetics of H2 formation on interstellar dust grains is presented using rate equations. It is shown that semi-empirical expressions that appeared in the literature represent two different physical regimes. In particular, it is shown that the expression given by Hollenbach, Werner and Salpeter [ApJ, 163, 165 (1971)] applies when high flux, or high mobility, of H atoms on the surface of a grain, makes it very unlikely that H atoms evaporate before they meet each other and recombine. The expression of Pirronello et al.\ [ApJ, 483, L131 (1997)] -- deduced on the basis of accurate measurements on realistic dust analogue -- applies to the opposite regime (low coverage and low mobility). The implications of this analysis for the understanding of the processes dominating in the Interstellar Medium are discussed.

astro-ph↗

arXiv1998

Symmetries and Universality Classes in Conservative Sandpile Models

The symmetry properties which determine the critical exponents and universality classes in conservative sandpile models are identified. This is done by introducing a set of models, including all possible combinations of abelian vs. non-abelian, deterministic vs. stochastic and isotropic vs. anisotropic toppling rules. The universality classes are determined by an extended set of critical exponents, scaling functions and geometrical features. Two universality classes are clearly identified: (a) the universality class of abelian models and (b) the universality class of stochastic models. In addition, it is found that non-abelian models with deterministic toppling rules exhibit non-universal behavior.

cond-mat↗

arXiv1998

Universality in Sandpile Models

A new classification of sandpile models into universality classes is presented. On the basis of extensive numerical simulations, in which we measure an extended set of exponents, the Manna two state model [S. S. Manna, J. Phys. A 24, L363 (1991)] is found to belong to a universality class of random neighbor models which is distinct from the universality class of the original model of Bak, Tang and Wiesenfeld [P. Bak, C. Tang and K. Wiensenfeld, Phys. Rev. Lett. 59, 381 (1987)]. Directed models are found to belong to a universality class which includes the directed model introduced and solved by Dhar

cond-mat.stat-mech↗

arXiv1998

Universality Classes in Isotropic, Abelian and non-Abelian, Sandpile Models

Universality in isotropic, abelian and non-abelian, sandpile models is examined using extensive numerical simulations. To characterize the critical behavior we employ an extended set of critical exponents, geometric features of the avalanches, as well as scaling functions describing the time evolution of average quantities such as the area and size during the avalanche. Comparing between the abelian Bak-Tang-Wiesenfeld model [P. Bak, C. Tang and K. Wiensenfeld, Phys. Rev. Lett. 59, 381 (1987)], and the non-abelian models introduced by Manna [S. S. Manna, J. Phys. A. 24, L363 (1991)] and Zhang [Y. C. Zhang, Phys. Rev. Lett. 63, 470 (1989)] we find strong indications that each one of these models belongs to a distinct universality class.

cond-mat.stat-mech↗