Efficient simulations of gas-grain chemistry in interstellar clouds

Chemical reactions on dust grains are of crucial importance in interstellar chemistry because they produce molecular hydrogen and various organic molecules. Due to the submicron size of the grains and the low flux, the surface populations of reactive species are small and strongly fluctuate. Under these conditions rate equations fail and the master equation is needed for modeling these reactions. However, the number of equations in the master equation grows exponentially with the number of reactive species, severely limiting its feasibility. Here we present a method which dramatically reduces the number of equations, thus enabling the incorporation of the master equation in models of interstellar chemistry.Comment: 10 pages + 3 figure

Binomial moment equations for stochastic reaction systems

A highly efficient formulation of moment equations for stochastic reaction networks is introduced. It is based on a set of binomial moments that capture the combinatorics of the reaction processes. The resulting set of equations can be easily runcated to include moments up to any desired order. The number of equations is dramatically reduced compared to the master equation. This formulation enables the simulation of complex reaction networks, involving a large number of reactive species much beyond the feasibility limit of any existing method. It provides an equation-based paradigm to the analysis of stochastic networks, complementing the commonly used Monte Carlo simulations.Comment: 3 figure

Groverian Entanglement Measure of Pure Quantum States with Arbitrary Partitions

The Groverian entanglement measure of pure quantum states of $n$ qubits is generalized to the case in which the qubits are divided into any $m \le n$ parties and the entanglement between these parties is evaluated. To demonstrate this measure we apply it to general states of three qubits and to symmetric states with any number of qubits such as the Greenberg-Horne-Zeiliner state and the W state.Comment: 5 pages, 2 figures, 1 tabl

Generic Emergence of Power Law Distributions and L\'evy-Stable Intermittent Fluctuations in Discrete Logistic Systems

The dynamics of generic stochastic Lotka-Volterra (discrete logistic) systems of the form \cite{Solomon96a} $w_i (t+1) = \lambda(t) w_i (t) + a {\bar w (t)} - b w_i (t) {\bar w(t)}$ is studied by computer simulations. The variables $w_i$, $i=1,...N$, are the individual system components and ${\bar w (t)} = {1\over N} \sum_i w_i (t)$ is their average. The parameters $a$ and $b$ are constants, while $\lambda(t)$ is randomly chosen at each time step from a given distribution. Models of this type describe the temporal evolution of a large variety of systems such as stock markets and city populations. These systems are characterized by a large number of interacting objects and the dynamics is dominated by multiplicative processes. The instantaneous probability distribution $P(w,t)$ of the system components $w_i$, turns out to fulfill a (truncated) Pareto power-law $P(w,t) \sim w^{-1-\alpha}$. The time evolution of ${\bar w (t)}$ presents intermittent fluctuations parametrized by a truncated L\'evy distribution of index $\alpha$, showing a connection between the distribution of the $w_i$'s at a given time and the temporal fluctuations of their average.Comment: 18 pages and 5 figures (in one zipped file); [email protected], [email protected], [email protected], [email protected], http://shum.huji.ac.il/~sori

Theoretical Analysis and Simulations of the Generalized Lotka-Volterra Model

The dynamics of generalized Lotka-Volterra systems is studied by theoretical techniques and computer simulations. These systems describe the time evolution of the wealth distribution of individuals in a society, as well as of the market values of firms in the stock market. The individual wealths or market values are given by a set of time dependent variables $w_i$, $i=1,...N$. The equations include a stochastic autocatalytic term (representing investments), a drift term (representing social security payments) and a time dependent saturation term (due to the finite size of the economy). The $w_i$'s turn out to exhibit a power-law distribution of the form $P(w) \sim w^{-1-\alpha}$. It is shown analytically that the exponent $\alpha$ can be expressed as a function of one parameter, which is the ratio between the constant drift component (social security) and the fluctuating component (investments). This result provides a link between the lower and upper cutoffs of this distribution, namely between the resources available to the poorest and those available to the richest in a given society. The value of $\alpha$ %as well as the position of the lower cutoff is found to be insensitive to variations in the saturation term, that represent the expansion or contraction of the economy. The results are of much relevance to empirical studies that show that the distribution of the individual wealth in different countries during different periods in the 20th century has followed a power-law distribution with $1 < \alpha < 2$

Scaling Range and Cutoffs in Empirical Fractals

Fractal structures appear in a vast range of physical systems. A literature survey including all experimental papers on fractals which appeared in the six Physical Review journals (A-E and Letters) during the 1990's shows that experimental reports of fractal behavior are typically based on a scaling range $\Delta$ which spans only 0.5 - 2 decades. This range is limited by upper and lower cutoffs either because further data is not accessible or due to crossover bends. Focusing on spatial fractals, a classification is proposed into (a) aggregation; (b) porous media; (c) surfaces and fronts; (d) fracture and (e) critical phenomena. Most of these systems, [except for class (e)] involve processes far from thermal equilibrium. The fact that for self similar fractals [in contrast to the self affine fractals of class (c)] there are hardly any exceptions to the finding of $\Delta \le 2$ decades, raises the possibility that the cutoffs are due to intrinsic properties of the measured systems rather than the specific experimental conditions and apparatus. To examine the origin of the limited range we focus on a class of aggregation systems. In these systems a molecular beam is deposited on a surface, giving rise to nucleation and growth of diffusion-limited-aggregation-like clusters. Scaling arguments are used to show that the required duration of the deposition experiment increases exponentially with $\Delta$. Furthermore, using realistic parameters for surfaces such as Al(111) it is shown that these considerations limit the range of fractal behavior to less than two decades in agreement with the experimental findings. It is conjectured that related kinetic mechanisms that limit the scaling range are common in other nonequilibrium processes which generate spatial fractals.Comment: 15 pages, 8 figures, 1 table. This paper also contains the histograms relevant for "The limited Scaling Range of Empirical Fractals": http://xxx.lanl.gov/ps/cond-mat/980103

The distribution of first hitting times of random walks on directed Erd\H{o}s-R\'enyi networks

We present analytical results for the distribution of first hitting times of random walkers (RWs) on directed Erd\H{o}s-R\'enyi (ER) networks. Starting from a random initial node, a random walker hops randomly along directed edges between adjacent nodes in the network. The path terminates either by the retracing scenario, when the walker enters a node which it has already visited before, or by the trapping scenario, when it becomes trapped in a dead-end node from which it cannot exit. The path length, namely the number of steps, $d$, pursued by the random walker from the initial node up to its termination, is called the first hitting time. Using recursion equations, we obtain analytical results for the tail distribution of first hitting times, $P(d>\ell)$. The distribution $P(d>\ell)$ can be expressed as a product of an exponential distribution and a Rayleigh distribution. We obtain expressions for the mean, median and standard deviation of this distribution in terms of the network size and its mean degree. We also calculate the distribution of last hitting times, namely the path lengths of self-avoiding walks on directed ER networks, which do not retrace their paths. The last hitting times are found to be much longer than the first hitting times. The results are compared to those obtained for undirected ER networks. It is found that the first hitting times of RWs in a directed ER network are much longer than in the corresponding undirected network. This is due to the fact that RWs on directed networks do not exhibit the backtracking scenario, which is a dominant termination mechanism of RWs on undirected networks. It is shown that our approach also applies to a broader class of networks, referred to as semi-ER networks, in which the distribution of in-degrees is Poisson, while the out-degrees may follow any desired distribution with the same mean as the in-degree distribution.Comment: 30 pages, 9 figures. arXiv admin note: text overlap with arXiv:1609.08375, arXiv:1606.0156

Pattern Formation and a Clustering Transition in Power-Law Sequential Adsorption

A new model that describes adsorption and clustering of particles on a surface is introduced. A {\it clustering} transition is found which separates between a phase of weakly correlated particle distributions and a phase of strongly correlated distributions in which the particles form localized fractal clusters. The order parameter of the transition is identified and the fractal nature of both phases is examined. The model is relevant to a large class of clustering phenomena such as aggregation and growth on surfaces, population distribution in cities, plant and bacterial colonies as well as gravitational clustering.Comment: 4 pages, 5 figure

Entanglement of Periodic States, the Quantum Fourier Transform and Shor's Factoring Algorithm

The preprocessing stage of Shor's algorithm generates a class of quantum states referred to as periodic states, on which the quantum Fourier transform is applied. Such states also play an important role in other quantum algorithms that rely on the quantum Fourier transform. Since entanglement is believed to be a necessary resource for quantum computational speedup, we analyze the entanglement of periodic states and the way it is affected by the quantum Fourier transform. To this end, we derive a formula that evaluates the Groverian entanglement measure for periodic states. Using this formula, we explain the surprising result that the Groverian entanglement of the periodic states built up during the preprocessing stage is only slightly affected by the quantum Fourier transform.Comment: 21 pages, 3 figure

The distribution of path lengths of self avoiding walks on Erd\H{o}s-R\'enyi networks

We present an analytical and numerical study of the paths of self avoiding walks (SAWs) on random networks. Since these walks do not retrace their paths, they effectively delete the nodes they visit, together with their links, thus pruning the network. The walkers hop between neighboring nodes, until they reach a dead-end node from which they cannot proceed. Focusing on Erd\H{o}s-R\'enyi networks we show that the pruned networks maintain a Poisson degree distribution, $p_t(k)$, with an average degree, $\langle k \rangle_t$, that decreases linearly in time. We enumerate the SAW paths of any given length and find that the number of paths, $n_T(\ell)$, increases dramatically as a function of $\ell$. We also obtain analytical results for the path-length distribution, $P(\ell)$, of the SAW paths which are actually pursued, starting from a random initial node. It turns out that $P(\ell)$ follows the Gompertz distribution, which means that the termination probability of an SAW path increases with its length.Comment: 24 pages, 11 figure