First passage and first hitting times of Lévy flights and Lévy walks

Abstract For both Lévy flight and Lévy walk search processes we analyse the full distribution of first-passage and first-hitting (or first-arrival) times. These are, respectively, the times when the particle moves across a point at some given distance from its initial position for the first time, or when it lands at a given point for the first time. For Lévy motions with their propensity for long relocation events and thus the possibility to jump across a given point in space without actually hitting it (‘leapovers’), these two definitions lead to significantly different results. We study the first-passage and first-hitting time distributions as functions of the Lévy stable index, highlighting the different behaviour for the cases when the first absolute moment of the jump length distribution is finite or infinite. In particular we examine the limits of short and long times. Our results will find their application in the mathematical modelling of random search processes as well as computer algorithms

Nonstationary distributions and relaxation times in a stochastic model of memristor

We propose a stochastic model for a memristive system by generalizing known approaches and experimental results. We validate our theoretical model by experiments carried out on a memristive device based on multilayer structure. In the framework of the proposed model we obtain the exact analytic expressions for stationary and nonstationary solutions. We analyze the equilibrium and non-equilibrium steady-state distributions of the internal state variable of the memristive system and study the influence of fluctuations on the resistive switching, including the relaxation time to the steady-state. The relaxation time shows a nonmonotonic dependence, with a minimum, on the intensity of the fluctuations. This paves the way for using the intensity of fluctuations as a control parameter for switching dynamics in memristive devices

Stochastic model of memristor based on the length of conductive region

We propose a stochastic model of a voltage controlled bipolar memristive system, which includes the properties of widely used dynamic SPICE models and takes into account the fluctuations inherent in memristors. The proposed model is described by rather simple equations of Brownian diffusion, does not require significant computational resources for numerical modeling, and allows obtaining the exact analytical solutions in some cases. The noise-induced transient bimodality phenomenon, arising under resistive switching, was revealed and investigated theoretically and experimentally in a memristive system, by finding a quite good qualitatively agreement between theory and experiment. Based on the proposed model, the mathematical apparatus of Markov processes for the first passage time of the boundaries can be used to analyse the temporal characteristics of resistive switching

Oligogermanes containing only electron-withdrawing substituents: Synthesis and properties

A series of germanes Ar3GeX, containing electron-withdrawing substituents [Ar = p-FC6H4, 1a–d, 1a (X = Cl), 1b (X = Br), 1c (X = H), 1d (X = NMe2); p-F3CC6H4, 2a–d, 2a (X = Cl), 2b (X = Br), 2c (X = H), 2d (X = NMe2)], was synthesized and used to prepare symmetrical digermanes Ar3Ge–GeAr3, (p-FC6H4)3GeGe(C6H4F-p)3 (3), and (p-F3CC6H4)3GeGe(C6H4CF3-p)3 (4) and trigermane [(p-F3CC6H4)3Ge]2Ge(C6F5)2 (5) by hydrogermolysis reaction. The properties of all compounds were investigated by multinuclear NMR and for oligogermanes by UV/vis and fluorescence spectroscopy, as well as by electrochemical methods. In addition, the molecular structures of 1a, 1b, 2b, 2c, and 3–5 were studied by X-ray diffraction analysis. Compound 5 showed a significantly shifted UV/visible absorption to the red field in comparison with previously described derivatives

Compounds of Group 14 Elements with an Element–Element (E = Si, Ge, Sn) Bond: Effect of the Nature of the Element Atom

Two series of germanium compounds, (<i>p</i>-Tol)₃Ge-MMe₃ (M = Si (<b>1</b>), Ge (<b>2</b>), Sn (<b>3</b>)) and (Me₃Si)₃Ge-MPh₃ (M = Ge (<b>4</b>), Sn (<b>5</b>)), were prepared using lithium or potassium intermediates. The changing of the reaction conditions results in trigermane Ph₃Ge-Ge­(SiMe₃)₂-GePh₃ (<b>6</b>). The molecular structures of <b>1</b>, <b>2</b>, and <b>6</b> were investigated by X-ray analysis. By UV/visible spectroscopy it is established that introduction of a tin atom results in a significant bathochromic absorption shift. Furthermore, according to cyclic voltammetry, oxidation potentials decrease in the order <b>1 </b>> <b>2</b> > <b>3</b>. The electronic structures of <b>1</b>–<b>4</b> and related (Me₃Si)₃Ge-SiPh₃ were investigated by DFT calculations. Fluorescence properties of <b>1</b>–<b>3</b> were studied in the solid state and in solution; for compound <b>3</b> phosphorescence (lifetime is 4.58 ms) is observed in the solid state