Long-range Jumps Research Articles

Recurrence and transience of symmetric random walks with long-range jumps

Recurrence and transience of symmetric random walks with long-range jumps

Infinite ergodic theory for three heterogeneous stochastic models with application to subrecoil laser cooling.

We compare ergodic properties of the kinetic energy for three stochastic models of subrecoil-laser-cooled gases. One model is based on a heterogeneous random walk (HRW), another is an HRW with long-range jumps (the exponential model), and the other is a mean-field-like approximation of the exponential model (the deterministic model). All the models show an accumulation of the momentum at zero in the long-time limit, and a formal steady state cannot be normalized, i.e., there exists an infinite invariant density. We obtain the exact form of the infinite invariant density and the scaling function for the exponential and deterministic models, and we devise a useful approximation for the momentum distribution in the HRW model. While the models are kinetically nonidentical, it is natural to wonder whether their ergodic properties share common traits, given that they are all described by an infinite invariant density. We show that the answer to this question depends on the type of observable under study. If the observable is integrable, the ergodic properties, such as the statistical behavior of the time averages, are universal as they are described by the Darling-Kac theorem. In contrast, for nonintegrable observables, the models in general exhibit nonidentical statistical laws. This implies that focusing on nonintegrable observables, we discover nonuniversal features of the cooling process, which hopefully can lead to a better understanding of the particular model most suitable for a statistical description of the process. This result is expected to hold true for many other systems, beyond laser cooling.

Quenched invariance principle for a class of random conductance models with long-range jumps

We study random walks on \({\mathbb {Z}}^d\) (with \(d\ge 2\)) among stationary ergodic random conductances \(\{C_{x,y}:x,y\in {\mathbb {Z}}^d\}\) that permit jumps of arbitrary length. Our focus is on the quenched invariance principle (QIP) which we establish by a combination of corrector methods, functional inequalities and heat-kernel technology assuming that the p-th moment of \(\sum _{x\in {\mathbb {Z}}^d}C_{0,x}|x|^2\) and q-th moment of \(1/C_{0,x}\) for x neighboring the origin are finite for some \(p,q\ge 1\) with \(p^{-1}+q^{-1}<2/d\). In particular, a QIP thus holds for random walks on long-range percolation graphs with connectivity exponents larger than 2d in all \(d\ge 2\), provided all the nearest-neighbor edges are present. Although still limited by moment conditions, our method of proof is novel in that it avoids proving everywhere-sublinearity of the corrector. This is relevant because we show that, for long-range percolation with exponents between \(d+2\) and 2d, the corrector exists but fails to be sublinear everywhere. Similar examples are constructed also for nearest-neighbor, ergodic conductances in \(d\ge 3\) under the conditions complementary to those of the recent work of Bella and Schäffner (Ann Probab 48(1):296–316, 2020). These examples elucidate the limitations of elliptic-regularity techniques that underlie much of the recent progress on these problems.

Mathematical home burglary model with stochastic long crime trips and patrolling: Applied to Mexico City

Mathematical models for predicting the geographical distribution of areas with high rates of home burglary and numerical experiments to evaluate the effectiveness of police patrol routing strategies in reducing crime can support security policymakers in planning more effective patrol routes. We give an overview of the model formulated by Jones et al. (2010) to study the effects of the presence of law enforcement on the formation of home-burglary hotspots. We propose an extension of the model to contemplate that a small proportion of burglars travel far away from their awareness space to burgle houses. We incorporate these trips in the form of long-range stochastic jumps that are biased towards more attractive sites. Simulations suggest that hotspots lose attractiveness as police more influences via deterrence on burglars’ decision making about returning home rather than burglarizing a house. The results indicate that a higher proportion of burglars making long crime journeys might counteract the deterrent effect form the police presence. A case study is carried out to predict at the zone and citywide levels the distribution of areas with the highest home burglary rates in Mexico City. The L-BFGS method is applied to get the values for the parameters in both the Jones et al. model and our model; we use data from home burglaries reported, patrol zones and prison admissions. The minimum of the objective function is slightly lower for our model than for the Jones et al. model. Results obtained through numerical simulations using our model better fit the spatial statistical distribution of home-burglary hotspots for the entire city than for the zones.

Muon Spectroscopy for Investigating Diffusion in Energy Storage Materials

We review recent applications of positive muon spin relaxation (μSR) spectroscopy as an active probe of ion diffusion in energy storage materials. μSR spectroscopy allows the study of ionic diffusion in solid-state materials on a time scale between 10−5 and 10−8 s where most long-range and consecutive short-range jumps of ions between interstitial sites occur. μSR also allows one to probe and model ionic diffusion in materials that contain magnetic ions, since both electronic and nuclear contributions to the muon depolarization can be separated, making μSR an excellent technique for the microscopic study of the ionic motions in crystalline materials. We highlight a series of battery materials for which μSR has provided insight into intrinsic ionic conduction and magnetic properties without interference of external factors, such as the presence of magnetic ions, macroscopic particle morphologies, or elaborate measurement setups.

The fractional p-Laplacian emerging from homogenization of the random conductance model with degenerate ergodic weights and unbounded-range jumps

We study a general class of discrete p-Laplace operators in the random conductance model with long-range jumps and ergodic weights. Using a variational formulation of the problem, we show that under the assumption of bounded first moments and a suitable lower moment condition on the weights, the homogenized limit operator is a fractional p-Laplace operator. Under strengthened lower moment conditions, we can apply our insights also to the spectral homogenization of the discrete Laplace operator to the continuous fractional Laplace operator.

An SIS epidemic model with vaccination in a dynamical contact network of mobile individuals with heterogeneous spatial constraints

Network-based epidemic models have been extensively employed to understand the spread of infectious diseases, but have generally overlooked the fact that most realistic networks are dynamical rather than static. In this paper, we study a susceptible-infected-susceptible epidemic model with vaccination in a dynamical contact network of moving individuals, where we regard mobile individuals as random walkers that are allowed to perform long-range jumps. Different from previous studies of epidemics in a random walk network with a constant interaction radius, we consider the scenario where the individuals have a heterogeneous distribution of interaction radius r and infected individuals are vaccinated with a probability depending on the interaction radius distribution. We derive the basic reproduction number R0, which is strongly related to the interaction radius distribution and is proportional to the second order moment of interaction radius ⟨r2⟩ in the special case of a constant vaccination rate. We argue that if R0<1 then the disease-free equilibrium is locally asymptotically stable, whereas if R0>1 then there is a unique endemic equilibrium which is locally asymptotically stable and uniformly persistent. In addition, we provide a sufficient condition for the global stability of the disease-free equilibrium. Both theoretical and simulation results reveal that the distribution of individual interaction radius has significant effects on the basic reproduction number and the final epidemic prevalence. In general, the disease will break out more readily in the population with a more heterogeneous radius distribution, while it will end in a lower epidemic prevalence. Interestingly, the results suggest that an optimal vaccination intervention for disease prevention and control is achievable regardless of the radius distribution. Furthermore, some interesting results on the structure of the underlying contact network are shown to have strong correlation with the epidemic dynamics. This study provides potential implications for developing efficient containment measures against infectious disease while considering the spatial constraints of moving individuals.

CoCo-MD: A Simple and Effective Method for the Enhanced Sampling of Conformational Space.

CoCo ("complementary coordinates") is a method for ensemble enrichment based on principal component analysis (PCA) that was developed originally for the investigation of NMR data. Here we investigate the potential of the CoCo method, in combination with molecular dynamics simulations (CoCo-MD), to be used more generally for the enhanced sampling of conformational space. Using the alanine penta-peptide as a model system, we find that an iterative workflow, interleaving short multiple-walker MD simulations with long-range jumps through conformational space informed by CoCo analysis, can increase the rate of sampling of conformational space up to 10 times for the same computational effort (total number of MD timesteps). Combined with the reservoir-REMD method, free energies can be readily calculated. An alternative, approximate but fast and practically useful, alternative approach to unbiasing CoCo-MD generated data is also described. Applied to cyclosporine A, we can achieve far greater conformational sampling than has been reported previously, using a fraction of the computational resource. Simulations of the maltose binding protein, begun from the "open" state, effectively sample the "closed" conformation associated with ligand binding. The PCA-based approach means that optimal collective variables to enhance sampling need not be defined in advance by the user but are identified automatically and are adaptive, responding to the characteristics of the developing ensemble. In addition, the approach does not require any adaptations to the associated MD code and is compatible with any conventional MD package.

The velocity of 1d Mott variable-range hopping with external field

Mott variable-range hopping is a fundamental mechanism for low–temperature electron conduction in disordered solids in the regime of Anderson localization. In a mean field approximation, it reduces to a random walk (shortly, Mott random walk) on a random marked point process with possible long-range jumps. We consider here the one-dimensional Mott random walk and we add an external field (or a bias to the right). We show that the bias makes the walk transient, and investigate its linear speed. Our main results are conditions for ballisticity (positive linear speed) and for sub-ballisticity (zero linear speed), and the existence in the ballistic regime of an invariant distribution for the environment viewed from the walker, which is mutually absolutely continuous with respect to the original law of the environment. If the point process is a renewal process, the aforementioned conditions result in a sharp criterion for ballisticity. Interestingly, the speed is not always continuous as a function of the bias. Keywords: random walk in random environment, disordered media, ballisticity, environment viewed from the walker, electron transport in disordered solids.

Spatio-temporal patterns of highly pathogenic avian influenza virus subtype H5N8 spread, France, 2016 to 2017.

IntroductionFrance is one of Europe’s foremost poultry producers and the world’s fifth largest producer of poultry meat. In November 2016, highly pathogenic avian influenza (HPAI) virus subtype H5N8 emerged in poultry in the country. As of 23 March 2017, a total of 484 confirmed outbreaks were reported, with consequences on animal health and socio-economic impacts for producers. Methods: We examined the spatio-temporal distribution of outbreaks that occurred in France between November 2016 and March 2017, using the space–time K-function and space–time permutation model of the scan statistic test. Results: Most outbreaks affected duck flocks in south-west France. A significant space–time interaction of outbreaks was present at the beginning of the epidemic within a window of 8 km and 13 days. This interaction disappeared towards the epidemic end. Five spatio-temporal outbreak clusters were identified in the main poultry producing areas, moving sequentially from east to west. The average spread rate of the epidemic front wave was estimated to be 5.5 km/week. It increased from February 2017 and was negatively associated with the duck holding density. Conclusion: HPAI-H5N8 infections varied over time and space in France. Intense transmission events occurred at the early stages of the epidemic, followed by long-range jumps in the disease spread towards its end. Findings support strict control strategies in poultry production as well as the maintenance of high biosecurity standards for poultry holdings. Factors and mechanisms driving HPAI spread need to be further investigated.

Navigation by anomalous random walks on complex networks.

Anomalous random walks having long-range jumps are a critical branch of dynamical processes on networks, which can model a number of search and transport processes. However, traditional measurements based on mean first passage time are not useful as they fail to characterize the cost associated with each jump. Here we introduce a new concept of mean first traverse distance (MFTD) to characterize anomalous random walks that represents the expected traverse distance taken by walkers searching from source node to target node, and we provide a procedure for calculating the MFTD between two nodes. We use Lévy walks on networks as an example, and demonstrate that the proposed approach can unravel the interplay between diffusion dynamics of Lévy walks and the underlying network structure. Moreover, applying our framework to the famous PageRank search, we show how to inform the optimality of the PageRank search. The framework for analyzing anomalous random walks on complex networks offers a useful new paradigm to understand the dynamics of anomalous diffusion processes, and provides a unified scheme to characterize search and transport processes on networks.

Effective-medium approximation for lattice random walks with long-range jumps.

We consider the random walk on a lattice with random transition rates and arbitrarily long-range jumps. We employ Bruggeman's effective-medium approximation (EMA) to find the disorder-averaged (coarse-grained) dynamics. The EMA procedure replaces the disordered system with a cleverly guessed reference system in a self-consistent manner. We give necessary conditions on the reference system and discuss possible physical mechanisms of anomalous diffusion. In the case of a power-law scaling between transition rates and distance, lattice variants of Lévy-flights emerge as the effective medium, and the problem is solved analytically, bearing the effective anomalous diffusivity. Finally, we discuss several example distributions and demonstrate very good agreement with numerical simulations.

Comparative effects of avoidance and immunization on epidemic spreading in a dynamic small-world network with community structure

Considering the actual behavior of people’s short-term travel, we propose a dynamic small-world community network model with tunable community strength which has constant local links and time varying long-range jumps. Then an epidemic model of susceptible-infected-recovered is established based on the mean-field method to evaluate the inhibitory effects of avoidance and immunization on epidemic spreading. And an approximate formula for the epidemic threshold is obtained by mathematical analysis. The simulation results show that the epidemic threshold decreases with the increase of inner-community motivation rate and inter-community long-range motivation rate, while it increases with the increase of immunization rate or avoidance rate. It indicates that the inhibitory effect on epidemic spreading of immunization works better than that of avoidance.

Differential Microscopic Mobility of Components within a Deep Eutectic Solvent.

From macroscopic measurements of deep eutectic solvents such as glyceline (1:2 molar ratio of choline chloride to glycerol), the long-range translational diffusion of the larger cation (choline) is known to be slower compared to that of the smaller hydrogen bond donor (glycerol). However, when the diffusion dynamics are analyzed on the subnanometer length scale, we find that the displacements associated with the localized diffusive motions are actually larger for choline. This counterintuitive diffusive behavior can be understood as follows. The localized diffusive motions confined in the transient cage of neighbor particles, which precede the cage-breaking long-range diffusion jumps, are more spatially constrained for glycerol than for choline because of the stronger hydrogen bonds the former makes with chloride anions. The implications of such differential localized mobility of the constituents should be especially important for applications where deep eutectic solvents are confined on the nanometer length scale and their long-range translational diffusion is strongly inhibited (e.g., within microporous media).

Dynamic blockage in an exclusion process

We study an asymmetric exclusion model with one dynamic roadblock particle. The roadblock particle is allowed to move diffusively as well as by long-range jumps mimicking an unbinding/rebinding process. Using Monte Carlo simulations and analytical arguments, the two types of roadblock moves are considered both separately and in combination. Several interesting dynamic phenomena are observed. The long-range jumps of the roadblock lead to a current that depends on the number of particles in the system rather than on the particle density, and thus scales linearly with the system size (up to a critical size). This behavior can be explained by a collective motion of all particles following the roadblock. The diffusive roadblock movements on the other hand lead to a ratcheting motion with the active (driven) particles pushing the roadblock forward.

Fractional Diffusion Equations for Lattice and Continuum: Grünwald-Letnikov Differences and Derivatives Approach

Fractional diffusion equations for three-dimensional lattice models based on fractional-order differences of the Grünwald-Letnikov type are suggested. These lattice fractional diffusion equations contain difference operators that describe long-range jumps from one lattice site to another. In continuum limit, the suggested lattice diffusion equations with noninteger order differences give the diffusion equations with the Grünwald-Letnikov fractional derivatives for continuum. We propose a consistent derivation of the fractional diffusion equation with the fractional derivatives of Grünwald-Letnikov type. The suggested lattice diffusion equations can be considered as a new microstructural basis of space-fractional diffusion in nonlocal media.

Fractional Liouville equation on lattice phase-space

In this paper we propose a lattice analog of phase-space fractional Liouville equation. The Liouville equation for phase-space lattice with long-range jumps of power-law types is suggested. We prove that the continuum limit transforms this lattice equation into Liouville equation with conjugate Riesz fractional derivatives of non-integer orders with respect to coordinates of continuum phase-space. An application of the fractional Liouville equation with these Riesz fractional derivatives to describe properties of plasma-like nonlocal media is considered.

Renormalization group solution of the Chutes & Ladder model

We analyze a semi-infinite one-dimensional random walk process with a biased motion that is incremental in one direction and long-range in the other. On a network with a fixed hierarchy of long-range jumps, we find with exact renormalization group calculations that there is a dynamical transition between a localized adsorption phase and an anomalous diffusion phase in which the mean-square displacement exponent depends non-universally on the Bernoulli coin. We relate these results to similar findings of unconventional phase behavior in hierarchical networks.

Acceleration of evolutionary spread by long-range dispersal.

The spreading of evolutionary novelties across populations is the central element of adaptation. Unless populations are well mixed (like bacteria in a shaken test tube), the spreading dynamics depend not only on fitness differences but also on the dispersal behavior of the species. Spreading at a constant speed is generally predicted when dispersal is sufficiently short ranged, specifically when the dispersal kernel falls off exponentially or faster. However, the case of long-range dispersal is unresolved: Although it is clear that even rare long-range jumps can lead to a drastic speedup--as air-traffic-mediated epidemics show--it has been difficult to quantify the ensuing stochastic dynamical process. However, such knowledge is indispensable for a predictive understanding of many spreading processes in natural populations. We present a simple iterative scaling approximation supported by simulations and rigorous bounds that accurately predicts evolutionary spread, which is determined by a trade-off between frequency and potential effectiveness of long-distance jumps. In contrast to the exponential laws predicted by deterministic "mean-field" approximations, we show that the asymptotic spatial growth is according to either a power law or a stretched exponential, depending on the tails of the dispersal kernel. More importantly, we provide a full time-dependent description of the convergence to the asymptotic behavior, which can be anomalously slow and is relevant even for long times. Our results also apply to spreading dynamics on networks with a spectrum of long-range links under certain conditions on the probabilities of long-distance travel: These are relevant for the spread of epidemics.

Solid Electrolytes Based on Fast Fluorine Ion Motions — 19F T1ρ NMR Relaxation vs Conductivity Measurements of Mechanosynthesized Nanocrystalline Ba0.6La0.4F2.4

The use of mechanosynthesis for the preparation of ion conductors provides access to new compounds whose properties are governed by chemical metastability, defects introduced and size effects present. The benefits of this valuable synthesis method manifest in enhanced ion transport properties of the prepared nanocrystalline ceramics. Such materials are useful candidates to act as solid electrolytes, e.g., in fluorine ion batteries [1]. Here, the introduction of large amounts of La3+ into the cubic structure of BaF2 was carried out by using high-energy ball-milling. The replacement of Ba2+ with a trivalent cation leads to the formation of an increased number fraction of point defects in the metastable crystal structure, which resulted in an increase of the dc-conductivity when compared to pure, nano-sized BaF2 [2]. Until now, only little is known about activation energies and jump rates of the elementary hopping processes of such materials. In the present study, we used broadband impedance spectroscopy and 19F NMR relaxometry to get to the bottom of ion jump diffusion proceeding on short-range and long-range length scales in Ba0.6La0.4F2.4. The dielectric measurements revealed an activation energy of 0.57 eV for macroscopic ion transport, whilst NMR, which is sensitive to both long-range motion and localized jumps, gave much smaller activation energies [3]. This was confirmed by high-frequency impedance measurements carried out at frequencies as high as 3 GHz that enabled us to compare the results from complementary methods applied in the same time window.[1] C. Rongeat, M. A. Reddy, R. Witter, and M. Fichtner, J. Phys. Chem. C, 117, 4943 (2013).[2] A. Düvel, J. Bednarcik, V. Šepelák, and P. Heitjans, J. Phys. Chem. C, in press (2014), DOI: 10.1021/jp410018t.[3] F. Preishuber-Pflügl, and M. Wilkening, Phys. Chem. Chem. Phys., in press (2014), DOI: 10.1039/C4CP00422A.