L\'evy walks on lattices as multi-state processes

Continuous-time random walks combining diffusive scattering and ballistic propagation on lattices model a class of L\'evy walks. The assumption that transitions in the scattering phase occur with exponentially-distributed waiting times leads to a description of the process in terms of multiple states, whose distributions evolve according to a set of delay differential equations, amenable to analytic treatment. We obtain an exact expression of the mean squared displacement associated with such processes and discuss the emergence of asymptotic scaling laws in regimes of diffusive and superdiffusive (subballistic) transport, emphasizing, in the latter case, the effect of initial conditions on the transport coefficients. Of particular interest is the case of rare ballistic propagation, in which case a regime of superdiffusion may lurk underneath one of normal diffusion.Comment: 27 pages, 4 figure

Activated Random Walkers: Facts, Conjectures and Challenges

We study a particle system with hopping (random walk) dynamics on the integer lattice $\mathbb Z^d$. The particles can exist in two states, active or inactive (sleeping); only the former can hop. The dynamics conserves the number of particles; there is no limit on the number of particles at a given site. Isolated active particles fall asleep at rate $\lambda > 0$, and then remain asleep until joined by another particle at the same site. The state in which all particles are inactive is absorbing. Whether activity continues at long times depends on the relation between the particle density $\zeta$ and the sleeping rate $\lambda$. We discuss the general case, and then, for the one-dimensional totally asymmetric case, study the phase transition between an active phase (for sufficiently large particle densities and/or small $\lambda$) and an absorbing one. We also present arguments regarding the asymptotic mean hopping velocity in the active phase, the rate of fixation in the absorbing phase, and survival of the infinite system at criticality. Using mean-field theory and Monte Carlo simulation, we locate the phase boundary. The phase transition appears to be continuous in both the symmetric and asymmetric versions of the process, but the critical behavior is very different. The former case is characterized by simple integer or rational values for critical exponents ($\beta = 1$, for example), and the phase diagram is in accord with the prediction of mean-field theory. We present evidence that the symmetric version belongs to the universality class of conserved stochastic sandpiles, also known as conserved directed percolation. Simulations also reveal an interesting transient phenomenon of damped oscillations in the activity density

Going from microscopic to macroscopic on nonuniform growing domains

Throughout development, chemical cues are employed to guide the functional specification of underlying tissues while the spatiotemporal distributions of such chemicals can be influenced by the growth of the tissue itself. These chemicals, termed morphogens, are often modeled using partial differential equations (PDEs). The connection between discrete stochastic and deterministic continuum models of particle migration on growing domains was elucidated by Baker, Yates, and Erban [ Bull. Math. Biol. 72 719 (2010)] in which the migration of individual particles was modeled as an on-lattice position-jump process. We build on this work by incorporating a more physically reasonable description of domain growth. Instead of allowing underlying lattice elements to instantaneously double in size and divide, we allow incremental element growth and splitting upon reaching a predefined threshold size. Such a description of domain growth necessitates a nonuniform partition of the domain. We first demonstrate that an individual-based stochastic model for particle diffusion on such a nonuniform domain partition is equivalent to a PDE model of the same phenomenon on a nongrowing domain, providing the transition rates (which we derive) are chosen correctly and we partition the domain in the correct manner. We extend this analysis to the case where the domain is allowed to change in size, altering the transition rates as necessary. Through application of the master equation formalism we derive a PDE for particle density on this growing domain and corroborate our findings with numerical simulations

L\'evy walks

Random walk is a fundamental concept with applications ranging from quantum physics to econometrics. Remarkably, one specific model of random walks appears to be ubiquitous across many fields as a tool to analyze transport phenomena in which the dispersal process is faster than dictated by Brownian diffusion. The L\'{e}vy walk model combines two key features, the ability to generate anomalously fast diffusion and a finite velocity of a random walker. Recent results in optics, Hamiltonian chaos, cold atom dynamics, bio-physics, and behavioral science demonstrate that this particular type of random walks provides significant insight into complex transport phenomena. This review provides a self-consistent introduction to L\'{e}vy walks, surveys their existing applications, including latest advances, and outlines further perspectives.Comment: 50 page

Strong disorder RG approach of random systems

There is a large variety of quantum and classical systems in which the quenched disorder plays a dominant r\^ole over quantum, thermal, or stochastic fluctuations : these systems display strong spatial heterogeneities, and many averaged observables are actually governed by rare regions. A unifying approach to treat the dynamical and/or static singularities of these systems has emerged recently, following the pioneering RG idea by Ma and Dasgupta and the detailed analysis by Fisher who showed that the Ma-Dasgupta RG rules yield asymptotic exact results if the broadness of the disorder grows indefinitely at large scales. Here we report these new developments by starting with an introduction of the main ingredients of the strong disorder RG method. We describe the basic properties of infinite disorder fixed points, which are realized at critical points, and of strong disorder fixed points, which control the singular behaviors in the Griffiths-phases. We then review in detail applications of the RG method to various disordered models, either (i) quantum models, such as random spin chains, ladders and higher dimensional spin systems, or (ii) classical models, such as diffusion in a random potential, equilibrium at low temperature and coarsening dynamics of classical random spin chains, trap models, delocalization transition of a random polymer from an interface, driven lattice gases and reaction diffusion models in the presence of quenched disorder. For several one-dimensional systems, the Ma-Dasgupta RG rules yields very detailed analytical results, whereas for other, mainly higher dimensional problems, the RG rules have to be implemented numerically. If available, the strong disorder RG results are compared with another, exact or numerical calculations.Comment: review article, 195 pages, 36 figures; final version to be published in Physics Report

Ordering in voter models on networks: Exact reduction to a single-coordinate diffusion

We study the voter model and related random-copying processes on arbitrarily complex network structures. Through a representation of the dynamics as a particle reaction process, we show that a quantity measuring the degree of order in a finite system is, under certain conditions, exactly governed by a universal diffusion equation. Whenever this reduction occurs, the details of the network structure and random-copying process affect only a single parameter in the diffusion equation. The validity of the reduction can be established with considerably less information than one might expect: it suffices to know just two characteristic timescales within the dynamics of a single pair of reacting particles. We develop methods to identify these timescales, and apply them to deterministic and random network structures. We focus in particular on how the ordering time is affected by degree correlations, since such effects are hard to access by existing theoretical approaches.Comment: 37 pages, 10 figures. Revised version with additional discussion and simulation results to appear in J Phys

Reaction rates for a generalized reaction-diffusion master equation

It has been established that there is an inherent limit to the accuracy of the reaction-diffusion master equation. Specifically, there exists a fundamental lower bound on the mesh size, below which the accuracy deteriorates as the mesh is refined further. In this paper we extend the standard reaction-diffusion master equation to allow molecules occupying neighboring voxels to react, in contrast to the traditional approach in which molecules react only when occupying the same voxel. We derive reaction rates, in two dimensions as well as three dimensions, to obtain an optimal match to the more fine-grained Smoluchowski model, and show in two numerical examples that the extended algorithm is accurate for a wide range of mesh sizes, allowing us to simulate systems intractable with the standard reaction-diffusion master equation. In addition, we show that for mesh sizes above the fundamental lower limit of the standard algorithm, the generalized algorithm reduces to the standard algorithm. We derive a lower limit for the generalized algorithm, which, in both two dimensions and three dimensions, is on the order of the reaction radius of a reacting pair of molecules

Non-Equilibrium Properties of Open Quantum Systems

We study two classes of open systems: discrete-time quantum walks (a type of Floquet-engineered discrete quantum map) and the Lindblad master equation (a general framework of dissipative quantum systems), focusing on the non-equilibrium properties of these systems. We study localization and delocalization phenomena, soliton-like excitations, and quasi-stationary properties of open quantum systems