Properties of the solutions of delocalised coagulation and inception problems with outflow boundaries

Well posedness is established for a family of equations modelling particle populations undergoing delocalised coagulation, advection, inflow and outflow in a externally specified velocity field. Very general particle types are allowed while the spatial domain is a bounded region of $d$-dimensional space for which every point lies on exactly one streamline associated with the velocity field. The problem is formulated as a semi-linear ODE in the Banach space of bounded measures on particle position and type space. A local Lipschitz property is established in total variation norm for the propagators (generalised semi-groups) associated with the problem and used to construct a Picard iteration that establishes local existence and global uniqueness for any initial condition. The unique weak solution is shown further to be a differentiable or at least bounded variation strong solution under smoothness assumptions on the parameters of the coagulation interaction. In the case of one spatial dimension strong differentiability is established even for coagulation parameters with a particular bounded variation structure in space. This one dimensional extension establishes the convergence of the simulation processes studied in [Patterson, Stoch. Anal. Appl. 31, 2013] to a unique and differentiable limit

On the beneficial role of noise in resistive switching

We study the effect of external noise on resistive switching. Experimental results on a manganite sample are presented showing that there is an optimal noise amplitude that maximizes the contrast between high and low resistive states. By means of numerical simulations, we study the causes underlying the observed behavior. We find that experimental results can be related to general characteristics of the equations governing the system dynamics.Comment: 4 pages, 5 figure

Bilinear Coagulation Equations

We consider coagulation equations of Smoluchowski or Flory type where the total merge rate has a bilinear form $\pi(y)\cdot A\pi(x)$ for a vector of conserved quantities $\pi$, generalising the multiplicative kernel. For these kernels, a gelation transition occurs at a finite time $t_\mathrm{g}\in (0,\infty)$, which can be given exactly in terms of an eigenvalue problem in finite dimensions. We prove a hydrodynamic limit for a stochastic coagulant, including a corresponding phase transition for the largest particle, and exploit a coupling to random graphs to extend analysis of the limiting process beyond the gelation time.Comment: Generalises the previous version to focus on general coagulation processes of bilinear type, without restricting to the single example of the previous version. The previous results are mentioned as motivation, and all results of the previous version can be obtained from this more general versio

Kinetic Theory of Cluster Dynamics

In a Newtonian system with localized interactions the whole set of particles is naturally decomposed into dynamical clusters, defined as finite groups of particles having an influence on each other's trajectory during a given interval of time. For an ideal gas with short-range intermolecular force, we provide a description of the cluster size distribution in terms of the reduced Boltzmann density. In the simplified context of Maxwell molecules, we show that a macroscopic fraction of the gas forms a giant component in finite kinetic time. The critical index of this phase transition is in agreement with previous numerical results on the elastic billiard

Traffic flow densities in large transport networks

We consider transport networks with nodes scattered at random in a large domain. At certain local rates, the nodes generate traffic flowing according to some navigation scheme in a given direction. In the thermodynamic limit of a growing domain, we present an asymptotic formula expressing the local traffic flow density at any given location in the domain in terms of three fundamental characteristics of the underlying network: the spatial intensity of the nodes together with their traffic generation rates, and of the links induced by the navigation. This formula holds for a general class of navigations satisfying a link-density and a sub-ballisticity condition. As a specific example, we verify these conditions for navigations arising from a directed spanning tree on a Poisson point process with inhomogeneous intensity function.Comment: 20 pages, 7 figure

Large deviations in relay-augmented wireless networks

We analyze a model of relay-augmented cellular wireless networks. The network users, who move according to a general mobility model based on a Poisson point process of continuous trajectories in a bounded domain, try to communicate with a base station located at the origin. Messages can be sent either directly or indirectly by relaying over a second user. We show that in a scenario of an increasing number of users, the probability that an atypically high number of users experiences bad quality of service over a certain amount of time, decays at an exponential speed. This speed is characterized via a constrained entropy minimization problem. Further, we provide simulation results indicating that solutions of this problem are potentially non-unique due to symmetry breaking. Also two general sources for bad quality of service can be detected, which we refer to as isolation and screening.Comment: 28 pages, 5 figures; corrected several misprint

Large-deviation principles for connectable receivers in wireless networks

We study large-deviation principles for a model of wireless networks consisting of Poisson point processes of transmitters and receivers, respectively. To each transmitter we associate a family of connectable receivers whose signal-to-interference-and-noise ratio is larger than a certain connectivity threshold. First, we show a large-deviation principle for the empirical measure of connectable receivers associated with transmitters in large boxes. Second, making use of the observation that the receivers connectable to the origin form a Cox point process, we derive a large-deviation principle for the rescaled process of these receivers as the connection threshold tends to zero. Finally, we show how these results can be used to develop importance-sampling algorithms that substantially reduce the variance for the estimation of probabilities of certain rare events such as users being unable to connectComment: 29 pages, 2 figure

Convergence of stochastic particle systems undergoing advection and coagulation

The convergence of stochastic particle systems representing physical advection, inflow, outflow and coagulation is considered. The problem is studied on a bounded spatial domain such that there is a general upper bound on the residence time of a particle. The laws on the appropriate Skorohod path space of the empirical measures of the particle systems are shown to be relatively compact. The paths charged by the limits are characterised as solutions of a weak equation restricted to functions taking the value zero on the outflow boundary. The limit points of the empirical measures are shown to have densities with respect to Lebesgue measure when projected on to physical position space. In the case of a discrete particle type space a strong form of the Smoluchowski coagulation equation with a delocalised coagulation interaction and an inflow boundary condition is derived. As the spatial discretisation is refined in the limit equations, the delocalised coagulation term reduces to the standard local Smoluchowski interaction