The model of binary aggregation with constant kernel is subjected to stochastic resetting: aggregates of any size explode into monomers at independent stochastic times. These resetting times are Poisson distributed, and the rate of the process is called the resetting rate. The master equation yields a Bernoulli-type equation in the generating function of the concentration of aggregates of any size, which can be solved exactly. This resetting prescription leads to a non-equilibrium steady state for the densities of aggregates, which is a function of the size of the aggregate, rescaled by a function of the resetting rate. The steady-state density of aggregates of a given size is maximised if the resetting rate is set to the quotient of the aggregation rate by the size of the aggregate (minus one).
We consider a continuous-space and continuous-time diffusion process under resetting with memory. A particle resets to a position chosen from its trajectory in the past according to a memory kernel. Depending on the form of the memory kernel, we show analytically how different asymptotic behaviours of the variance of the particle position emerge at long times. These range from standard diffusive ($σ^2 \sim t$) all the way to anomalous ultraslow growth $σ^2 \sim \ln \ln t$.
Heterogeneous planar diffusion with axisymmetric power-law decaying diffusion coefficient under stochastic resetting
Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts（马其顿科学院与艺术院计算机科学与信息技术研究中心）；Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril and Methodius University（圣西里尔和梅托迪乌斯大学自然科学与数学学院物理研究所）；Department of Physics, Korea University（高丽大学物理系）；Laboratoire de Physique et Chimie Théoriques (CNRS UMR 7019), Université de Lorraine Nancy（洛林大学南锡分校理论与物理化学实验室）；Centro de Física Teórica e Computacional, Universidade de Lisboa（里斯本大学理论与计算物理中心）；Departamento de Física, Universidade Estadual de Maringá（马里纳州立大学物理系）
We study the dynamics of predator-prey systems where prey are confined to a single region of space and where predators move randomly according to a power-law (Lévy) dispersal kernel. Site fidelity, an important feature of animal behaviour, is incorporated in the model through a stochastic resetting dynamics of the predators to the prey patch. We solve in the long time limit the rate equations of Lotka-Volterra type that describe the evolution of the two species densities. Fixing the demographic parameters and the Lévy exponent, the total population of predators can be maximized for a certain v…
Probabilities of rare events in product kernel aggregation: An exact formula and phase diagram
The Institute of Mathematical Sciences（数学科学研究所）；Homi Bhabha National Institute（霍米·巴巴国立学院）；Tata Institute of Fundamental Research（塔塔基础科学研究院）；Mathematics Institute, University of Warwick（华威大学数学研究所）
In this paper we show how the method of Zakharov transformations may be used to analyze the stationary solutions of the Smoluchowski aggregation equation for arbitrary homogeneous kernel. The resulting massdistributions are of Kolmogorov type in the sense that they carry a constant flux of mass from small masses to large. We derive a ``locality criterion'', expressed in terms of the asymptotic properties of the kernel, that must be satisfied in order for the Kolmogorov spectrum to be an admissiblesolution. Whether a given kernel leads to a gelation transition or not can be determined by comput…
It is proposed to revisit the inverse problem associated with Smoluchowski's coagulation equation. The objective is to reconstruct the functional form of the collision kernel from observations of the time evolution of the cluster size distribution. A regularised least squares method originally proposed by Wright and Ramkrishna (1992) based on the assumption of self-similarity is implemented and tested on numerical data generated for a range of different collision kernels. This method expands the collision kernel as a sum of orthogonal polynomials and works best when the kernel can be expressed…
We consider the scaling solutions of Smoluchowski's equation of irreversible aggregation, for a non gelling collision kernel. The scaling mass distribution f(s) diverges as s^{-tau} when s->0. tau is non trivial and could, until now, only be computed by numerical simulations. We develop here new general methods to obtain exact bounds and good approximations of $τ$. For the specific kernel KdD(x,y)=(x^{1/D}+y^{1/D})^d, describing a mean-field model of particles moving in d dimensions and aggregating with conservation of ``mass'' s=R^D (R is the particle radius), perturbative and nonperturbative…
We study the asymptotic properties of the steady state mass distribution for a class of collision kernels in an aggregation-shattering model in the limit of small shattering probabilities. It is shown that the exponents characterizing the large and small mass asymptotic behavior of the mass distribution depend on whether the collision kernel is local (the aggregation mass flux is essentially generated by collisions between particles of similar masses), or non-local (collision between particles of widely different masses give the main contribution to the mass flux). We show that the non-local r…
In a process of aggregation, a finite number of particles merge irreversibly to create growing clusters. In this work, impact of particular initial conditions: monodisperse, power-law, exponential, and inspired by condensation nuclei was tested against several aggregation processes: constant, additive, product, electrorheological, anti-social, Berry's, Brownian, shear, and gravitational kernels. Coagulating systems consisting of a thousand monomers were observed in the late time of the evolution, for the moment when only fifty clusters were left in the system. In this way, the impact of partic…
For this paper, we studied the time evolution of a system of coagulating particles under a generalized electrorheological (ER) kernel with real power, $K\left(i,j\right) = \left( \frac{1}{i}+\frac{1}{j} \right)^α$, and monodisperse initial conditions. We used a combinatorial framework in which time and cluster sizes were discrete and the binary aggregation governed the time evolution of the system. We modified a previously-known solution for the constant kernel to cover the generalized ER kernel and used it in the framework to obtain the exact expression for the cluster size distribution (the …
We observe never-ending oscillations in systems undergoing aggregation and collision-controlled shattering. Specifically, we investigate aggregation-shattering processes with aggregation kernels K_{i,j} = (i/j)^a+(j/i)^a and shattering kernels F_{i,j}=λK_{i,j}, where i and j are cluster sizes and parameter λquantifies the strength of shattering. When 0<a<1/2, there are no oscillations and the system monotonically approaches to a steady state for all values of λ; in this region we obtain an analytical solution for the stationary cluster size distribution. Numerical solutions of the rate equatio…
In a recent series of papers, an exact combinatorial solution was claimed for a variant of the so-called Marcus--Lushnikov model of aggregation. In this model, a finite number of aggregates, are initially assumed to be present in the form of monomers. At each time step, two aggregates are chosen according to certain size-dependent probabilities and irreversibly joined to form an aggregate of higher mass. The claimed result given an expression for the full probability distribution over all possible size distributions in terms of the so-called Bell polynomials. In this paper, we develop the asym…
The evolution of the Erdős-Rényi (ER) network by adding edges can be viewed as a cluster aggregation process. Such ER processes can be described by a rate equation for the evolution of the cluster-size distribution with the connection kernel $K_{ij}\sim ij$, where $ij$ is the product of the sizes of two merging clusters. Here, we study more general cases in which $K_{ij}$ is sub-linear as $K_{ij}\sim (ij)^ω$ with $0 \le ω< 1/2$; we find that the percolation transition (PT) is discontinuous. Moreover, PT is also discontinuous when the ER dynamics evolves from proper initial conditions. The rate…
In a non-equilibrium system, a Constant Flux Relation (CFR) expresses the fact that a constant flux of a conserved quantity exactly determines the scaling of the particular correlation function linked to the flux of that conserved quantity. This is true regardless of whether mean--field theory is applicable or not. We focus on cluster--cluster aggregation and discuss the consequences of mass conservation for the steady state of aggregation models with a monomer source in the diffusion-limited regime. We derive the CFR for the flux-carrying correlation function for binary aggregation with a gen…
Transient dynamics leading to the synchrony of pulse-coupled oscillators has previously been studied as an aggregation process of synchronous clusters, and a rate equation for the cluster size distribution has been proposed. However, the evolution of the cluster size distribution for general cluster sizes has not been solved yet. In this paper, we study the evolution of the cluster size distribution from the perspective of a percolation model by regarding the number of aggregations as the number of attached bonds. Specifically, we derive the scaling form of the cluster size distribution with s…
