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We generalize the ordinary aggregation process to allow for choice. In ordinary aggregation, two random clusters merge and form a larger aggregate. In our implementation of choice, a target cluster and two candidate clusters are randomly selected, and the target cluster merges with the larger of the two candidate clusters. We study the long-time asymptotic behavior, and find that as in ordinary aggregation, the size density adheres to the standard scaling form. However, aggregation with choice exhibits a number of novel features. First, the density of the smallest clusters exhibits anomalous scaling. Second, both the small-size and the large-size tails of the density are overpopulated, at the expense of the density moderate-size clusters. We also study the complementary case where the smaller candidate clusters participates in the aggregation process, and find abundance of moderate clusters at the expense of small and large clusters. Additionally, we investigate aggregation processes with choice among multiple candidate clusters, and a symmetric implementation where the choice is between two pairs of clusters.
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In this paper we present a study of the early stages of unstable state evolution of systems with spatial symmetry changes. In contrast to the early time linear theory of unstable evolution described by Cahn, Hilliard, and Cook, we develop a generalized theory that predicts two distinct stages of the early evolution for symmetry breaking phase transitions. In the first stage the dynamics is dominated by symmetry preserving evolution. In the second stage, which shares some characteristics with the Cahn-Hilliard-Cook theory, noise driven fluctuations break the symmetry of the initial phase on a time scale which is large compared to the first stage for systems with long interaction ranges. To test the theory we present the results of numerical simulations of the initial evolution of a long-range antiferromagnetic Ising model quenched into an unstable region. We investigate two types of symmetry breaking transitions in this system: disorder-to-order and order-to-order transitions. For the order-to-order case, the Fourier modes evolve as a linear combination of exponentially growing or decaying terms with different time scales.
We study diffusion-controlled single-species annihilation with sparse initial conditions. In this random process, particles undergo Brownian motion, and when two particles meet, both disappear. We focus on sparse initial conditions where particles occupy a subspace of dimension $delta$ that is embedded in a larger space of dimension $d$. We find that the co-dimension $Delta=d-delta$ governs the behavior. All particles disappear when the co-dimension is sufficiently small, $Deltaleq 2$; otherwise, a finite fraction of particles indefinitely survive. We establish the asymptotic behavior of the probability $S(t)$ that a test particle survives until time $t$. When the subspace is a line, $delta=1$, we find inverse logarithmic decay, $Ssim (ln t)^{-1}$, in three dimensions, and a modified power-law decay, $Ssim (ln t),t^{-1/2}$, in two dimensions. In general, the survival probability decays algebraically when $Delta <2$, and there is an inverse logarithmic decay at the critical co-dimension $Delta=2$.
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).
Consider a planner who has to decide whether or not to introduce a new policy to a certain local population. The planner has only limited knowledge of the policys causal impact on this population due to a lack of data but does have access to the publicized results of intervention studies performed for similar policies on different populations. How should the planner make use of and aggregate this existing evidence to make her policy decision? Building upon the paradigm of `patient-centered meta-analysis proposed by Manski (2020; Towards Credible Patient-Centered Meta-Analysis, Epidemiology), we formulate the planners problem as a statistical decision problem with a social welfare objective pertaining to the local population, and solve for an optimal aggregation rule under the minimax-regret criterion. We investigate the analytical properties, computational feasibility, and welfare regret performance of this rule. We also compare the minimax regret decision rule with plug-in decision rules based upon a hierarchical Bayes meta-regression or stylized mean-squared-error optimal prediction. We apply the minimax regret decision rule to two settings: whether to enact an active labor market policy given evidence from 14 randomized control trial studies; and whether to approve a drug (Remdesivir) for COVID-19 treatment using a meta-database of clinical trials.
We review understanding of kinetics of fluid phase separation in various space dimensions. Morphological differences, percolating or disconnected, based on overall composition in a binary liquid or density in a vapor-liquid system, have been pointed out. Depending upon the morphology, various possible mechanisms and corresponding theoretical predictions for domain growth are discussed. On computational front, useful models and simulation methodologies have been presented. Theoretically predicted growth laws have been tested via molecular dynamics simulations of vapor-liquid transitions. In case of disconnected structure, the mechanism has been confirmed directly. This is a brief review on the topic for a special issue on coarsening dynamics, expected to appear in Comptes Rendus Physique.
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