使用蒙特卡罗模拟计算反应-扩散系统中的宏观反应速率
Computing macroscopic reaction rates in reaction-diffusion systems using Monte Carlo simulations
AI总结：
本文提出一种基于蒙特卡罗模拟的数值方法，通过事件计数统计直接计算反应-扩散系统中的宏观反应速率，并在多个模型上验证，揭示了微观概率与宏观速率间的粗粒化关系。
摘要：
随机反应-扩散模型被用于表示许多复杂的物理、生物、社会和生态系统。描述此类系统中大尺度动力学的宏观反应速率是有效的、依赖于尺度的参数，需要通过实验测量或使用微观模型计算。在随机反应-扩散系统的蒙特卡罗模拟中，特定事件发生的微观概率作为输入控制参数。发现涌现的宏观速率对微观概率的函数依赖是一个困难的问题，并且没有系统的分析方法来实现这一目标。因此，我们引入了一种直接的数值方法，利用蒙特卡罗模拟通过直接获得每个模拟时间步内发生的事件数量的计数统计来评估宏观反应速率。我们的技术首先在众所周知的基本例子上进行测试，即受限出生过程、扩散限制的两粒子凝聚以及双物种对湮灭动力学。接下来，我们利用由此获得的经验来研究在更复杂的模型系统中，微观算法概率如何粗粒化为有效的宏观速率，例如用于捕食者-猎物竞争和共存的Lotka-Volterra模型，以及石头-剪刀-布或循环Lotka-Volterra模型及其May-Leonard变体，这些模型捕捉具有循环优势基序的种群动态。由此，我们对空间扩展随机反应系统中的粗粒化以及相关微观和宏观模型参数之间的非平凡关系有了更深入的理解。所提出的技术通常应提供一种有用的手段，以更好地将蒙特卡罗模拟结果拟合到实验或观测数据。
英文摘要：
Stochastic reaction-diffusion models are employed to represent many complex physical, biological, societal, and ecological systems. The macroscopic reaction rates describing the large-scale kinetics in such systems are effective, scale-dependent parameters that need to be either measured experimentally or computed using a microscopic model. In a Monte Carlo simulation of stochastic reaction-diffusion systems, microscopic probabilities for specific events to happen serve as the input control parameters. Finding the functional dependence of emergent macroscopic rates on the microscopic probabilities is a difficult problem, and there is no systematic analytical way to achieve this goal. Therefore, we introduce a straightforward numerical method of using Monte Carlo simulations to evaluate the macroscopic reaction rates by directly obtaining the count statistics of how many events occur per simulation time step. Our technique is first tested on well-understood fundamental examples, namely restricted birth processes, diffusion-limited two-particle coagulation, and two-species pair annihilation kinetics. Next we utilize the thus gained experience to investigate how the microscopic algorithmic probabilities become coarse-grained into effective macroscopic rates in more complex model systems such as the Lotka--Volterra model for predator-prey competition and coexistence, as well as the rock-paper-scissors or cyclic Lotka--Volterra model as well as its May--Leonard variant that capture population dynamics with cyclic dominance motifs. Thereby we achieve a deeper understanding of coarse-graining in spatially extended stochastic reaction systems and the nontrivial relationships between the associated microscopic and macroscopic model parameters. The proposed technique should generally provide a useful means to better fit Monte Carlo simulation results to experimental or observational data.
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