A Many-Body Field Theory Approach to Stochastic Models in Population Biology
A Many-Body Field Theory Approach to Stochastic Models in Population Biology
复制标题
群体生物学中随机模型的多体场论方法
作者：
P. Dodd;N. Ferguson
背景理论生态学或数学流行病学中使用的许多模型是随机的，也可能是空间明确的。量子场论的技术以前曾用于反应扩散系统，主要是研究它们的临界行为。在这里，我们认为，他们使许多计算更容易，是一个可能的起点，新的近似。方法我们回顾了马尔可夫过程的多体场形式主义，并说明如何将其应用到一个“布朗虫”人口模型，和一个流行病模型。我们展示了如何主方程和时刻层次都可以写在特别紧凑的形式。引入功能的方法允许系统计算的有效行动，它给出了平均数量的动态。我们得到了一般（空间）平移不变系统的有效作用量的1-loop近似，从而近似了平均场的非平衡动力学。结论空间随机系统的主方程在多体场形式中通常具有较简洁的形式。人们可以写下动力学来生成物理相关的矩的泛函，相当于整个矩层次。平均场的单圈动力学与特定矩闭包的单圈动力学相同。
Background Many models used in theoretical ecology, or mathematical epidemiology are stochastic, and may also be spatially-explicit. Techniques from quantum field theory have been used before in reaction-diffusion systems, principally to investigate their critical behavior. Here we argue that they make many calculations easier and are a possible starting point for new approximations. Methodology We review the many-body field formalism for Markov processes and illustrate how to apply it to a ‘Brownian bug’ population model, and to an epidemic model. We show how the master equation and the moment hierarchy can both be written in particularly compact forms. The introduction of functional methods allows the systematic computation of the effective action, which gives the dynamics of mean quantities. We obtain the 1-loop approximation to the effective action for general (space-) translation invariant systems, and thus approximations to the non-equilibrium dynamics of the mean fields. Conclusions The master equations for spatial stochastic systems normally take a neater form in the many-body field formalism. One can write down the dynamics for generating functional of physically-relevant moments, equivalent to the whole moment hierarchy. The 1-loop dynamics of the mean fields are the same as those of a particular moment-closure.