主动切换粒子群体的全局密度方程
Global density equations for a population of actively switching particles
AI总结：
本文研究主动切换粒子群体的密度演化，分别针对共同随机环境与独立切换情形推导 hSPDE、hPDE、矩方程、快速切换极限及路径积分表示。
摘要：
细胞生物学中有许多过程可以用主动切换粒子来建模。连续自由度根据混合随机微分方程（hSDE）演化，其漂移项取决于离散的内部或环境状态，而该状态按照连续时间马尔可夫链切换。本文中，我们为非相互作用的主动切换粒子群体推导了全局密度方程，这些粒子要么独立切换，要么处于共同随机切换的环境中。在随机环境情形下，我们表明全局粒子密度按照混合随机偏微分方程（hSPDE）演化。对高斯噪声过程取平均后，可得到关于单粒子密度的混合偏微分方程（hPDE）。我们利用相应的泛函 Chapman–Kolmogorov 方程推导单粒子密度的矩方程，并说明随机切换环境如何诱发统计相关性。我们还讨论了粒子相互作用的影响，这种相互作用在 hSPDE 和 hPDE 两个层面都会产生矩封闭问题。前者可以通过取平均场极限来处理，但由此得到的 hPDE 成为单粒子密度的非线性泛函。随后，我们为独立切换粒子发展了类似构造。我们引入一组由单粒子内部状态索引的离散全局密度，通过对切换过程取期望推导这些密度的 SPDE，然后利用慢/快分析，在快速切换极限下将该 SPDE 约化为标量随机 Fokker–Planck 方程。最后，我们在无相互作用情况下推导全局密度的路径积分，并将其与近期关于 Brownian 气体和 run-and-tumble 粒子的研究联系起来。
英文摘要：
There are many processes in cell biology that can be modelled in terms of an actively switching particle. The continuous degrees of freedom evolve according to a hybrid stochastic differential equation (hSDE) whose drift term depends on a discrete internal or environmental state that switches according to a continuous time Markov chain. In this paper we derive global density equations for a population of non-interacting actively switching particles, either independently switching or subject to a common randomly switching environment. In the case of a random environment, we show that the global particle density evolves according to a hybrid stochastic partial differential equation (hSPDE). Averaging with respect to the Gaussian noise processes yields a hybrid partial differential equation (hPDE) for the one-particle density. We use the corresponding functional Chapman-Kolmogorov equation to derive moment equations for the one-particle density and show how a randomly switching environment induces statistical correlations. We also discuss the effects of particle interactions, which generate moment closure problems at both the hSPDE and hPDE levels. The former can be handled by taking a mean field limit, but the resulting hPDE is now a nonlinear functional of the one-particle density. We then develop the analogous constructions for independently switching particles. We introduce a discrete set of global densities that are indexed by the single-particle internal states. We derive an SPDE for the densities by taking expectations with respect to the switching process, and then use a slow/fast analysis to reduce the SPDE to a scalar stochastic Fokker-Planck equation in the fast-switching limit. We end by deriving path integrals for the global densities in the absence of interactions and relate this to recent studies of Brownian gases and run-and-tumble particles.
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