标题： 随机过程中的精确结果，涉及分裂、死亡和扩散：空间关联、边缘熵产生和宏观电流 显示英文标题

标题： Exact Results in Stochastic Processes with Division, Death, and Diffusion: Spatial Correlations, Marginal Entropy Production, and Macroscopic Currents

摘要： 我们考虑一个通用的随机粒子模型类，其在某一时刻的状态由一组连续自由度（例如位置）描述，且由于出生-死亡过程，该集合的长度随时间随机变化。 使用主方程形式，我们写出相应（无限）概率分布集的动力学：这表现为带有与模型相关的源项和汇项的耦合福克-普朗克方程。 我们推导了该模型类熵产生率的一般表达式，该表达式基于路径不可逆性。 为了展示该框架的实际应用，我们分析了一个生物启发模型，该模型包含分裂、死亡和扩散，其中空间相关性通过分裂过程产生。 通过系统地消除多余自由度，我们得到边缘概率分布，从而能够精确计算关键统计特性，如平均密度和关联函数。 我们通过数值布朗动力学模拟验证了我们的分析结果，发现理论与模拟之间有很好的一致性。 因此，我们的方法为解决之前未解决的随机出生-死亡动力学问题提供了一个强大的工具。 显示英文摘要

摘要： We consider a generic class of stochastic particle-based models whose state at an instant in time is described by a set of continuous degrees of freedom (e.g. positions), and the length of this set changes stochastically in time due to birth-death processes. Using a master equation formalism, we write down the dynamics of the corresponding (infinite) set of probability distributions: this takes the form of coupled Fokker-Planck equations with model-dependent source and sink terms. We derive the general expression of entropy production rate for this class of models in terms of path irreversibility. To demonstrate the practical use of this framework, we analyze a biologically motivated model incorporating division, death, and diffusion, where spatial correlations arise through the division process. By systematically integrating out excess degrees of freedom, we obtain the marginal probability distribution, enabling exact calculations of key statistical properties such as average density and correlation functions. We validate our analytical results through numerical Brownian dynamics simulations, finding excellent agreement between theory and simulation. Our method thus provides a powerful tool for tackling previously unsolved problems in stochastic birth-death dynamics.