梯度流之外的变分结构：宏观涨落理论视角
Variational structures beyond gradient flows: a macroscopic fluctuation-theory perspective
AI总结：
本研究从宏观涨落理论出发，通过引入通量拓展密度描述，构建抽象框架将变分结构从满足细致平衡的耗散系统推广至更广泛的随机粒子系统，给出分量分解与广义正交关系，并验证了理论的适用性。
摘要：
由满足细致平衡的随机粒子系统（称为耗散系统或梯度流）导出的宏观方程具有天然的变分结构，该结构可从粒子系统密度的大偏差率泛函推导而来。尽管大偏差可在相当广泛的范围内研究，但这些变分结构通常局限于满足细致平衡的系统。本研究借助宏观涨落理论的洞见，通过引入能编码非耗散效应的通量来拓展密度描述，旨在将这种变分关联推广至耗散系统之外。我们的核心贡献是一个抽象框架：针对给定的通量-密度代价函数与拟势，该框架可将其分解为耗散分量与非耗散分量，并给出二者之间的广义正交关系。随后我们将该抽象理论应用于多种随机粒子系统，包括跳过程的独立副本、零程过程、满足复平衡的化学反应网络以及格气模型。
英文摘要：
Macroscopic equations arising out of stochastic particle systems in detailed balance (called dissipative systems or gradient flows) have a natural variational structure, which can be derived from the large-deviation rate functional for the density of the particle system. While large deviations can be studied in considerable generality, these variational structures are often restricted to systems in detailed balance. Using insights from macroscopic fluctuation theory, in this work we aim to generalise this variational connection beyond dissipative systems by augmenting densities with fluxes, which encode non-dissipative effects. Our main contribution is an abstract framework, which for a given flux-density cost and a quasipotential, provides a decomposition into dissipative and non-dissipative components and a generalised orthogonality relation between them. We then apply this abstract theory to various stochastic particle systems -- independent copies of jump processes, zero-range processes, chemical-reaction networks in complex balance and lattice-gas models.
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