随机向量丛的接触大偏差

中文摘要 AI 辅助

大偏差理论缺乏其速率函数和涨落对称性的几何基础。本文在随机向量丛上发展了接触大偏差理论，其中速率函数、缩放累积生成函数以及Gallavotti-Cohen型涨落对偶性均源自接触1-形式。约束函数充当广义拉格朗日量；最小约束原理产生动力学，接触势从主方程逐阶构建，产生由不变密度、漂移和涨落张量控制的耦合Hamilton-Jacobi-输运系统。接触路径测度满足大偏差原理，其速率函数由约束作用量给出；缩放累积生成函数满足具有Donsker-Varadhan变分特征的稳态特征值方程。熵产生率、涨落-耗散组合$e=\tfrac12 g^T Ag-\sigma$是物理可观测量；时间反转对合$J:(t,y,\phi)\mapsto(t,y,-\phi-\nabla\ln\rho)$，具有反转漂移$v^{\mathrm{rev}}=-v+Ag$，产生Gallavotti-Cohen型对偶性$\lambda_{\mathrm{fwd}}(q)=\lambda_{\mathrm{rev}}(q+1)+\lambda_{\mathrm{fwd}}(-1)$。在可逆情形$Ag=0$下，恢复了经典的一维GC对称性。

英文摘要

Large deviation theory lacks a geometric foundation for its rate functions and fluctuation symmetries. This paper develops a contact large deviation theory on stochastic vector bundles, in which the rate function, the scaled cumulant generating function, and a Gallavotti--Cohen-type fluctuation duality all follow from the contact 1-form. The constraint function acts as a generalized Lagrangian; the least constraint principle yields the dynamics, and the contact potential is built order by order from the master equation, producing a coupled Hamilton--Jacobi--transport system governed by the invariant density, drift, and fluctuation tensor. The contact path measure satisfies a large deviation principle with rate function given by the constraint action; the scaled cumulant generating function obeys a stationary eigenvalue equation with a Donsker--Varadhan variational characterization. The entropy production rate, the fluctuation--dissipation combination $e=\tfrac12 g^T Ag-σ$, is the physical observable; the time-reversal involution $J:(t,y,ϕ)\mapsto(t,y,-ϕ-\nabla\lnρ)$ with reversed drift $v^{\mathrm{rev}}=-v+Ag$ yields a Gallavotti--Cohen-type duality $λ_{\mathrm{fwd}}(q)=λ_{\mathrm{rev}}(q+1)+λ_{\mathrm{fwd}}(-1)$. The classical one-dimensional GC symmetry is recovered in the reversible case $Ag=0$.

发表机构

• State Key Laboratory of Hydroscience and Engineering, Tsinghua University（清华大学水沙科学国家重点实验室）
• Department of Hydraulic Engineering, Tsinghua University（清华大学水利工程系）

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