离散电磁学中的陈-西蒙斯涨落与信息几何
Chern--Simons Fluctuations and Information Geometry in Discrete Electromagnetism
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中文总结 AI 辅助
该研究在闭合定向三维流形的离散电磁场中，构造受螺旋度约束的统计态，推导其相关统计量公式，利用陈-西蒙斯泛函方差控制态的局部可区分性，为离散电磁学的统计推断提供新方法。
中文摘要 AI 辅助
我们在闭合定向三维流形上为惠特尼离散化电磁场构造了受螺旋度约束的统计态。单纯 de Rham 复形提供了精确的离散规范对称性，而惠特尼内积将正合、调和与余正合部分分离。空间阿贝尔陈-西蒙斯泛函具有规范不变性，仅依赖于余正合势。在固定调和模式后，我们在约化电磁相空间上引入受螺旋度偏置的高斯系综，并推导了其容许参数、配分函数、平均螺旋度、相对熵及费舍尔信息的显式公式。该分布在规定平均螺旋度约束下唯一最小化相对熵，其螺旋度磁化率等于离散陈-西蒙斯泛函的方差，控制相邻统计态的局部可区分性。由于在无约束麦克斯韦动力学下磁螺旋度通常不守恒，这些态代表的是约束推断而非动力学平衡。
英文摘要
We construct helicity-conditioned statistical states for a Whitney-discretized electromagnetic field on a closed oriented three-manifold. The simplicial de Rham complex provides exact discrete gauge symmetry, while the Whitney inner product separates exact, harmonic, and coexact sectors. The spatial Abelian Chern--Simons functional is gauge invariant and depends only on the coexact potential. After fixing harmonic modes, we introduce a helicity-biased Gaussian ensemble on the reduced electromagnetic phase space and derive explicit formulas for its admissible parameters, partition function, mean helicity, relative entropy, and Fisher information. The distribution uniquely minimizes relative entropy under a prescribed mean-helicity constraint. Its helicity susceptibility equals the variance of the discrete Chern--Simons functional and controls the local distinguishability of neighboring statistical states. Because magnetic helicity is generally not conserved under unconstrained Maxwell dynamics, these states represent conditioned inference rather than dynamical equilibrium.
