发表机构
Université Paris-Saclay, CNRS（巴黎萨克雷大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究通过映射到有向聚合物和复制方法，分析了贝叶斯监测一维扩散粒子时的熵演化，揭示了恒定监测下的有界熵及幂律衰减监测下的两种渐近对数增长区域。
AI 中文摘要
我们研究对沿直线扩散的单个粒子的贝叶斯监测，观测者使用每个空间点上的噪声测量连续跟踪该粒子。与粒子位置后验分布相关的平均香农熵 $\overline{S(t)}$ 量化了在根据测量序列进行条件化后仍存在的不确定性。我们将该问题映射为计算 $1 + 1$ 维有向聚合物配分函数的矩。对于恒定监测速率，熵在任意有限测量强度下保持有界，并且可以使用 Kardar-Parisi-Zhang 方程的精确结果进行量化，提供饱和值和长时间渐近行为。对于监测强度按幂律 $\sim t^{-\alpha}$ 衰减的情况，出现两个不同的渐近区域：当 $\alpha > 1/2$ 时，$\overline{S(t)} \simeq \tfrac12 \ln t$；当 $0 < \alpha < 1/2$ 时，$\overline{S(t)} \simeq \alpha \ln t$。这两个结果均通过复制方法获得，该方法将问题映射到具有时间相关耦合的吸引性 Lieb-Liniger 哈密顿量：这两个区域可以分别围绕自由扩散和吸引性 Lieb-Liniger 基态的微扰展开来理解。我们讨论了极限情况 $\alpha = 1/2$，并将预测与离散高斯和对数伽马聚合物模型的模拟进行了比较。
英文摘要
We study the Bayesian monitoring of a single particle diffusing along a line, while an observer tracks it continuously using noisy measurements at each spatial point. The average Shannon entropy $\overline{S(t)}$ associated with the posterior distribution of the particle's position quantifies the uncertainty that remains after conditioning on the sequence of measurements. We map the problem to the calculation of the moments of the partition functions of directed polymers in $1 + 1$ dimensions. For a constant monitoring rate, the entropy remains bounded for any finite measurement intensity and can be quantified using exact results from the Kardar-Parisi-Zhang equation, providing both the saturation value and the asymptotic behavior for long times. For a monitoring intensity that decreases according to a power law $\sim t^{-α}$, two distinct asymptotic regimes emerge: $\overline{S(t)} \simeq \tfrac12 \ln t$ for $α> 1/2$ and $\overline{S(t)} \simeq α\ln t$ for $0 < α< 1/2$. Both results are obtained using the replica method, which maps the problem onto an attractive Lieb-Liniger Hamiltonian with a time-dependent coupling: the two regimes can be understood from a perturbative expansion around free diffusion and around the attractive Lieb-Liniger ground state, respectively. We discuss the limiting case $α= 1/2$ and compare the predictions with simulations of discrete Gaussian and log-gamma polymer models.
Comments6 pages (main text) + 2 pages (End Matter) + 15 pages (Supplemental Material)
