亨尼-格林斯坦随机游走的首次返回统计与点扩散函数——远场稳定定律
First-return statistics and the point-spread function of a Henyey-Greenstein random walk - a far-field stable law
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中文总结 AI 辅助
针对亨尼-格林斯坦随机游走，研究远场稳定定律下的首次返回统计，发现半空间出射粒子的首次返回飞行次数分布、横向位置标度，以及远场点扩散函数的二维柯西尾，还验证了吸收对核结构的影响及经验特征函数的标度指数，通过蒙特卡洛模拟验证结果。
中文摘要 AI 辅助
一个粒子法向入射到半空间，经亨尼-格林斯坦（HG）相位函数散射后，在首次返回时间后出射，其飞行次数n趋近于无分布的斯帕尔-安德森指数，即f(n) ~ n^{-3/2}，振幅由入射条件而非定理决定。在给定n且经过关联瞬态后，横向出射位置服从高斯分布，方差与n成正比，我们直接验证了该标度关系。将该高斯位移从属到单侧1/2-稳定首次返回时间，在扩散极限下得到各向同性1-稳定标度极限：法向入射铅笔束的远场点扩散函数（PSF）尾为二维柯西尾p(r) ~ r^{-3}，这是半空间泊松核的大半径极限。指数α=1是中心极限定理与无分布首次返回指数的乘积，因此，任何具有相同扩散极限（有限步方差、短程角关联）的相位函数都无法改变该指数。吸收以exp(-μ_a L)权重每条路径的总长度L，截断从属过程，产生法雷尔-帕特森-威尔逊（1992）扩散偶极子每个源项携带的屏蔽径向结构(μ_eff + 1/r)exp(-μ_eff r)/r²；本文未推导有限深度双源偶极子。保留尾计算舍弃的入射深度因子，使同一核延伸至半径为1.7 l*的有限核，无额外拟合参数。经验特征函数得到的标度依赖指数α_eff追踪相似参数γ，在一个输运平均自由程附近，将g₁匹配的相位函数以0.15-0.22的差值区分。结果为g=0至0.9时的蒙特卡洛模拟，最多包含10⁶个游走粒子。
英文摘要
A particle launched normally into a half space, scattering by the Henyey-Greenstein (HG) phase function, exits after a first-return time whose count of flights n approaches the distribution-free Sparre Andersen exponent, f(n) ~ n^{-3/2}, with an amplitude fixed by the launch rather than by the theorem. Conditioned on n, and past a correlation-set transient, the lateral exit position is Gaussian with variance proportional to n, a scaling we test directly. Subordinating that Gaussian displacement to the one-sided 1/2-stable first-return time gives, in the diffusion limit, an isotropic 1-stable scaling limit: the far tail of the point-spread function (PSF) for a normally incident pencil beam is the two-dimensional Cauchy tail p(r) ~ r^{-3}, the large-radius limit of the half-space Poisson kernel. The index alpha = 1 is the product of the central limit theorem and the distribution-free first-return exponent, so no phase function admitting the same diffusion limit - finite step variance, short-range angular correlation - can move it. Absorption weights each path by exp(-mu_a L) in its total length L, truncating the subordinator and producing the screened radial structure (mu_eff + 1/r) exp(-mu_eff r)/r^2 carried by each source term of the Farrell-Patterson-Wilson (1992) diffusion dipole; the finite-depth two-source dipole is not derived here. Keeping the launch-depth factor that the tail calculation discards continues the same kernel into a finite core of radius 1.7 l*, with no additional fitted parameter. A scale-dependent index alpha_eff from the empirical characteristic function tracks the similarity parameter gamma, separating phase functions matched in g_1 by 0.15-0.22 near one transport mean free path. Results are Monte Carlo at g = 0 to 0.9 with up to 10^6 walkers.
