弹跳硬币抛掷与三面骰子问题的精确边缘落地概率
Exact edge landing probability for the bouncing coin toss and the three-sided die problem
AI总结：
本文结合经典碰撞方程与统计力学方法，首次推导出部分非弹性硬币抛掷三种落地结果的精确解析概率，并通过模拟与实验验证，最终提出公平三面骰子的最优几何设计。
摘要：
你是否曾通过抛硬币并查看其落地的一面来做出有争议的决定？这种古老的“正面或反面”做法在面对未决选择时仍被广泛使用，因为它依赖于等概率的直观公平性。然而，它严重忽视了一个有趣的第三种结果：硬币可能以边缘着地并静止。鉴于这一明显却难以捉摸的可能性，此前的工作已致力于捕捉厚硬币的三种落地概率，但仅在计算上取得了成功，因为一个精确且完全有依据的解析解仍然未被物理学界获得。在本工作中，我们将经典碰撞方程与统计力学方法相结合，推导出部分非弹性硬币抛掷的精确解析结果概率。我们通过将理论预测与先前报告的模拟和实验数据进行比较来验证该理论预测；我们讨论了在高度非弹性区域出现的适度偏差；我们描述了与先前近似模型的差异；最后，基于当前结果，我们提出了公平的圆柱形三面骰子的最优几何形状。
英文摘要：
Have you ever taken a disputed decision by tossing a coin and checking its landing side? This ancestral "heads or tails" practice is still widely used when facing undecided alternatives since it relies on the intuitive fairness of equiprobability. However, it critically disregards an interesting third outcome: the possibility of the coin coming at rest on its edge. Provided this evident yet elusive possibility, previous works have already focused on capturing all three landing probabilities of thick coins, but have only succeeded computationally as an exact and fully grounded analytical solution still evades the physics community. In this work we combine the classical equations of collisions with a statistical-mechanics approach to derive the exact analytical outcome probabilities of the partially inelastic coin toss. We validate the theoretical prediction by comparing it to previously reported simulations and experimental data; we discuss the moderate discrepancies arising at the highly inelastic regime; we describe the differences with previous, approximate models; and we finally propose the optimal geometry for the fair, cylindrical three-sided die based on the current results.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
