徐宝,吴鸿业,于悦

• 1:包头师范学院物理科学与技术学院

摘要(Abstract)：

理想Galton板模型的出口分布函数是一维扩散方程的解,可用后者检验出口分布函数的正确性。由于最大熵原理不考虑边界反弹效应,存在侧壁反弹时,从最大熵原理导出的出口分布函数不满足一维扩散方程。为了严格考虑侧壁反弹效应,可以采用周期折叠方案。得到的出口分布函数满足一维扩散方程。最大熵原理得到的出口分布函数是周期折叠方案的零级近似。采用Monte Carlo数值方法产生Galton板出口分布函数,比较两种方案对统计数据的表达能力,结果表明:对于存在侧壁反弹效应的出口分布函数,周期折叠方案的一级近似(三峰分布)比零级近似(最大熵原理得到的单峰分布)具有更好的表达能力。在保持Galton板宽度不变的前提下,随着钉板层数增大,出口分布函数从正态分布逐渐转变到均匀分布。以单峰分布为例,给出了确定转变点的方法。

关键词(KeyWords)： Galton板模型;侧壁反弹效应;最大熵原理;周期折叠方案

Abstract:

Keywords:

基金项目(Foundation): 教育部高等学校物理类专业教学指导委员会2023年度教学研究项目（JZW-23-GT-11）;; 包头师范学院一流本科课程项目（固体物理学）;包头师范学院教改项目（BSJG24Y14）资助

作者(Author): 徐宝,吴鸿业,于悦

参考文献(References)：

• [1]KUNERT J, MONTAG A, POHLMANN S. The quincunx:history and mathematics[J]. Statistical Papers,2001, 42:143-169.
• [2]李天成,陈言,张新超,等.一种双向伽尔顿板[P].中国:CN219759000 U, 2023-09-26.LI T C, CHEN Y, ZHANG X C, et al. A bidirectional Galton board[P]. China:CN 219759000 U, 2023-09-26.(in Chinese)
• [3]彭芳麟.伽尔顿板实验的计算机模拟[J].大学物理,2005,24(1):45-49.PENG F L. Computational simulation of Galton's experiment[J]. College Physics,2005, 24(1):45-49.(in Chinese)
• [4]谢名春. Galton板实验中的小球数分布函数[J].四川师范大学学报(自然科学版), 1996,19(3):93-97.XIE M C. The distributional function of numbers of the little globelet in narrow troughs in Galton Board experiment[J]. Journal of Sichuan Normal University(Natural Science), 1996, 19(3):93-97.(in Chinese)
• [5]郭竞达,史旭光.蒙特卡罗模拟在物理与生物学中的应用[J].物理与工程,2021, 31(4):21-26.GUO J D, SHI X G. Application of Monte Carlo simulation in physics and Biology[J]. 2021, 31(4):21-26.(in Chinese)
• [6]ZHANG Y, WANG Q. Simulation and analysis of two toy models[DB/OL]. 2023:ar Xvi:2310.02646.
• [7]BRUNO L, IPPOLITO I, CALVO A. Granular mixing in a Galton board[J]. Granular Matter, 2001, 3:83-86.
• [8]BENITO J G, IPPOLITO I, VIDALES A M. Improving mixture of grains by using bi-dimensional Galton boards[J].Physica A, 2008, 387:5371-5380.
• [9]BRUNO L, CALVO A, IPPOLITO I. Dispersive flow of disks through a two-dimensional Galton board[J]. European Physical Journal E, 2003, 11:131-140.
• [10]王碧玉,黄智贤,郑辉东,等.基于Galton板的固定床内示踪剂浓度分布[J].化工学报, 2013, 64(12):4283-4289.WANG B Y, HUANG Z X, ZHENG H D, et al. Tracer concentration distribution in fixed bed based on Galton board[J]. Journal of Chemical Industry and Engineering,2013, 64(12):4283-4289.(in Chinese)
• [11]CHEPELIANSKII A D, SHEPELYANSKY D L. Dynamical turbulent flow on the Galton board with friction[J].Physical Review Letters, 2001, 87(3):034101.
• [12]KOZLOV V V, MITROFANOVA Y M. Galton board[J]. Regular and Chaotic Dynamics, 2003, 8(4):431-439.
• [13]ROSATO A D, BLACKMORE D, BUCKLEY L, et al.Experimental, simulation and nonlinear dynamics of Galton's board[J]. International Journal of Nonlinear Sciences and Numerical Simulation, 2004, 5(4):289-312.
• [14]DAUD A A M. An improved mathematical model of Galton board with velocity dependent restitution[J]. Journal of Computational and Nonlinear Dynamics, 2017, 12(6):060901.
• [15]DAUD A A M. Mathematical modelling and symbolic dynamics analysis of three new Galton board models[J].Communications in Nonlinear Science and Numerical Simulation, 2014, 19(10):3476-3491.
• [16]BOUWMEESTER D, MARZOLI I, KARMAN G P, et al. Optical Galton board[J]. Physical Review A, 1999,61:013410.
• [17]SANZ-HERN??NDEZ D, MASSOURAS M, REYREN N,et al. A nano-magnetic Galton board[DB/OL]. 2020, arxiv:2010.10389.
• [18]晋宏营,刘美云.伽尔顿板实验小球分布的研究[J].物理与工程,2012,22(6):31-34.JIN H Y, LIU M Y. Study on balls distribution in Galton board experiment[J]. Physics and Engineering,2012,22(6):31-34.(in Chinese)
• [19]柯朗·R,希尔伯特·D.数学物理方法(卷Ⅱ)[M].熊振翔,杨应辰,译.北京:科学出版社, 1977:159-162.COURANT R, HILBERT D. Methods of mathematical physics(VolumeⅡ)[M]. Translated by XIONG Z X,YANG Y C. Beijing:Science Press, 1977:159-162.(in Chinese)