摘要翻译：
我们详细描述了布朗运动的历史，以及爱因斯坦、萨瑟兰、斯莫卢乔夫斯基、巴切利耶、佩兰和朗之万对布朗运动理论的贡献。布朗运动理论在物理学中一直是一个重要的话题，最近的生物物理实验说明了布朗运动理论的重要性，例如，它用于测量单个DNA分子上的拉力。在第二部分，我们强调布朗运动理论在数学上的重要性，并通过两个选择的例子加以说明。本文用初等方法解释了牛顿势的布朗运动的经典表示。我们最后描述了平面布朗曲线几何学的最新进展。它的核心是保形不变性和多重分形性的概念，与布朗曲线本身的位势理论相关联。
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英文标题：
《Brownian Motion, "Diverse and Undulating"》
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作者：
Bertrand Duplantier
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最新提交年份：
2007
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分类信息：

一级分类：Physics        物理学
二级分类：Statistical Mechanics        统计力学
分类描述：Phase transitions, thermodynamics, field theory, non-equilibrium phenomena, renormalization group and scaling, integrable models, turbulence
相变，热力学，场论，非平衡现象，重整化群和标度，可积模型，湍流
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一级分类：Physics        物理学
二级分类：Soft Condensed Matter        软凝聚态物质
分类描述：Membranes, polymers, liquid crystals, glasses, colloids, granular matter
膜，聚合物，液晶，玻璃，胶体，颗粒物质
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一级分类：Physics        物理学
二级分类：Mathematical Physics        数学物理
分类描述：Articles in this category focus on areas of research that illustrate the application of mathematics to problems in physics, develop mathematical methods for such applications, or provide mathematically rigorous formulations of existing physical theories. Submissions to math-ph should be of interest to both physically oriented mathematicians and mathematically oriented physicists; submissions which are primarily of interest to theoretical physicists or to mathematicians should probably be directed to the respective physics/math categories
这一类别的文章集中在说明数学在物理问题中的应用的研究领域，为这类应用开发数学方法，或提供现有物理理论的数学严格公式。提交的数学-PH应该对物理方向的数学家和数学方向的物理学家都感兴趣；主要对理论物理学家或数学家感兴趣的投稿可能应该指向各自的物理/数学类别
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一级分类：Mathematics        数学
二级分类：History and Overview        历史和概况
分类描述：Biographies, philosophy of mathematics, mathematics education, recreational mathematics, communication of mathematics, ethics in mathematics
传记、数学哲学、数学教育、娱乐数学、数学交流、数学伦理
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一级分类：Mathematics        数学
二级分类：Mathematical Physics        数学物理
分类描述：math.MP is an alias for math-ph. Articles in this category focus on areas of research that illustrate the application of mathematics to problems in physics, develop mathematical methods for such applications, or provide mathematically rigorous formulations of existing physical theories. Submissions to math-ph should be of interest to both physically oriented mathematicians and mathematically oriented physicists; submissions which are primarily of interest to theoretical physicists or to mathematicians should probably be directed to the respective physics/math categories
math.mp是math-ph的别名。这一类别的文章集中在说明数学在物理问题中的应用的研究领域，为这类应用开发数学方法，或提供现有物理理论的数学严格公式。提交的数学-PH应该对物理方向的数学家和数学方向的物理学家都感兴趣；主要对理论物理学家或数学家感兴趣的投稿可能应该指向各自的物理/数学类别
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一级分类：Mathematics        数学
二级分类：Probability        概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Physics        物理学
二级分类：Chemical Physics        化学物理学
分类描述：Experimental, computational, and theoretical physics of atoms, molecules, and clusters - Classical and quantum description of states, processes, and dynamics; spectroscopy, electronic structure, conformations, reactions, interactions, and phases. Chemical thermodynamics. Disperse systems. High pressure chemistry. Solid state chemistry. Surface and interface chemistry.
原子、分子和团簇的实验、计算和理论物理-状态、过程和动力学的经典和量子描述；光谱学，电子结构，构象，反应，相互作用和相。化学热力学。分散系统。高压化学。固态化学。表面和界面化学。
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一级分类：Physics        物理学
二级分类：History and Philosophy of Physics        物理学的历史与哲学
分类描述：History and philosophy of all branches of physics, astrophysics, and cosmology, including appreciations of physicists.
物理学、天体物理学和宇宙学的所有分支的历史和哲学，包括物理学家的赏析。
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英文摘要：
  We describe in detail the history of Brownian motion, as well as the contributions of Einstein, Sutherland, Smoluchowski, Bachelier, Perrin and Langevin to its theory. The always topical importance in physics of the theory of Brownian motion is illustrated by recent biophysical experiments, where it serves, for instance, for the measurement of the pulling force on a single DNA molecule.   In a second part, we stress the mathematical importance of the theory of Brownian motion, illustrated by two chosen examples. The by-now classic representation of the Newtonian potential by Brownian motion is explained in an elementary way. We conclude with the description of recent progress seen in the geometry of the planar Brownian curve. At its heart lie the concepts of conformal invariance and multifractality, associated with the potential theory of the Brownian curve itself.
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PDF链接：
https://arxiv.org/pdf/705.1951