Tuesday, December 24, 2013

Three-Dimensional Brownian Motion and the Golden Ratio Rule 数学上的一维布朗运动模型称为无规行走,三维模型称为无规飞行。屏幕是二维平面,我用键盘敲出的字码也象布朗颗粒一样毫无规律可言,无以名之,姑且称为无规滑行。

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Intuitive metaphor[edit]

Consider a large balloon of 100 metres in diameter. Imagine this large balloon in a football stadium. The balloon is so large that it lies on top of many members of the crowd. Because they are excited, these fans hit the balloon at different times and in different directions with the motions being completely random. In the end, the balloon is pushed in random directions, so it should not move on average. Consider now the force exerted at a certain time. We might have 20 supporters pushing right, and 21 other supporters pushing left, where each supporter is exerting equivalent amounts of force. In this case, the forces exerted towards the left and the right are imbalanced in favor of the left; the balloon will move slightly to the left. This type of imbalance exists at all times, and it causes random motion of the balloon. If we look at this situation from far above, so that we cannot see the supporters, we see the large balloon as a small object animated by erratic movement.
Brownian motion model of the trajectory of a particle of dye in water.
Consider the particles emitted by Brown's pollen grain moving randomly in water: we know that a water molecule is about 0.1 by 0.2 nm in size, whereas the particles which Brown observed were of the order of a few micrometres in size (these are not to be confused with the actual pollen particle which is about 100 micrometres). So a particle from the pollen may be likened to the balloon, and the water molecules to the fans, except that in this case the balloon is surrounded by fans. The Brownian motion of a particle in a liquid is thus due to the instantaneous imbalance in the combined forces exerted by collisions of the particle with the much smaller liquid molecules (which are in random thermal motion) surrounding it.
An animation of the Brownian motion concept is available as a Java applet.
  1. Brownian motion - Wikipedia, the free encyclopedia

    en.wikipedia.org/wiki/Brownian_motion
    This is a simulation of the Brownian motion of a big particle (dust particle) that ... A single realisation of three-dimensional Brownian motion for times 0 ≤ t ≤ 2.
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    Three-Dimensional Brownian Motion and the Golden Ratio Rule

    www.maths.manchester.ac.uk/~goran/golden.pdf
    by K Glover - ‎2013 - ‎Cited by 5 - ‎Related articles
    σ2(y) L (y) dy staying strictly above the curve h(i) = L−1(L(i)/2) for i > 0 . In particular, when. X is the radial part of three-dimensional Brownian motion, we find that.
  3. Glover , Hulley , Peskir : Three-dimensional Brownian motion and ...

    projecteuclid.org/euclid.aoap/1362684849
    by K Glover - ‎2013 - ‎Cited by 5 - ‎Related articles
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  4. Theory of Brownian motion in three dimensions. Application to ...

    pubs.acs.org/doi/abs/10.1021/j150605a029
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  5. Three-dimensional Brownian motion and the golden ratio rule

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    by K Glover - ‎2013 - ‎Cited by 5 - ‎Related articles
    Mar 12, 2013 - Abstract: Let $X=(X_t)_{t\ge0}$ be a transient diffusion process in $(0,\infty)$ with the diffusion coefficient $\sigma>0$ and the scale function ...
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    Brownian Motion and Stochastic Calculus

    www.ntu.edu.sg/home/nprivault/MA5182/chapter4.pdf
    Next we turn to simulations of 2 dimensional and 3 dimensional Brownian motions. ... 4.3: Sample paths of a three-dimensional Brownian motion. 4.2 Wiener ...
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    BROWNIAN MOTION A tutorial - IPAM

    www.ipam.ucla.edu/publications/rstut/rstut_6958.ppt
    Brownian motion is the unique process. with the following ... d-dimensional Brownian. motion. Feynman-Kac formula (2). Invariance (3). The d-dimensional ...
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    BROWNIAN MOTION AND HAUSDORFF DIMENSION Contents ...

    www.math.uchicago.edu/~may/VIGRE/VIGRE2011/.../Hansen.pdf
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  9. Three-dimensional tracking of Brownian motion of ... - Optics InfoBase

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    Positive recurrence of reflecting Brownian motion in three dimensions

    www.isye.gatech.edu/~dai/publications/draft_bramsonDaiHarrison09.pdf
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