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The Random Walk and Diffusion

Macquarie University 2004. 2. The Random Walk. Also known as The Drunken Walk.Uses random effects to simulate real-world dynamics.Basis of understanding the process of diffusion in nature:heat,aromas,ink drop in water.. Macquarie University 2004. 3. The Physics. Liquids and gases contain many

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The Random Walk and Diffusion

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    1. Macquarie University 2004 1 The Random Walk and Diffusion PHYS220 2004 by Lesa Moore DEPARTMENT OF PHYSICS

    2. Macquarie University 2004 2 The Random Walk Also known as The Drunken Walk. Uses random effects to simulate real-world dynamics. Basis of understanding the process of diffusion in nature: heat, aromas, ink drop in water.

    3. Macquarie University 2004 3 The Physics Liquids and gases contain many freely-moving particles. Particles change directions when they collide. Each particle takes a random walk in three dimensions, moving a random distance in one direction till a collision scatters it into a new, random direction.

    4. Macquarie University 2004 4 Review of Lab Week 5 You have modelled the random walk for 49 paths over 15 (random-size, random-direction) steps and looked at distribution of path lengths

    5. Macquarie University 2004 5 Distribution of Path Lengths

    6. Macquarie University 2004 6 Compared with Normal (Gaussian) Distribution

    7. Macquarie University 2004 7 Normal (Gaussian) Distribution Estimate of a parent distribution based on mean m and standard deviation s of a set of measurements. Not bounded on either side, symmetrical. The formula is:

    8. Macquarie University 2004 8 Average Displacement m Mean displacement averaged over all trajectories (paths). May be positive or negative in 1D example. Use =AVERAGE(range) in Excel.

    9. Macquarie University 2004 9 Standard Deviation s Measures the spread away from the starting point. s2=mean of squares <x2> - square of mean<x>2: Use =STDEV(range) in Excel. Actual formula assumes data is sample of population:

    10. Macquarie University 2004 10 At half-width equal to sigma, height of curve is e^ -1/2 of peak height fwhm = 2.354 sigma 68% within 1 sigma 95% within 2 sigmaAt half-width equal to sigma, height of curve is e^ -1/2 of peak height fwhm = 2.354 sigma 68% within 1 sigma 95% within 2 sigma

    11. Macquarie University 2004 11 The Random Walk: further modelling Now, use steps of fixed size but in a random direction (one step per time interval). May be modelled as lattice in 1, 2 or 3 dimensions. Want to study behaviour over time. Begin with 1D

    12. Macquarie University 2004 12 Walk in One Dimension (1D)

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