What does non-Gaussian mean?
What does non-Gaussian mean?
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\( \alpha_2 = \frac{\left< x^4 \right> }{3 \left< x^2 \right> } - 1 \) is the "non-Gaussian parameter"
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For a Gaussian distribution, \( \left< x^4 \right> = 3\left< x^2 \right> \), so \( \alpha_2 = 0 \)
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Can be calculated from particle-tracking
What does non-Gaussian mean?
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A random walk (diffusion) is Gaussian at any given time:
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\( P(x, t) = \frac{1}{2 \sqrt{\pi D t}} - e ^ \frac{x^2}{4 D t} \)
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So if the particles aren’t diffusing, then \( \alpha_2 \neq 0 \)
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For caged particles, \( \alpha_2 \approx -\frac{1}{5} \)
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Closely related to caging and cage-breaking behavior
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In a cage, particles move very small distances
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Cage-breaking would involve much larger jumps
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The distribution of "step sizes" would be very non-Gaussian
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What does non-Gaussian mean?
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\( \alpha_2 \) measures how "not gaussian" the distribution is
Diffusion |
Mixed |
Caging |
\( \alpha_2 = 0 \) |
\( \alpha_2 > 0 \) |
\( \alpha_2 = -\frac{1}{5} \) |
Data
What kind of step distributions do we actually get?
What kind of step distributions do we actually get?
What kind of step distributions do we actually get?
The Non-Gaussian Parameter \( \alpha_2 \)
α₂ goes to ∞?
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Line drawn is \(A \left(\phi^\star - \phi\right)^n\), where
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\( \phi^\star = 0.5999 \pm 0.0011\) (jamming is at \(\phi^J = 0.634\pm0.004\) )
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\( n = -1.21 \pm 0.05 \)
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α₂ goes to ∞?
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Provocative, but inconclusive
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\( \phi^\star = 0.5999 \pm 0.0011\) is an unusual density
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There is less than two orders of magnitude on this plot
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3.5 is a long ways from ∞
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Maximizing \( \alpha_2 \)
Maximizing \( \alpha_2 \)
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Start with the sum of two gaussians \( P(r) \propto A r ^ 2 \sigma ^ 2 e ^ {-\frac{r ^ 2}{\sigma ^ 2}} + B r ^ 2 e ^ {-r^2} \)
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Increasing σ while decreasing \( \frac{A}{B} \) gives a larger \( \alpha_2 \)
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More specifically: For a given σ, \( \frac{A}{B} = \frac{\sigma ^ 2}{1 + \sigma^2} \) yields tme maximum \( \alpha_2 = \frac{\left(\delta ^ 2-1\right)^2}{4 \delta ^2} \)
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Back to the Step Distributions
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As we increase density, we get an increased separation
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As time varies, the ratio \( \frac{A}{B} \) varies
Approximating \( \alpha_2 \) with Aging
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Prepare a state at \( \phi_0 = 0.55 \) at equilibrium
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Fast quench it to some density \( \phi \)
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Calculate \( \max_{\Delta t} \alpha_2 \) as a function of time
Aging \( \alpha_2 \)
Cartoon
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Simulation
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Other Directions
Fitting The Step Distributions
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Fit the step distributions to the sum of two gaussians
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Figure out how that scales with time and ϕ
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This is hard.
Coming from Above
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What happens at packing fractions just below jamming?
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What about floaters?
That’s all.
Thanks!
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Corey O’Hern, Mark Shattuck, Christine Jacobs-Wagner
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Brad Parry, Ivan Surovtsev, Eric Dufresne, and everyone I talked to
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Sackler, PEB, and HHMI