Understanding Non-Gaussian Behavior

What does non-Gaussian mean?

What does non-Gaussian mean?

  • \( \alpha_2 = \frac{\left< x^4 \right> }{3 \left< x^2 \right> } - 1 \) is the "non-Gaussian parameter"

  • For a Gaussian distribution, \( \left< x^4 \right> = 3\left< x^2 \right> \), so \( \alpha_2 = 0 \)

  • Can be calculated from particle-tracking

What does non-Gaussian mean?

  • A random walk (diffusion) is Gaussian at any given time:

    • \( P(x, t) = \frac{1}{2 \sqrt{\pi D t}} - e ^ \frac{x^2}{4 D t} \)

  • So if the particles aren’t diffusing, then \( \alpha_2 \neq 0 \)

    • For caged particles, \( \alpha_2 \approx -\frac{1}{5} \)

  • Closely related to caging and cage-breaking behavior

    • In a cage, particles move very small distances

    • Cage-breaking would involve much larger jumps

    • The distribution of "step sizes" would be very non-Gaussian

What does non-Gaussian mean?

  • \( \alpha_2 \) 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 ∞?

  • Line drawn is \(A \left(\phi^\star - \phi\right)^n\), where

    • \( \phi^\star = 0.5999 \pm 0.0011\) (jamming is at \(\phi^J = 0.634\pm0.004\) )

    • \( n = -1.21 \pm 0.05 \)

α₂ goes to ∞?

  • Provocative, but inconclusive

    • \( \phi^\star = 0.5999 \pm 0.0011\) is an unusual density

    • There is less than two orders of magnitude on this plot

    • 3.5 is a long ways from ∞

Maximizing \( \alpha_2 \)

Maximizing \( \alpha_2 \)

  • Start with the sum of two gaussians \( P(r) \propto A r ^ 2 \sigma ^ 2 e ^ {-\frac{r ^ 2}{\sigma ^ 2}} + B r ^ 2 e ^ {-r^2} \)

    • Increasing σ while decreasing \( \frac{A}{B} \) gives a larger \( \alpha_2 \)

    • More specifically: For a given σ, \( \frac{A}{B} = \frac{\sigma ^ 2}{1 + \sigma^2} \) yields tme maximum \( \alpha_2 = \frac{\left(\delta ^ 2-1\right)^2}{4 \delta ^2} \)

Back to the Step Distributions

  • As we increase density, we get an increased separation

  • As time varies, the ratio \( \frac{A}{B} \) varies

Approximating \( \alpha_2 \) with Aging

  • Prepare a state at \( \phi_0 = 0.55 \) at equilibrium

  • Fast quench it to some density \( \phi \)

  • Calculate \( \max_{\Delta t} \alpha_2 \) as a function of time

Cartoon

Aging \( \alpha_2 \)

Cartoon
Simulation

Other Directions

Fitting The Step Distributions

  • Fit the step distributions to the sum of two gaussians

  • Figure out how that scales with time and ϕ

Cartoon

  • This is hard.

Coming from Above

  • What happens at packing fractions just below jamming?

  • What about floaters?

That’s all.

Thanks!

  • Corey O’Hern, Mark Shattuck, Christine Jacobs-Wagner

  • Brad Parry, Ivan Surovtsev, Eric Dufresne, and everyone I talked to

  • Sackler, PEB, and HHMI