Non-Gaussian Behavior in Hard Spheres

Wendell Smith

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", used as a measure of dynamical heterogeneities

  • 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 \( \alpha_2 \)

System Size Dependence of \( \alpha_2 \)

Lines drawn are \(A \left(\phi^\star - \phi\right)^n\), and \(\phi^\star \) is fitted

System Size Dependence of \( \alpha_2 \)

α₂ goes to ∞?

  • Provocative, but inconclusive

    • \( \phi^\star = 0.600 \pm 0.001\) 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 the 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

Close to Jamming

Close to Jamming

  • Glassy behavior starts at \( \phi \approx 0.55 – 0.59 \)

  • Jamming is at \( \phi \approx 0.64 \)

  • What happens if we go really close to jamming?

    • Permanent Caging

    • Floaters

Step Distributions Close to Jamming

Close to Jamming:

Glassy:

Peak at \( 10^{-2} \) : Tight Cages

Peak at \( 10^{-1} \) : Caged Floaters

Peak at \( 10^{-1} \) : Caged Particles

Peak at \( 10^{0} \) : Rearrangements

Step Distributions Close to Jamming

Solid: Backbone

Dashed: Floaters

Thin Dotted: Both

Rearrangements

  • Each dot represents a single system

  • At 1000 timepoints over the course of the simulation, a snapshot was taken, and the structure was "minimized" to find the inherent structures visited

  • Systems seem able to access either a very limited number of inherent structures, or a very large number

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.

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