Non-Gaussian Behavior in Hard Spheres

$\alpha_2 = \frac{\left< x^4 \right> }{3 \left< x^2 \right> } - 1$ 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

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

So if the particles aren’t diffusing, then $\alpha_2 \neq 0$

Closely related to caging and cage-breaking behavior

$\alpha_2$ measures how "not gaussian" the distribution is

Provocative, but inconclusive

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

As we increase density, we get an increased separation

As time varies, the ratio $\frac{A}{B}$ varies

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

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?

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

Fit the step distributions to the sum of two gaussians

Figure out how that scales with time and ϕ

This is hard.

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

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

Sackler, PEB, and HHMI