Extrema statistics in the dynamics of a non-Gaussian random field

When the equations that govern the dynamics of a random field are nonlinear, the field can develop with time non-Gaussian statistics even if its initial condition is Gaussian. Here, we provide a general framework for calculating the effect of the underlying nonlinear dynamics on the relative densities of maxima and minima of a two-dimensional field. Using this simple geometrical probe, we can identify the size of the non-Gaussian contributions in the random field, or alternatively the magnitude of the nonlinear terms in the underlying equations of motion. We demonstrate our approach by applying it to an initially Gaussian field that evolves according to the deterministic KPZ equation, which models surface growth and shock dynamics.

Figure 1

A two-dimensional (excluding the $y$ direction) geometrical interpretation of the second term of the KPZ equation [Eq. (18)] applied to a growing surface. The surface is assumed to grow perpendicularly at a constant rate $\lambda$. Measured vertically, the growth rate is $dh/dt=\lambda \sqrt{1+{(dh/dx)}^2}$. In three dimensions, the derivative is replaced by a gradient [see Eq. (19)].

Figure 2

The imbalance between maxima and minima $\Delta n$ of $h(\vec{r},t)$, where $h$ obeys the KPZ equation (with $\lambda /4\nu =0.1$), as a function of time. At $t=0$, $h\left(\vec{r}\right)$ was taken to be a Gaussian field with (a) a Gaussian spectrum $A\left(k\right)\propto \exp [-k^2/\left(4k_0^2\right)]$; (b) a ring spectrum $A\left(k\right)\propto \delta (k-k_0)$. Shown are our theoretical perturbative result [Eq. (17)] and data from simulations.