Statistics of shocks in a toy model with heavy tails

We study the energy minimization for a particle in a quadratic well in the presence of short-ranged heavy-tailed disorder, as a toy model for an elastic manifold. The discrete model is shown to be described in the scaling limit by a continuum Poisson process model which captures the three universality classes. This model is solved in general, and we give, in the present case (Frechet class), detailed results for the distribution of the minimum energy and position, and the distribution of the sizes of the shocks (i.e., switches in the ground state) which arise as the position of the well is varied. All these distributions are found to exhibit heavy tails with modified exponents. These results lead to an “exotic regime” in Burgers turbulence decaying from a heavy-tailed initial condition.

Figure 1

Particle in a random potential landscape confined by an elastic force (i.e., a quadratic potential centered at $r$). $u\left(r\right)$ is the position with minimal total energy $H\left(r\right)$. Its fluctuations from sample to sample scale as $u_m\sim m^{-\zeta }$.

Figure 2

Comparison of the PDF for the position $u$ as given in Eq. (17) (black dashed lines) with numerical simulations (red solid lines). The algebraic tails are clearly distinguishable as straight lines on the semilogarithmic plot. From the narrowest shape to the broadest (corresponding to the fatter tails), $\mu =15$, 10, and 4. The sample size is $N=5\times {10}^5$.

Figure 3

Parabola construction for the minimization problem: when the center $r$ of the parabola is shifted from $r_1$ to $r_2$, the position of the particle moves from $u_1$ to $u_2$. For given $r_1$ and $r_2$, the intersection of both the parabola is called $u^{*}$. More details are displayed in Appendix pp5.

Figure 4

Discontinuous motion of the particle can be decomposed in shocks. Those shocks occur (here in $r_s$) while the parabola is shifted and touches the potential at two positions $u_1$ and $u_2$, as depicted. The size of the shock is denoted $s=u_2-u_1$.

Figure 5

PDF $\rho \left(s\right)$ of the shock size, plotted from Eq. (37) for $\mu =3/2$. In black dotted lines are the asymptotics for small and large $s$ as given by Eqs. (38) and (39).