Universal Statistics of Branched Flows

Even very weak correlated disorder potentials can cause extreme fluctuations in Hamiltonian flows. In two dimensions this leads to a pronounced branching of the flow. Although present in a great variety of physical systems, a quantitative theory of the branching statistics is lacking. Here, we derive an analytical expression for the number of branches valid for all distances from a source. We also derive the scaling relations that make this expression universal for a wide range of random potentials. Our theory has possible applications in many fields ranging from semiconductor to geophysics.

Figure 1

(a) Branched flow (dark gray is high intensity) from a plane source, caused by a weak disorder potential (low potential in green, high potential in yellow, with standard deviation $ϵ=6\%$ of the kinetic energy of the particles). We want to count the number of branches per unit length (along the red line) at some distance away from the source. The number of branches is (to a very good approximation) equal to half the number of caustics (red dots). (b) Intensity profile $\rho (y)$ along the red line. Branches, bounded by two caustics and with an increased density in the region between them, show up clearly. The initial density is normalized to one, which is drawn as a dashed black line, together with a line for zero density. (c) Histogram of the values of the random potential used in (a) with the same color code. The potential is clearly very weak compared to the energy of the flow, which is here normalized to one (indicated by the dashed vertical line).

Figure 2

(a) Second moments of $m(t)$ (circles are numerical values, solid line is the analytical prediction) and $n(t)$ (stars are numerical values, solid line is the analytical prediction). (b) Probability to reach a caustic along a trajectory $P_c(t)$, numerical calculation (stars) and composite analytical solution (solid line). All calculations in this figure are performed using an ensemble of Gaussian correlated disorder potentials with $ϵ=0.04E_0$ and ${\ell }_c=0.1$.

Figure 3

Short- and long-term asymptotics as well as compound solution of $N_b(t)$, numerical data for $ϵ=0.04E_0$ and with ${\ell }_c=0.1$.

Figure 4

(a) Number of branches $N_b(t)$ for different sets of parameters of the random potential ($ϵ$ in percent of the total particle energy), different correlation functions, and with a fully two-dimensional simulation (2D). We note again that $t$ corresponds to the distance away from the source. The type of correlation function is indicated by Roman numerals: Gaussian (I), exponential (II), and power law with $\alpha =1,2,3,4$ (III, IV, V, VI) (cf. text for details). (b) Same curves as in (a), but $t$ scaled by $t_0$ and $N_b(t)$ by ${\sigma }_2/{\sigma }_1$, together with analytical prediction (solid black line).