Statistics of Branched Flow in a Weak Correlated Random Potential

Recent images of electron flow through a two-dimensional electron gas device show branching behavior that is reproduced in numerical simulations of motion in a correlated random potential [M. A. Topinka et al., Nature 410, 183 (2001)]. We show how such branching arises from caustics in the classical flow and find a simple scaling behavior of the branching under variation of the random potential strength. Analytic results describing statistical properties of the branching are confirmed by classical and quantum numerical tests.

Figure 1

Number of visible branches as a function of dimensionless scaled time $t/t_0$, where the characteristic time scale $t_0$ depends on the strength $v_0$ of the random potential, Eq. (3). A visible branch is defined as an intensity enhancement of at least a factor of 10 above the background. The two dashed curves correspond to different values of the smoothing length $b$. The solid line represents the predicted exponential falloff at long times [Eq. (9)], with slope ${\alpha }^2/\beta =0.30$ ($\alpha =0.65$, $\beta =1.42$).

Figure 2

Exponential falloff of maximum branch intensity $I_b^{\mathrm{m}\mathrm{a}\mathrm{x}}$ with scaled time $t/t_0$. The three dashed curves correspond to classical simulations for several values of $v_0$ and smoothing scale $b$, and the solid curve is a quantum calculation. The power-law dependence of $I_b^{\mathrm{m}\mathrm{a}\mathrm{x}}$ on $b$ confirms that the peaks arise from simple caustics (folds). Finally, the straight line corresponds to the theoretical prediction of Eq. (10), with analytically predicted exponent $\gamma =0.17$ ($\alpha =0.65$, $\beta =1.42$).

Figure 3

Fraction of space $f^{\mathrm{b}\mathrm{r}}$ covered by branches taller than the arbitrary cutoff $I^{\mathrm{c}\mathrm{u}\mathrm{t}}=10$, as a function of scaled time $t/t_0$. Again, correct scaling is observed under changes of the potential strength $v_0$. The slope of the solid line is the analytical prediction $2\gamma =0.34$ of Eq. (11).

Figure 4

Cumulative probability distribution ${\int }_{I_b}^{\infty }d{I}_b^{\prime }P(I_b^{\prime })$ of intensities $I_b$ at a fixed scaled time $t/t_0=2.2$. The slope $-2$ of the solid line is the theoretical prediction of Eq. (12). Note that the power-law behavior breaks down as we approach the maximum intensity from Eq. (10), $I_b^{\mathrm{m}\mathrm{a}\mathrm{x}}\approx e^{4.5}$ for $b=0.0003$ and $I_b^{\mathrm{m}\mathrm{a}\mathrm{x}}\approx e^{5.5}$ for $b=0.00003$.