Being Heterogeneous Is Advantageous: Extreme Brownian Non-Gaussian Searches

Redundancy in biology may be explained by the need to optimize extreme searching processes, where one or few among many particles are requested to reach the target like in human fertilization. We show that non-Gaussian rare fluctuations in Brownian diffusion dominates such searches, introducing drastic corrections to the known Gaussian behavior. Our demonstration entails different physical systems and pinpoints the relevance of diversity within redundancy to boost fast targeting. We sketch an experimental context to test our results: polydisperse systems.

Figure 1

Comparison between Gaussian and non-Gaussian PDFs for subordination processes. The two PDFs share the same mean and standard deviation but the non-Gaussian one has an excess probability both in the tails and in the center part. The non-Gaussian PDF is obtained from the FSP model with $p=0.99$ (see text).

Figure 2

Extreme-MFPT for the DD model (orange or gray in the gray scale version) and the Gaussian one (black) vs $M$. Note that the ratio $\mathbb{E}[T_M]/{\tau }_{\mathrm{av}}$ does not depend on $\ell$ and $D_{\mathrm{av}}$. Symbols refer to numerical simulations, while solid lines are theoretical estimations from Eqs. (S8) (black) and (S16) (orange or gray in the gray scale version) of the Supplemental Material [48]. Simulations are with $d_{\mathit{Y}}=1$ and for different values of $\tau$ (details are reported in the Supplemental Material [48]). Inset: Crossover between the SS and the LD regime, attained upon changing $\tau$ at fixed $M$.

Figure 3

Extreme-MFPT for FSP model (cyan or gray in the gray scale version). In analogy to Fig. 2, symbols refer to numerical simulations, while solid lines are theoretical estimations from Eqs. (S8) (black) and (S23) (cyan or gray in the gray scale version) of the Supplemental Material [48]. Inset: behavior with respect to $p$ at fixed $M$; the solid line indicates the theoretical trend from Eq. (9). Simulation details are reported in [48].