Slow Kinetics of Brownian Maxima

We study extreme-value statistics of Brownian trajectories in one dimension. We define the maximum as the largest position to date and compare maxima of two particles undergoing independent Brownian motion. We focus on the probability $P(t)$ that the two maxima remain ordered up to time $t$ and find the algebraic decay $P\sim t^{-\beta }$ with exponent $\beta =1/4$. When the two particles have diffusion constants $D_1$ and $D_2$, the exponent depends on the mobilities, $\beta =(1/\pi )\arctan \sqrt{D_2/D_1}$. We also use numerical simulations to investigate maxima of multiple particles in one dimension and the largest extension of particles in higher dimensions.

Figure 1

Space-time diagram of the positions (thin lines) and the ordered maxima (thick lines) of two Brownian particles.

Figure 2

Sample trajectory of the composite particle. There is upward drift along the boundary $u=0$ and the boundary $v=0$ is absorbing.

Figure 3

The probability $P$ versus time $t$. Shown are Monte Carlo simulation results obtained from ${10}^8$ independent realizations (circles). Also shown as reference (line) is the theoretical prediction (6).

Figure 4

The exponent $\beta$ versus the ratio $D_1/D_2$. The line corresponds to the theoretical curve (1) and the dots to results of Monte Carlo simulations with ${10}^7$ independent realizations.