Slow Kinetics of Brownian Maxima
E. Ben-Naim and P.L. Krapivsky
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 ~ t^(-beta) with
exponent beta=1/4. When the two particles have diffusion constants D1
and D2, the exponent depends on the mobilities,
beta=(1/pi)arctan[sqrt(D2/D1)]. We also use numerical simulations to
investigate maxima of multiple particles in one dimension and the
largest extension of particles in higher dimensions.
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