Slow Kinetics of Brownian Maxima

Abstract

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.

arXiv comment5 pages, 4 figures

References in corpus (23)

• Persistence and First-Passage Properties in Non-equilibrium Systems
• Free Energy Functional for Nonequilibrium Systems: An Exactly Solvable Case
• Nontrivial Exponent for Simple Diffusion
• Persistent spins in the linear diffusion approximation of phase ordering and zeros of stationary gaussian processes
• Universal Record Statistics of Random Walks and Lévy Flights
• Random Convex Hulls and Extreme Value Statistics
• First-Passage Kinetic Monte Carlo method
• Exact distribution of the maximal height of p vicious walkers
• Theory of Single File Diffusion in a Force Field
• Go with the Winners: a General Monte Carlo Strategy
• Kinetics of a Diffusive Capture Process: Lamb Besieged by a Pride of Lions
• Record occurrence and record values in daily and monthly temperatures
• First-Passage Exponents of Multiple Random Walks
• Record-Breaking Avalanches in Driven Threshold Systems
• Single File Diffusion enhancement in a fluctuating modulated 1D channel
• Ordering of Random Walks: The Leader and the Laggard
• Strong Mobility in Weakly Disordered Systems
• Kinetics of First Passage in a Cone
• Statistics of Superior Records
• Life and Death at the Edge of a Windy Cliff
• Scaling Exponent for Incremental Records
• Survival of a Diffusing Particle in a Transverse Shear Flow: A First-Passage Problem with Continuously Varying Persistence Exponent
• Persistence of Random Walk Records

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• Scaling Exponents for Ordered Maxima
• Diffusive Persistence on Disordered Lattices and Random Networks
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• Fluctuations and Persistence in Quantum Diffusion on Regular Lattices