First-passage and extreme-value statistics of a particle subject to a constant force plus a random force

Details First-passage and extreme-value statistics of a particle subject to a constant force plus a random force First-passage and extreme-value statistics of a particle subject to a constant force plus a random force

2008-07-10

arxiv.org/abs/0807.1671v2

J. Stat. Phys. 133, 217 (2008)

Physics

Condensed Matter

Statistical Mechanics

Contains an analysis of the extreme-value statistics not included in first version

Scientific paper

10.1007/s10955-008-9615-y

We consider a particle which moves on the x axis and is subject to a constant force, such as gravity, plus a random force in the form of Gaussian white noise. We analyze the statistics of first arrival at point $x_1$ of a particle which starts at $x_0$ with velocity $v_0$. The probability that the particle has not yet arrived at $x_1$ after a time $t$, the mean time of first arrival, and the velocity distribution at first arrival are all considered. We also study the statistics of the first return of the particle to its starting point. Finally, we point out that the extreme-value statistics of the particle and the first-passage statistics are closely related, and we derive the distribution of the maximum displacement $m={\rm max}_t[x(t)]$.

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