Cumulative Damage Distributions

Abstract

The density of the first-passage time for a particle, with drift, undergoing Brownian motion was first obtained by E. Schrödinger [101, 1915]. He considered N particles in Brownian motion, all initially at zero and white. When one reaches a distance ℓ it becomes green. Let the probability of first passage beyond ℓ in the interval (t, t + Δt) be ∫ ^t+Δt_t with p(·) unknown. If NP_w(t) is the expected number of white particles at t > 0, then

$$ N\int_t^{t + \Delta t} {p\left( x \right)dx = N[NP_w \left( t \right) - NP_w \left( {t + \Delta t} \right)] implies p\left( t \right) = - P_w^\prime \left( t \right).} $$

From assumed Brownian motion of velocity v the density of particles, at position x at time t > 0, is

$$ \rho \left( {x,t} \right) = \frac{N} {{\sqrt {2\pi \sigma ^2 t} }}e^{ - \frac{{\left( {x - \nu t} \right)^2 }} {{2\sigma ^2 t}}} . $$

Make the transformation y = x − vt to obtain particles without drift, namely,

$$ \rho \left( {y,t} \right) = \frac{N} {{\sqrt {2\pi \sigma ^2 t} }}e^{ - \frac{{y^2 }} {{2\sigma ^2 t}}} . $$

Schrödinger recognizes this as the solution of the heat-diffusion equation, viz.,

$$ \frac{{\sigma ^2 }} {2}\frac{{\partial ^2 \rho }} {{\partial y^2 }} = \frac{{\partial \rho }} {{\partial t}} with b.c. \rho _w \left( {\ell - \nu t,t} \right) = 0. $$

He then obtained the solution for the density using the reflection principle, viz.,

$$ \rho _w \left( {y,t} \right) = \frac{N} {{\sqrt {2\pi \sigma ^2 t} }}\left[ {e^{ - \frac{{y^2 }} {{2\sigma ^2 t}}} - e^{2\ell \nu /\sigma ^2 } e - \frac{{\left( {y - \nu t} \right)^2 }} {{2\sigma ^2 t}}} \right]. $$