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 Incoming links: Survival probability from an absorbing transition density

Past exam of the theoretical physics II course of the University of Cambridge / 2015 / Paper 1 / 2 / e / Solution 2026-10-20

The survival probability from an absorbing transition density follows by integrating over final position. Since

\int_{- L}^{L} cos \frac{( 2 j + 1 ) π x}{2 L} d x = \frac{4 L ( - 1 ) ^{j}}{( 2 j + 1 ) π},

the exact survival probability for

t > 0

is

S (t) = \frac{4}{π} \sum_{j = 0}^{\infty} \frac{( - 1 ) ^{j}}{2 j + 1} e^{- (2 j + 1)^{2} u}, u = \frac{π ^{2} D t}{4 L ^{2}} .

The slowest Dirichlet mode dominates at large time:

S (t) \sim \frac{4}{π} exp (- \frac{π ^{2} D t}{4 L ^{2}}) .

The first relative correction is

- \frac{1}{3} e^{- 8 u}

. The approximation is a late-time result; extrapolating its prefactor to

t = 0

would incorrectly give a probability larger than one. The full alternating series tends to one there.

Figure 1.
Original absorbing-interval transition densities and the approach of the survival probability to its slowest-mode asymptote
. Details.

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