The Contact Field: Ray-Knight Laws and Reaction in a Distributed Reactive Medium

A particle diffusing through a reactive medium reacts not on first contact but when the contact it has accumulated crosses a threshold, so the quantity that decides its fate is a random field: the contact accumulated at every level it can react at. For one-dimensional Brownian motion that field has an exact law — the Ray-Knight theorems identify it, read in the space variable, as a squared Bessel process — and this paper applies that law to the functional physical reaction theory measures. The result is that the survival probability in an arbitrary reactivity profile $\hat v$ is the reciprocal of $W(a)$, where $W” = 2\hat v W$ with $W(0) = 1$ and $W'(0) = 0$: one linear equation with two initial conditions, the reactivity entering as a potential, with no eigenvalue problem and no geometry remaining.

Two cases fix the result. A medium of finite width $w$ and reduced reactivity $\hat v_0$ has the closed form $S = [\cosh kw + k\sinh kw\,(a-w)]^{-1}$ with $k = \sqrt{2\hat v_0}$; a reactive point of the same total strength has $S = (1 + 2\hat\kappa a)^{-1}$, which recovers Collins-Kimball kinetics and the Robin boundary condition. Between them, a medium of finite width behaves as a point whose reactivity is reduced by the factor $1 – w/2a$ to leading order, a deficit set by the geometry of the search and independent of the strength of the chemistry. At the parameters used throughout, simulation gives $0.67569 \pm 0.00113$ against an exact $0.676195$ for the medium and $0.66681 \pm 0.00117$ against an exact $2/3$ for the point, both scored on one batch of paths so that the gap between them is resolvable.

Because the same equation returns the transform at every argument, the whole law of the reacted contact is available, and with it the kinetics of a surface whose reaction threshold is not exponential — which no local boundary condition can express.