Random search for a partially reactive target by multiple diffusive searchers

We study the first-passage problem for a partially reactive target by N identical diffusive particles in a finite d-dimensional space, laying a focus on the effects of the partial reactivity when searchers are initially excluded from the target region. By solving the Fokker-Planck equation, we obtain the mean first-passage time that exhibits a power-law dependence on the number of searchers as τ_{N}∼N^{-α}, proving that the exponent α varies with dimensionality, reactivity, and the number of searchers, and specifying conditions in which crossovers between different exponents occur. We confirm the validity of our analytic results by performing Langevin dynamics simulations for various sets of system parameters.