Single-Species Reactions on a Random Catalytic Chain

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We present an exact solution for a catalytically-activated annihilation A + A → 0 reaction taking place on a one-dimensional chain in which some segments (placed at random, with mean concentration p) possess special, catalytic properties. Annihilation reaction takes place, as soon as any two A particles land from the reservoir onto two vacant sites at the extremities of the catalytic segment, or when any A particle lands onto a vacant site on a catalytic segment while the site at the other extremity of this segment is already occupied by another A particle. We find that the disorder-average pressure P(quen) per site of such a chain is given by P(quen) = P(lan) + β⁻¹ F, where P(lan) = β⁻¹ (1+z) is the Langmuir adsorption pressure, (z being the activity and β⁻¹ - the temperature), while β⁻¹ F is the reaction-induced contribution, which can be expressed, under appropriate change of notations, as the Lyapunov exponent for the product of 2 × 2 random matrices, obtained exactly by Derrida and Hilhorst (J. Phys. A 16, 2641 (1983)). Explicit asymptotic formulae for the particle mean density and the compressibility are also presented.