{'fraction_non_alphanumeric': 0.041425020712510356, 'fraction_numerical': 0.014913007456503728, 'mean_word_length': 4.69811320754717, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Single-species reactions on a random catalytic chain

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. An 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−1F, where P(Lan) = β−1 ln(1 + z) is the Langmuir adsorption pressure, (z being the activity and β−1 the temperature), while β−1F is the reaction-induced contribution, which can be expressed, under appropriate change of notation, as the Lyapunov exponent for the product of 2 × 2 random matrices, obtained exactly by Derrida and Hilhorst (1983 J. Phys. A: Math. Gen. 16 2641). Explicit asymptotic formulae for the particle mean density and the compressibility are also presented.