4 February 2000 Ž . Chemical Physics Letters 317 2000 481–489 www.elsevier.nlrlocatercplett Differential approach to the memory-function reaction kinetics P.A. Frantsuzov a,1 , O.A. Igoshin b , E.B. Krissinel c, ),2 a National Institute of Materials and Chemical Research, Tsukuba, Ibaraki 305-8565, Japan b Chemical Physics Department, Weizmann Institute of Science, RehoÕot 76100, Israel c Fakultat fur Chemie, UniÕersitat Konstanz, D-78437 Konstanz, Germany ¨ ¨ ¨ Received 13 October 1999; in final form 20 December 1999 Abstract Ž . A new formulation of the encounter theory for liquid-phase many-particle reacting systems MPRS is developed. This new formulation is based upon the notion of the pair distribution function and results in a set of partial differential equations equivalent to the integro-differential equations of the conventional approach. The solution is greatly improved by considering pair–particle interactions. The new formulation is applicable to arbitrary MPRS and allows for a general numerical solution. Applied to the Lotka–Volterra reaction, this approach revealed damping and increasing oscillations on the concentration, which cannot be seen in the framework of phenomenological chemical kinetics. q 2000 Elsevier Science B.V. All rights reserved. 1. Introduction A system of reacting particles in liquids is per- haps one of the most typical occurrences met in chemical research which still is a challenging prob- lem for theoretical investigation. Whenever, with the exception of a few special cases, consideration goes beyond the limits of a single reacting pair, there is no general recipe for obtaining an exact solution which would be valid for any reactant concentration. ) Corresponding author: Fax: q49-7531-88-3014; e-mail: eugene@sg7.chemie.uni-konstanz.de 1 On leave from the Institute of Chemical Kinetics and Com- bustion, Siberian Branch of the Russian Academy of Sciences, Institutskaya 3, Novosibirsk 630090 Russia. 2 On leave from the Institute for Water and Environmental Problems, Siberian Branch of the Russian Academy of Sciences, Papanintsev 105, Barnaul 656099, Russia. However, in the most typical cases, the concentra- tion of reactants may be considered as a small parameter since 4 3 j s p ac <1. 1.1 Ž . 0 3 Here c is the concentration of a reactant and a is 0 the characteristic size of the reaction zone, estimated as the sum of reactant radii. Most modern theoretical approaches to many-particle reacting systems Ž . Ž . MPRS in liquids imply assumption 1.1 . These approaches are represented by two general Ž w x. directions. The first direction cf. Refs. 1–5 is based on the Kirkwood superposition approximation Ž . SA . It attempts to describe MPRS kinetics in terms Ž . of time-dependent reaction rate constants that are to be obtained from the kinetic equations for the pair wx distribution functions. This may be considered 6 as a generalization of the Smoluchowski theory for 0009-2614r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. Ž . PII: S0009-2614 99 01440-2