We study binary one-dimensional exchange-mediated A +/?-• 0 reactions in the absence of diffusion. Using the Kirkwood superposition approximation to decouple the infinite hierarchy of many-particle densities a closed set of integrodifferential equations is derived. A comparison to simulation calculations shows that the approximation reproduces the correct long-time asymptotic decay of reactant concentration as well as the correct spatial correlations of the segregation pattern.PACS numbers: 82.20.-w, 05.40.+J Recent years have seen a drastic increase in interest inthe A +B-• 0 reaction scheme, following the realization that segregation of like particles occurs, 1 " 4 which leads to long-time behavior different from the classical kinetic scheme. A recent review article summarizes the findings. 4 In contrast to the A+A-+ A, A+A-^>0 9 and A+B-+B diffusion-limited reactions, for which in the last years several exact ID solutions have become available, 5 " 9 the analytic work for A +B-> 0 reactions is still based (even in ID) on approximate (although increasingly reliable) analytical methods. 4,10Most of the analyses have centered on diffusionlimited reactions in which the reactants move (either diffusionally or as a random walk) before reacting on contact. By contrast, far less attention has been paid to reactions in which the particles annihilate via longerranged interactions, such as exchange, in the absence of motion. Since motion and reaction often cause adverse effects on segregation, 1,2,10 we consider here only reactions with immobile reactants. Such reactions, especially when taking place in spaces of low dimensions, provide a stringent test of approximate analytical approaches, since possible deviations of the theory from the computer experiments are not obscured by diffusion effects or by the (often milder) behavior in higher dimensions.Interestingly, we find expressions which use the Kirkwood superposition approximation 11 for the three-center correlation functions to perform exceedingly well even for short-range exchange reactions in ID, and to provide both the correct long-time asymptotic behavior of the reaction, as well as the correct ^42?-particle segregation pattern. This fact is quite remarkable, since then the method may represent a powerful analytical approach to the A +B-• 0 problem, even when diffusion is involved. By contrast, neglect of the segregation aspect leads to (even qualitatively) wrong asymptotic forms.In the following we consider randomly distributed A and B particles, which react via exchange, w>(r) -w(r) = woexp( -r/ro), where w is the probability rate that A and B situated at distance r react and ro is a constant, which determines the interaction range. The exponential

Polymer science, an interdisciplinary science well-rooted in organic chemistry and in materials science, encompasses an inordinate number and diversity of substance classes and thus has far-reaching applications. Interestingly, polymers also represent a great challenge to the theoreticians, since their theoretical treatment often necessitates appropriate extensions of the classical methods from solid state physics and from statistical physics. Thus, new concepts often have to be invoked when considering the special properties of polymers. In this review we concentrate on one of the modern concepts in the theory of polymers, namely on scaling. Scaling is closely associated with new developments in the field of fractals and of hierarchical structures. Such concepts are invaluable for the modeling of complex geometries and for describing dynamical processes in polymeric materials. Here, we focus on a presentation of these ideas and we outline examples of recent research in which these concepts have been successfully applied.

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Hybrid treatment of spatiotemporal behavior in surface reactions with coexisting immobile and highly mobile reactantsWe study the bimolecular annihilation reaction A + B ---+ 0 with immobile reactants in arbitrary dimension. For this we derive a closed set of integrodifferential equations by using Kirkwood's superposition approximation to decouple the infinite hierarchy of equations for the manycenter correlation functions. We find that for exchange-type interactions and equal numbers of A and B species the reactant concentration n decays as n ~ Sd /2, where S is time-dependent, S = In t, and may be interpreted to be an effective reaction radius. A comparison to numerical simulations shows very good agreement to the theoretical expressions, both for the temporal evolution of particle concentrations and also for the pair-correlation functions; this confirms the validity of the superposition approximation.