As discussed in Part III, diffusion–limited reactions are generally studied through a variety of approximation and computer–simulation techniques, since a comprehensive exact method of analysis has not yet been suggested. For this reason, exactly solvable models are of extreme importance: they serve as benchmark tests for the existing approximation and simulation methods; the exact techniques may hint at a more general approach, and serve as a basis for better approximations; and they contribute enormously to our understanding of the field as a whole.

In Part IV, we discuss a particular example of an exactly solvable model – that of diffusion-limited coalescence, A + A→ A, in one dimension. The model has been studied extensively by numerous researches, who have thought up an impressive amount of imaginative, elegant, solutions. The description of these works would require an additional volume, so they are acknowledged only in the bibliography. Instead, we limit ourselves to the method of interparticle distribution functions (IPDF), merely because we took part in its development and we understand it best. It should likewise be noted that neither is our model of choice the only one which can be solved exactly (very few others exist, though). The coalescence model yields an astonishingly wide range of kinetic behavior, well beyond what might be suspected from its stark simplicity. The restriction of the model to one dimension commonly draws criticism. This is dictated by the need to find exact solutions. On the other hand, recall that diffusion–limited kinetics is more anomalous the lower the dimension, so there is an advantage in studying one–dimensional models, in which differences from mean–field classical behavior are most pronounced.