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.