Multiparticle diffusion limited [Formula: see text] reaction in small volumes

A multiparticle reaction model in which particles A annihilate in clusters of size k as [Formula: see text] is investigated analytically. The system is studied for arbitrary reaction order k > 2, dimension d, and size L. Particles diffuse with diffusion constant D, and annihilate with rate σ which depends on the positions of kA particles in the cluster prior to a reaction. The particles are assumed to be spatially extended objects with radius a. Exclusion effects are not taken into account since A particles are allowed to overlap. The master equation is rephrased in the language of a field theory which, in turn, is used to derive the equations of motion for many-point densities. An approximate form of the equations of motion was solved analytically in the diffusion-controlled limit (infinite reaction rate). An explicit expression for the effective reaction rate has been found in the form of the Laplace transform. It was shown that the number of particles saturates to a constant value for large times. The value is approached through an exponential decay. The exponential decay constant is the non-algebraic function of particle size a and system size L.