Gelation in coagulating systems

A review of recent achievements in the theory of sol–gel transitions is given. This paper outlines the basic ideas of a stochastic approach and describes its possible application as a basis for a theory of gelation in coagulating systems. The kinetics of coagulation is described in terms of the probability W to find in the system a given set of monomers, dimers, etc., at time t . This approach allows for a consideration of finite coagulating systems, i.e., those containing a finite number of monomeric units. The evolution equation for the generating functional of W is formulated and solved exactly for the coagulation kernel proportional to the product of masses of coagulating particles for which the traditional Smoluchowski’s scheme leads to a paradox related to the loss of the total mass of the coagulating particles. This paradox is resolved within the scheme described in this paper. It is shown that a single giant object (a gel) invisible in the thermodynamic limit arises after a critical time and begins to consume the mass of smaller but higher populated fraction (the sol). This approach is then applied for considering the time evolution of random graphs. The distribution of the linked components of a random graph over their orders and numbers of edges is found exactly. The paper is concluded with a short consideration of the sol–gel transition in systems with instantaneous sinks at large particle mass. It is demonstrated how the sol–gel transition manifests itself in such systems.