Field-theory methods in coagulation theory
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CLEI Theory
Field-Theory Methods In Coagulation Theory A. A. Lushnikov* Karpov Institute of Physical Chemistry, ul. Vorontsovo pole 10, Moscow, 103064 Russia Received November 16, 2010
Abstract—Coagulating systems are systems of chaotically moving particles that collide and coalesce, producing daughter particles of mass equal to the sum of the masses involved in the respective collision event. The present article puts forth basic ideas underlying the application of methods of quantum-field theory to the theory of coagulating systems. Instead of the generally accepted treatment based on the use of a standard kinetic equation that describes the time evolution of concentrations of particles consisting of a preset number of identical objects (monomers in the following), one introduces the probability W (Q, t) to find the system in some state Q at an instant t for a specific rate of transitions between various states. Each state Q is characterized by a set of occupation numbers Q = {n1 , n2 , . . . , ng , . . .}, where ng is the total number of particles containing precisely g monomers. Thereupon, one introduces the generating functional Ψ for the probability W (Q, t). The time evolution of Ψ is described by an equation that is similar to the ¨ Schrodinger equation for a one-dimensional Bose field. This equation is solved exactly for transition rates proportional to the product of the masses of colliding particles. It is shown that, within a finite time interval, which is independent of the total mass of the entire system, a giant particle of mass about the mass of the entire system may appear in this system. The particle in question is unobservable in the thermodynamic limit, and this explains the well-known paradox of mass-concentration nonconservation in classical kinetic theory. The theory described in the present article is successfully applied in studying the time evolution of random graphs. DOI: 10.1134/S1063778811080114
1. INTRODUCTION I am grateful to the destiny that Academician Arkadii Benediktovich Migdal (A.B., as everybody, but not I, called him) was my teacher. Now, I will also use this abbreviation. Very frequently, I recall one wise thought that I first heard from A.B. nearly half a century ago. Many times, he repeated that each researcher should have a study that he thinks to be of a greater value than all of his other achievements. For himself, Migdal deemed that his article on electron–phonon interaction in the Journal of Experimental and Theoretical Physics of 1957 [1] stands out among his contributions to science. Even though almost all of A.B.’s studies were brilliant, it was precisely this one that he appreciated higher than all others. Of course, I desired strongly to have such a study. Fortunately, I do have it [2]. What you will read below is my present view of those problems in coagulation theory that I came across more than three decades ago [2–5]. In solving the master kinetic equation of coagulation theory, it turned out that the total mass *
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