Deterministic and stochastic models for coalescence (aggregation and coagulation): A review of the mean-field theory for probabilists.

*(English)*Zbl 0930.60096Deterministic and stochastic models of coalescence (i.e. coagulation, gelation, aggregation, agglomeration, accretion, etc.) have been studied long times ago in the physical chemistry literature. M. Smoluchowski stated [Phys. Z. 17, 557-585 (1916)] the following coagulation-equations for the densities \(n(x,t)\):
\[
{d\over dt} n(x,t)={1\over 2} \sum^{x- 1}_{y= 1} K(y,x- y) n(y,t) n(x- y,t)- n(x,t) \sum^\infty_{y= 1} K(x,y) n(y,t)\quad(\text{discrete }x),\tag{1}
\]

\[ \begin{split}{d\over dt} n(x,t)={1\over 2} \int^x_0 K(y, x-y) n(y,t) n(x- y,t) dy-\\ n(x,t) \int^\infty_0 K(x,y) n(y,t) dy\quad (\text{continuous }x),\end{split}\tag{1\('\)} \] where \(x\) and \(y\) denote the masses of two arbitrary clusters. In the above equation (1) (resp. \((1')\)) we may regard the masses \(x\) (or \(y)= 1,2,3,\dots\) (resp. \(0< x\), \(y<\infty\)) as discrete (resp. real) and so in the discrete case (1) (resp. continuous case \((1')\)) the density \(n(x,t)\) is defined as the average number of clusters of mass \(x\) (resp. of mass belonging to \([x,x+ dx]\)) per unit volume at time \(t\). Next, there is an instantaneous rate at which the cluster of mass \(x\) merges with some cluster of mass \(y\), which is assumed to be proportional to the density \(n(y,t)\) with the constant of proportionality denoted by \(K(x,y)\). For instance, in Kingman’s coalescent in population genetics and component sizes in random graphs the special cases where \(K(x,y)= 1\) and \(K(x,y)= xy\) are investigated, respectively.

As mentioned by the author in the introduction: “General rate kernels \(K(x,y)\) are only now starting to be studied rigorously; so many interesting open problems appear” (namely 10 open problems are stated in the Sections 2 and 5 of the paper). As also explained by the author in the introduction: “The purpose of this survey is to bring the existence of this large body of scientific literature to the attention of theoretical and applied probabilists. We shall provide pointers to the science literature, outline some of the mathematical results developed therein, comment on the duality between coalescence and branching processes and pose some mathematical problems. That an opportunity arises in outline recent work of the author and colleagues is, of course, purely coincidental…”

Section 2 reviews the deterministic Smoluchowski coagulations equations…; this is the aspect of coalescence which has been most intensely studied in the scientific literature. This aspect is not “probabilistic”, but the remainder of the survey is. Section 3 gives “probabilistic” interpretations of some deterministic results about the Smoluchowski coagulations equations, using duality with branching-type processes. Here the idea of coalescence as the time reversal of splitting is used explicitly in the Subsection 3.3 to give a general construction for general kernels.

In the Sections 4 and 5 the main focus is on the “finite-volume mean field theory” of the Marcus-Lushnikov process. This is a stochastic model with \(N\) particles, which merge into clusters according to the following rule: a cluster of size \(x\) and a cluster of size \(y\) merge at (stochastic) rate \(K(x,y)/N\) and we seek to understand its large-\(N\) behaviour. The author emphasizes in Section 4 the three simplest specific kernels \(K\) (constant, additive and multiplicative), for which a rich and fairly explicit theory exists, with connections to other parts of mathematical probability (Kingman’s coalescent, discrete and continuum random trees; random graphs). The last Section 5 is deserved to the discussions of general kernels and some related open problems.

It is no doubt that the survey paper under review is a very substantial and concise document for both “theorem-proof” and “scientific modelling mathematics” communities, however the reviewer feels that it would be much more interesting for “mathematicians” if the author is more precise on some probabilistic terminologies. For instance, what is meant by “Feller property” (resp. “entrance boundary”) at page 36 (resp. 37); should we understand that in this case the semigroup of transition of the process should be Fellerian? (resp. that in this case the boundary is in the Martin’s sense?).

\[ \begin{split}{d\over dt} n(x,t)={1\over 2} \int^x_0 K(y, x-y) n(y,t) n(x- y,t) dy-\\ n(x,t) \int^\infty_0 K(x,y) n(y,t) dy\quad (\text{continuous }x),\end{split}\tag{1\('\)} \] where \(x\) and \(y\) denote the masses of two arbitrary clusters. In the above equation (1) (resp. \((1')\)) we may regard the masses \(x\) (or \(y)= 1,2,3,\dots\) (resp. \(0< x\), \(y<\infty\)) as discrete (resp. real) and so in the discrete case (1) (resp. continuous case \((1')\)) the density \(n(x,t)\) is defined as the average number of clusters of mass \(x\) (resp. of mass belonging to \([x,x+ dx]\)) per unit volume at time \(t\). Next, there is an instantaneous rate at which the cluster of mass \(x\) merges with some cluster of mass \(y\), which is assumed to be proportional to the density \(n(y,t)\) with the constant of proportionality denoted by \(K(x,y)\). For instance, in Kingman’s coalescent in population genetics and component sizes in random graphs the special cases where \(K(x,y)= 1\) and \(K(x,y)= xy\) are investigated, respectively.

As mentioned by the author in the introduction: “General rate kernels \(K(x,y)\) are only now starting to be studied rigorously; so many interesting open problems appear” (namely 10 open problems are stated in the Sections 2 and 5 of the paper). As also explained by the author in the introduction: “The purpose of this survey is to bring the existence of this large body of scientific literature to the attention of theoretical and applied probabilists. We shall provide pointers to the science literature, outline some of the mathematical results developed therein, comment on the duality between coalescence and branching processes and pose some mathematical problems. That an opportunity arises in outline recent work of the author and colleagues is, of course, purely coincidental…”

Section 2 reviews the deterministic Smoluchowski coagulations equations…; this is the aspect of coalescence which has been most intensely studied in the scientific literature. This aspect is not “probabilistic”, but the remainder of the survey is. Section 3 gives “probabilistic” interpretations of some deterministic results about the Smoluchowski coagulations equations, using duality with branching-type processes. Here the idea of coalescence as the time reversal of splitting is used explicitly in the Subsection 3.3 to give a general construction for general kernels.

In the Sections 4 and 5 the main focus is on the “finite-volume mean field theory” of the Marcus-Lushnikov process. This is a stochastic model with \(N\) particles, which merge into clusters according to the following rule: a cluster of size \(x\) and a cluster of size \(y\) merge at (stochastic) rate \(K(x,y)/N\) and we seek to understand its large-\(N\) behaviour. The author emphasizes in Section 4 the three simplest specific kernels \(K\) (constant, additive and multiplicative), for which a rich and fairly explicit theory exists, with connections to other parts of mathematical probability (Kingman’s coalescent, discrete and continuum random trees; random graphs). The last Section 5 is deserved to the discussions of general kernels and some related open problems.

It is no doubt that the survey paper under review is a very substantial and concise document for both “theorem-proof” and “scientific modelling mathematics” communities, however the reviewer feels that it would be much more interesting for “mathematicians” if the author is more precise on some probabilistic terminologies. For instance, what is meant by “Feller property” (resp. “entrance boundary”) at page 36 (resp. 37); should we understand that in this case the semigroup of transition of the process should be Fellerian? (resp. that in this case the boundary is in the Martin’s sense?).

Reviewer: Xuan Loc Nguyen (Hanoi)

##### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |