Post-gelation behavior of a spatial coagulation model

A coagulation model on a ﬁnite spatial grid is considered. Particles of discrete masses jump randomly between sites and, while located at the same site, stick together according to some coagulation kernel. The asymptotic behavior (for increasing particle numbers) of this model is studied in the situation when the coagulation kernel grows suﬃciently fast so that the phenomenon of gelation is observed. Weak accumulation points of an appropriate sequence of measure-valued processes are characterized in terms of solutions of a nonlinear equation. A natural description of the behavior of the gel is obtained by using the one-point compactiﬁcation of the size space. Two aspects of the limiting equation are of special interest. First, for a certain class of coagulation kernels, this equation diﬀers from a naive extension of Smoluchowski’s coagulation equation. Second, due to spatial inhomogeneity, an equation for the time evolution of the gel mass density has to be added. The jump rates are assumed to vanish with increasing particle masses so that the gel is immobile. Two diﬀerent gel growth mechanisms (active and passive gel) are found depending on the type of the coagulation kernel.


Introduction
We consider a particle system The state space of a single particle is where G is a finite set of (spatial) locations. Particles jump between sites (x, α) → (x, β) according to some rate function and, while located at the same site, stick together (x, α), (y, α) → (x + y, α) following stochastic rules determined by some coagulation kernel. The index N = 1, 2, . . . denotes the number of monomers (units of size 1) in the system so that The discrete (both in space and size) model described above was used in [27] as an approximation to the spatially continuous coagulation equation with diffusion ∂ ∂t c(t, k, r) = D(k) ∆ r c(t, k, r)+ (1.4) Theoretical investigations of the gelation phenomenon go back to the paper [10] on condensation polymerization. Flory studied the size distribution of polymers and established critical conditions (in terms of a parameter called "extent of reaction") for the formation of "infinitely large" molecules (gel). Developing this approach, Stockmayer [26] pointed out a connection of the polymer size distribution with equation (1.5), where (1.7) Polymeric molecules (k-mers) are composed of k monomeric units. Each monomeric unit carries f functional groups capable of reacting with each other. Thus, the kernel (1.7) represents the number of possible links between x-mers and y-mers. Note that an equation with the commonly used multiplicative kernel K(x, y) = x y (1.8) can be obtained from equation (1.5) with the kernel (1. 7) in the limit f → ∞ , when time is appropriately scaled. Stockmayer [26] argued with Flory about the correct post-gelation behavior and proposed a solution different from Flory's. Early reviews of the subject were given in [11] and [12,Ch. IX]. An extended discussion of different solutions after the gel point and corresponding modified equations can be found in [32] (f = 3) and [34] (f > 2). The paper [32] contains a rather complete list of relevant earlier references.
Rigorous results concerning the derivation of the spatially inhomogeneous coagulation equation (1.4) from systems of diffusing spherical particles, interacting at contact, were obtained in [18] (constant kernel) and [24] (kernel (1.6)). Stochastic models of coagulation in the spatially homogeneous case go back to [22], [14], [21]. In those papers the coagulation kernel, which contains the information about the microscopic behavior of the physical system, is postulated. An extended review of the subject was given in [2]. We also refer to the recent paper [13] studying the spatially homogeneous case with rather general gelling kernels. When combining the Marcus-Lushnikov approach with spatial inhomogeneity, particles coagulate with a certain rate when they are close enough to each other (e.g., in the same cell). Convergence results for such models with non-gelling kernels were obtained in [15] (bounded kernel) and [3] (sub-linear kernel). The two-site case of the van Dongen model described above (with D(k) = 1 and kernel (1.8)) was studied in [25]. Analytical results concerning the coagulation equation with diffusion (1.4) (and references to earlier studies) can be found, e.g., in [19] and [20] (see also [8,Section 8]).
Equation (1.4) with constant diffusion coefficients D(k) = 1 and the multiplicative kernel (1.8) was studied in [17]. The following equation for the gel mass density g(t, r) was suggested, ∂ ∂t g(t, r) = ∆ r g(t, r) + R , (1.9) where R = lim k→∞ k x=1 ∞ y=k−x+1 x 2 c(t, x, r) y c(t, y, r) (1.10) is a "Radon measure describing the rate of gel production". Considering diffusion coefficients D(k) vanishing sufficiently fast (with k → ∞) and the multiplicative kernel (1.8), van Dongen [27] proposed a modification of equation (1.4), namely ∂ ∂t c(t, k, r) = D(k) ∆ r c(t, k, r)+ (1.11) 1 2 x+y=k x y c(t, x, r) c(t, y, r) − k c(t, k, r) y c(t, y, r) + g(t, r) where the time evolution of the gel mass density is determined by the equation ∂ ∂t g(t, r) = g(t, r) ∞ k=1 k 2 c(t, k, r) . (1.12) The paper is organized as follows. In Section 2 the asymptotic behavior (as N → ∞) of the particle system (1.1) is studied. Weak accumulation points of an appropriately scaled sequence of measure-valued processes (based on (1.1)) are shown to be concentrated on the set of solutions of a nonlinear equation. The results cover the situation, when the coagulation kernel grows sufficiently fast so that the phenomenon of gelation is observed. Using the one-point compactification of the size space and considering mass density instead of number density leads to a natural description of the behavior of the gel under rather general assumptions on the coagulation kernel. Section 3 is concerned with properties of the limiting equation. Two aspects are of special interest. First, for a certain class of coagulation kernels, this equation differs from a naive extension of Smoluchowski's coagulation equation (as, e.g., (1.11) compared to (1.4)). Second, an equation for the time evolution of the gel mass density has to be added (as, e.g., (1.9) or (1.12)). Note that the second aspect is absent in the spatially homogeneous situation, since the gel mass density is determined just as the mass defect of the solution c(t, k) . In the spatially inhomogeneous situation the gel is distributed over different sites and gel equations are of interest. In general, they describe both the spatial motion and the growth of the gel. In this paper the case of vanishing diffusion coefficients is considered so that the gel is immobile. The growth behavior depends on the kernel and is determined by terms of the type occurring in (1.10) or (1.12). Finally, Section 4 contains most of the technical proofs.
2 Asymptotic behavior of the stochastic model We represent the particle system (1.1) in form of measures on the state space (1.2), where δ z denotes the delta-measure concentrated in z ∈ Z . The transition kernel of the corresponding jump process is where 1 B denotes the indicator function of a set B , δ α,β is Kronecker's symbol, κ and K are non-negative functions on {1, 2, . . .} × G 2 and {1, 2, . . .} 2 × G , respectively, and are jump transformations. The kernel (2.2) is defined on the state space of the process (2.1) (cf. (1.2)), It satisfies The pathwise behavior of the process in terms of particles is obtained from the kernel (2.2). The jump process is regular, since the kernel is bounded.
Consider the space Let P(Z ) denote the space of probability measures on Z equipped with the topology of weak convergence. For ϕ ∈ C(Z ) and µ ∈ P(Z ) , we introduce the notations and where F ϕ (x, y, α) = (2.8) K(x, y, α) x y (x + y) ϕ(x + y, α) − x ϕ(x, α) − y ϕ(y, α) , x, y < ∞ , (2.10) Let K be symmetric and such that where the sign ⇒ denotes convergence in distribution.
Then the processes (2.1) form a relatively compact sequence of random variables with values in D([0, ∞), P(Z )) , where D denotes the Skorokhod space of right-continuous functions with left limits. Every weak accumulation point X solves, almost surely, the limiting equation for all ϕ such that (cf. Remark 2.2) Moreover, X is almost surely continuous. Remark 2.5. The measure-valued process (2.1) represents the particle mass concentration. It counts the number of monomers (mass) of particles of a given size instead of the number of particles of a given size (particle number concentration) considered, e.g., in [5]. Note that the underlying particle process (1.1) has the "direct simulation" dynamics, not the "mass flow" dynamics considered, e.g., in [4] and [31,Sect. 3]. The process (2.1) turned out to be most appropriate for studying the post-gelation behavior.
Remark 2.6. Theorem 2.1 implies existence of solutions of equation (2.14). Very little is known about uniqueness of post-gelation solutions. However, in Section 3 properties are obtained for any solution of the limiting equation. One particular uniqueness result will be mentioned in Section 3.3.1.

Properties of the limiting equation
Let be any solution of equation (2.14) (cf. Remark 2.6 and (2.5)).

Derivation of strong equations
Expression (2.7) takes the form Taking into account symmetry of K (and therefore F ϕ ), (2.9) and (2.10), one obtains Remark 3.1. The solution (3.1) satisfies equation (2.14) for any test function ϕ of the form and

Properties of the gel solution
Taking into account (3.14) and (3.15), one concludes from (3.16) that According to (3.17), the growth of the gel may originate from two different sources. In the casẽ K = 0 (cf. (3.3)), the gel is "passive" and grows due to the "gel production term" which depends only on the sol solution. In the caseK > 0 , the gel "actively" collects mass from the sol solution, according to the term It turns out that the gel production term (3.18) vanishes in the active gel case (under some additional assumptions).

Estimates of the gel production term
Here we study the behavior of the term (3.18). The proofs of the lemmas will be given in Section 4.
First we find sufficient conditions assuring a vanishing gel production term.
Finally, we provide conditions assuring a non-vanishing gel production term.

Active gel case
Here we provide sufficient conditions assuring that the gel solution satisfies the equation Note that ν(t, ∞, γ) is non-decreasing in t , according to (3.17). Moreover, the gel solution satisfies Theorem 3.9. Let γ ∈ G . Assume the kernel satisfies for some C,C > 0 . Then the gel solution satisfies (3.31).
Theorem 3.10. Let γ ∈ G . Assume the kernel satisfies for some C,C > 0 and a ∈ [0, 1] . If the sol solution is such that then the gel solution satisfies (3.31).
The proof of the theorems is based on the following lemma.
Proof of Theorem 3.9.
Proof of Theorem 3.10.

Continuity
It follows from (3.1) that the functions ν(t, x, γ) are continuous in t , for any finite x and γ ∈ G .
Here we provide sufficient conditions for the continuity of ν(t, ∞, γ) .
Lemma 3.12 is an immediate consequence of the following slightly more general result.

Spatially homogeneous case
Let |G| denote the size of the grid. In the case |G| = 1 , when all particles are located at the same site, the sol equations (3.7) are sufficient to describe the evolution of ν , since In the case |G| > 1 (with κ > 0), equations for ν(t, ∞, γ) , γ ∈ G , are necessary, since there is mass exchange between different sites. However, even in the spatially homogeneous case the gel equation (3.16) provides additional insight into the gelation phenomenon.

Modified coagulation equations
Here we derive some versions of the limiting equation (2.14) that have been previously studied in the literature.
If the kernel has the form then the negative terms on the right-hand side of (3.46) are (in the case a ∈ [0, 1)) and (in the case a = 1) x c(t, x) .
x a c(t, x) − c(t, k) k a and (in the case a = 1)

Multiplicative kernel
Here we illustrate some of the results in the special case (1.8), which has been extensively studied in the literature.

Properties of the solution
The sol equations (3.7) take the form (cf. (3.45)) The unique solution of (3.51), with monodisperse initial conditions, is Using Stirling's formula Note that the function In particular, it follows that the moments and one obtains and, using (3.53) Thus, and it follows from Lemma 3.5 that which is consistent with equation (3.62). Moreover, one obtains

Active and passive gel
Here we consider kernels of the form which have been frequently studied in the literature. We assume that the initial condition is such that τ > 0 (cf. (3.32)). Using the results concerning the gel equation and, in particular, the gel production term, we discuss (on a heuristic level) the behavior of the sol solution.
Let the sol solution be such that (cf.    At t = τ , one might expect that the solution reaches algebraic growth with exponent (3.68). Correspondingly, moments (3.55) of the order For t > τ , the behavior is completely different in the cases of active and passive gel.

Active gel case
In the active gel case (a = 1), the first term in (3.65) takes control at t = τ (cf.

Passive gel case
In the passive gel case (a < 1), the first term in (3.65) disappears and gelation is controlled by the second term. The gel equation ( Thus, the solution keeps to be of the order (3.68), according to Lemma 3.5 and Lemma 3.7. For the kernel (3.63) with b = a and a ∈ (0.5, 1) , the critical order is β = −a − 1 2 . Correspondingly, moments of the order ε < a − 1 2 stay finite, while moments of the order ε ≥ a − 1 2 stay infinite. In particular, one concludes that m a (t) = ∞ . Note that the moments m ε (t) , ε > 0 , grow monotonically, which can be derived from the weak form of the equation.

Special initial conditions
Here we consider kernels of the form (3.63) and discuss initial conditions leading to τ = 0 (cf. (3.32)).

Slowly decaying initial distributions
In the case of the multiplicative kernel (1.8) it is known that [23, Th. 2.8] Thus, m 1 (0) = ∞ is a necessary and sufficient condition for τ = 0 , or, in other words, sufficiently slow decay of ν(0, x) in x leads to immediate gelation.

Initial gel
Consider the case ν(0, ∞) > 0 (cf. Remark 3.2). In the active gel case (a = 1), the gel mass starts growing immediately. Its slope depends on the corresponding moment, according to equation (3.70). Note that this moment should be integrable in any neighborhood of t = 0 (compare this with (3.59)). In the passive gel case (a < 1), the gel mass may remain constant and start growing later (dependent on the sol component of the initial condition). So, having in mind the passive gel case, it might be appropriate to define the gelation time as instead of (3.32). In the active gel case both definitions are equivalent (except the trivial case ν(0, ∞) = 1).
An interesting aspect of the initial gel case is that even the consideration of non-gelling kernels makes sense. In the passive gel case (e.g., K(x, y) = 1), the sol and the gel develop independently. However, in the active gel case the initial gel starts growing immediately. The linear kernel K(x, y) = x + y is an example of a non-gelling kernel, for which the gel is active. One obtains K(∞, y) = 1 and equation (3.70) takes the form Note that the sol equations are modified in the initial gel case.

Comments
Here we give some comments concerning the two spatially inhomogeneous gelation models mentioned in the introduction.

The van Dongen model
The sol equations (3.7), with the notations ν(t, k, γ) = k c(t, k, γ) , take the form which is a spatially discrete version of (1.11), when the multiplicative kernel (1.8) is chosen. Moreover, equation (3.31) holds, according to Theorem 3.9, and provides a spatially discrete version of (1.12).

Formal extensions of Smoluchowski's coagulation equation
Note that (3.73) is a spatially discrete version of (1.4), wheñ In this case, equation (3.16) can be formally transformed into which is a spatially discrete version of (1.9), (1.10) (without the gradient term), when the multiplicative kernel (1.8) is chosen.
However, condition (3.74) is not fulfilled for the kernel (1.8). Before commenting on this point, we illustrate the situation in the spatially homogeneous case. When skipping the term containing K , equations (3.73) take the form xc(t, x) .  Turning to the model (1.9), (1.10), the form of the gel production term can be explained now by analogy with the passive gel case. However, in the spatially inhomogeneous situation the spatial behavior of the gel has to be described, in addition to its growth properties. Simply adding a diffusion term seems to be another formal extension of Smoluchowski's coagulation equation. It is not clear if this model is of any practical relevance, since the gel would be expected to behave randomly, even if a truncation of the kernel was used. The asymptotic behavior of the gel is determined by the assumptions on the diffusion coefficients. For non-vanishing D(k) , a stochastic limit was predicted in [27]. We also refer to the corresponding discussion in [25].
and test functions of the form Note that |Φ(µ)| ≤ ϕ ∞ . The usual starting point for deriving a limiting equation is the martingale representation Helpful properties are and (for any k ≥ 0) Estimates for the generator Lemma 4.1. If ϕ has the form (2.15), then (4.7) Proof. If x ≤x(ϕ) and y >x(ϕ) , then the left-hand side of (4.7) takes the form |x c 0 (ϕ, α) − x ϕ(x, α)| . Other cases are treated analogously.
Estimates for the martingale term for any ϕ satisfying (2.15).
To prove relative compactness of the sequence (X N ) we apply [9, Theorem 3.7.6] with E = P(Z ) and the metric (cf. Lemma 4.4) where (ϕ k ) denote the reordered elements of the set (4.10). The compact containment condition is trivial, since the space P(Z ) is compact. The remaining condition to be checked is where the modulus of continuity is defined for δ, T > 0 and µ ∈ D([0, ∞), E) . Here {t i } ranges over all partitions of the form 0 = t 0 < t 1 < · · · < t n−1 < T ≤ t n with min 1≤i≤n (t i − t i−1 ) > δ and n ≥ 1 .
Proof. This is a consequence of the continuity of G(ϕ, µ) with respect to µ (cf. Remark 2.3). so that the assertion follows.
As a consequence of Lemma 4.9, one obtains Assumption (2.13) and (4.26) imply X(0) = ν 0 almost surely, so that equation (2.14) is fulfilled for all functions (ϕ k ) . Finally, any function of the form (2.15) can be approximated by linear combinations (with rational coefficients) of functions (ϕ k ) in such a way that the corresponding values of G converge.
This completes the proof of Theorem 2.1.