SOC in a microscopic model for Smoluchowski’s equation 

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Equivalence between the system and the PDE

Smoluchowski’s equation (2.3) is solved thanks to PDE (2.17).
Proposition 4.7. (i) Let (ct)t2[0,T ) be a solution to Smoluchowski’s equation (2.3), and let gt be its generating function, defined by gt(x, y, z) := hct, xaybzmi. Then for all z 2 [0, 1], (t, x, y) 7! gt(x, y, z) is a regular solution to the PDE (2.17) on [0, T ^ 􀀀1) × (0, 1)2, with initial conditions g0(., ., z).
(ii) Conversely, let (ct(p))p2S, t 2 [0, T), be a family of differentiable functions. Let gt(x, y, z) be its generating function and assume it is defined for t 2 [0, T), (x, y) 2 (0, 1)2 and z 2 [0, 1]. Assume that for every z 2 [0, 1], gt(., ., z) is a regular solution to the PDE (2.17) with initial conditions g0(., ., z).
• for all p 2 S and t 2 [0, T), ct(p) 0.
• (ct) is a solution to Smoluchowski’s equation (2.3) for t 2 [T ^Tgel), with initial conditions c0.

Limiting concentrations and Galton-Watson processes

In this section, we will study the limiting concentrations. Similarly to what happens in the oriented and symmetric model of [3], we expect the concentrations to converge when the time tends to +1, whenever gelation does not occur. Physically, this would mean that the system converges to a terminal state where all arms have been used (otherwise, further coagulations “should” occur). This is actually true, and this is an easy consequence of the preceding results.

Connection with two-type Galton-Watson processes

In [3], Bertoin shows that for monodisperse initial conditions and when gelation does not occur, the limiting concentrations can be described in terms of Galton-Watson processes.
The same kind of analogy is observed in our case. Precisely, we start from initial conditions c0(a, b,m) = μ(a, b)1{m=1} for a measure μ on N × N with hμ, ai = hμ, bi = 1. We may then define the probability measures
• μi(a, b) = μ(a, b)/hμ, 1i, with generating function i(x, y) := g0(x, y, 1)/g0(1, 1, 1).
• μm(a, b) = (b + 1)μ(a, b + 1), with generating function m(x, y) := @g0 @y (x, y, 1).
• μf (a, b) = (a + 1)μ(a + 1, b), with generating function f (x, y) := @g0 @x (x, y, 1).

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The gelation phase transition

The reason why Bertoin’s equation (3.2) can be solved, is that it can be transformed (see [3]) into a solvable PDE involving the generating function of (ct). For the standard Smoluchowski equation (3.1), this transformation is also possible for several particular choices of the kernel (m,m0), namely when is constant, additive or multiplicative: see e.g. [7]. However, the mass is a parameter of this PDE, so it is easy to solve only when the mass is known. But if large clusters can coagulate sufficiently fast, then one may observe in finite time the gelation phenomenon, which is interpreted as the formation of clusters of infinite mass, the gel. From the gelation time, the mass starts to decrease.
Existence and uniqueness of solutions of (3.1) are thus easy up to gelation, since in this regime the total mass Mt := X m1 mct(m).

Table of contents :

1 Introduction 
1 Content of the thesis
2 A short history of Smoluchowski’s equation
3 Smoluchowski’s equation of limited aggregations
4 A two-type limited aggregation model
5 Post-gelation uniqueness of coagulation equations
6 A microscopic model for Smoluchowski’s equation
2 A model for coagulation with mating 
1 Introduction
2 Setting and results
3 Preliminary results
4 Proof of the theorem
5 Explicit formulas
6 Limiting concentrations and Galton-Watson processes
7 Microscopic model
3 Post-gelation uniqueness of coagulation equations 
1 Introduction
2 Smoluchowski’s equation
3 Flory’s equation
4 The model with limited aggregations
5 The modified version
6 Limiting concentrations
4 SOC in a microscopic model for Smoluchowski’s equation 
1 Introduction
2 Presentation of the model and of the results
3 Convergence to Smoluchowski’s equation
4 Pre-gelation results
5 Preliminary results
6 Subcriticality
7 Tightness
8 Self-organized criticality
9 Asymptotic independence
10 Evolution in time
A Exploration process of a configuration model
B Tightness
C Mass in Smoluchowski’s and Flory’s models
D Exploration process of a two-type uniform pairing
Bibliography

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