The Exchange-Driven Growth Model: Basic Properties and Longtime Behavior

Abstract

The exchange-driven growth model describes a process in which pairs of clusters interact through the exchange of single monomers. The rate of exchange is given by an interaction kernel K which depends on the size of the two interacting clusters. Well-posedness of the model is established for kernels growing at most linearly and arbitrary initial data. The longtime behavior is established under a detailed balance condition on the kernel. The total mass density $\varrho$, determined by the initial data, acts as an order parameter, in which the system shows a phase transition. There is a critical value $\varrho _c\in (0,\infty ]$ characterized by the rate kernel. For $\varrho \le \varrho _c$, there exists a unique equilibrium state $\omega ^\varrho$ and the solution converges strongly to $\omega ^\varrho$. If $\varrho > \varrho _c$, the solution converges only weakly to $\omega ^{\varrho _c}$. In particular, the excess $\varrho - \varrho _c$ gets lost due to the formation of larger and larger clusters. In this regard, the model behaves similarly to the Becker–Döring equation. The main ingredient for the longtime behavior is the free energy acting as Lyapunov function for the evolution. It is also the driving functional for a gradient flow structure of the system under the detailed balance condition.

Appendix A. Lemma of de la Vallée-Poussin in \(\mathscr {P}^\varrho \)

Lemma A.1

For $\varrho >0$ and $\mathscr {C}= \{{{\bar{c}}^i}\}_{i\in I} \subset \mathscr {P}^\varrho$, the following are equivalent:

• (1) The family $\mathscr {C}$ is uniform integrable, that is

  $$\begin{aligned} \lim _{l\rightarrow \infty } \sup _{{\bar{c}}\in \mathscr {C}} \sum _{k\ge l} (1+k)\, {\bar{c}}_k \rightarrow 0. \end{aligned}$$

• (2) There exists a positive increasing superlinear sequence $(g_k)_{k\ge 0}$ such that

  $$\begin{aligned} \sup _{{\bar{c}} \in \mathscr {C}}\sum _{k\ge 0} g_k\, {\bar{c}}_k < \infty . \end{aligned}$$

  (A.1)

  Moreover, $(g_k)_{k\ge 0}$ can be chosen to satisfy the bound

  $$\begin{aligned} 0< (k+1)\,(g_{k+1} - g_k) \le 2\, g_k \qquad \text {for } k\ge 0 \end{aligned}$$

  (A.2)

Proof

The implication (2)$\Rightarrow$(1) is a straightforward consequence of the superlinear growth of $(g_k)_{k\ge 0}$ implying for any ${\bar{c}}\in \mathscr {C}$

$$\begin{aligned} \sum _{k\ge l} (1+k)\, {\bar{c}}_k \le \frac{1+l}{g_l} \sum _{k\ge l} g_k {\bar{c}}_k \le \frac{1+l}{g_l} \sum _{k\ge 0} g_k {\bar{c}}_k. \end{aligned}$$

Taking the $\sup$ over $\mathscr {C}$ and letting $l\rightarrow \infty$ proves the implication.

For the proof of (1)$\Rightarrow$(2), the construction from Cañizo (2006) based on Dellacherie and Meyer (Dellacherie and Meyer 1978, Theorem 22) is modified to satisfy condition (A.2). The tail distribution of ${\bar{c}}^i$ for $i\in I$ is defined by

$$\begin{aligned} C_k^i = \sum _{l\ge k} (l+1)\, {\bar{c}}_l^i \qquad \text { for } k\ge 0. \end{aligned}$$

Two auxiliary increasing sequences $(a_n)_{n\ge 1}$ and $(\ell _n)_{n\ge 0}$ are defined by

$$\begin{aligned} a_n = \inf \left\{ k\ge 0\,\bigg |\, \sup _{i\in I} C_k^i \le \frac{1}{n^2} \right\} \quad \text { for } n \ge 1 , \end{aligned}$$

and inductively $\ell _{n+1} = \max \{{ \ell _n+1 , a_{n+1}+1}\}$ starting with $\ell _0 = 0$. Then by construction, it holds $\sup _{i \in I} C_{\ell _n}^i \le \frac{1}{n^2}$ for $n\ge 1$ and $C_0 = 1+\varrho$. One more auxiliary sequence $(\varphi _k)_{k\ge 0}$ is given by

$$\begin{aligned} \varphi _k = n+1 \qquad \text {for } k\in [\ell _n, \ell _{n+1}). \end{aligned}$$

By construction $\varphi _k\rightarrow \infty$ as $k\rightarrow \infty$, since $\varphi _k \ge n+1$ for $k\ge \ell _n$. Then, the candidate for $g_k$ is the sequence $\varphi _k\, (k+1)$. Indeed, it holds for any $i\in I$

$$\begin{aligned} \sum _{k\ge 0} \varphi _k\, (k+1)\, {\bar{c}}_k^i= & {} \sum _{n\ge 0} (n+1) \sum _{k=\ell _n}^{\ell _{n+1}-1} (k+1), {\bar{c}}_k^i = \sum _{n\ge 0} \sum _{k\ge \ell _n} (k+1)\, {\bar{c}}_k^i \\= & {} \sum _{n\ge 0} C_{\ell _n}^i \le 1+\varrho + \sum _{n\ge 1} \frac{1}{n^2} < \infty . \end{aligned}$$

To verify condition (A.2), it is necessary to regularize the sequence $(\varphi _k)_{k\ge 1}$ by defining the following interpolation: $d_0=1$ and $\Phi _0=0$ and inductively for $n\ge 0$

$$\begin{aligned} d_{n+1}= & {} \min \left\{ { d_n , \frac{n+1 - \Phi _{\ell _n}}{\ell _{n+1} - \ell _n}, \frac{1}{\ell _{n+ 1}}}\right\} \ ; \\ \Phi _k= & {} \Phi _{\ell _n} + d_n ({k - \ell _n}) \qquad \text {for } k\in [\ell _n , \ell _{n+1}) . \end{aligned}$$

The construction ensures that $\Phi _k$ is an increasing sequence with $\Phi _k \le \varphi _k$. Hence, $g_k = \Phi _k\, (k+1) + 1$ still satisfies (A.1) with an additional constant $1+\varrho$ on the right-hand side. Finally to show (A.2), note that for $k\in [\ell _n, \ell _{n+1})$ it holds

$$\begin{aligned} \Phi _{k+1} - \Phi _k \le d_{n+1} \le \frac{1}{\ell _{n+1}} \le \frac{1}{k+1}. \end{aligned}$$

Hence, estimate (A.1) follows from

$$\begin{aligned} (k+1)\,({g_{k+1} -g_k}) = (k+1)\, \big ({ (k+1) \Phi _{k+1} - k \Phi _k}\big ) \le (k+1)\, ({ 1 + \Phi _k}) \le 2\, g_k , \end{aligned}$$

where the lower bound $\Phi _k \ge 1$ was used in the last step. $\square$