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 ̺, determined by the initial data, acts as an order parameter, in which the system shows a phase transition. There is a critical value ̺ c ∈ (0, ∞] characterized by the rate kernel. For ̺ ≤ ̺ c , there exists a unique equilibrium state ω ̺ and the solution converges strongly to ω ̺ . If ̺ > ̺ c the solution converges only weakly to ω ̺ c . In particular, the excess ̺ − ̺ 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.

Large time behavior of exchange-driven growth

“…Part of the results of this paper, namely those in Section 3, overlap with some of the results in [24] which were independently obtained. Actually the results in [24] cover a class of kernels wider than those considered in Section 3 of this paper.…”

“…This behavior was first demonstrated numerically in [8], where it was observed that the seemingly innocuous change in the kernel (K(j, 0) > 0) fundamentally alters the dynamical behavior, driving the system, towards a unique equilibrium (BD-like) instead of indefinite growth where the cluster densities eventually vanish (Smoluchowski-like when K(j, 0) = 0). For a large class of kernels this observation was recently proven in [24].…”

“…Despite its importance, rigorous results on the properties and behavior of the corresponding equations (existence, uniqueness, asymptotic behavior etc.) are few and have been obtained only very recently [7], [24]. It is the purpose of this article to continue the mathematical study of the EDG systems focusing on the large time asymptotic properties of solutions with explicit convergence rates where possible.…”

“…In particular, for general non-symmetric kernels whose growth is bounded as K(j, k) ≤ Cjk (for large j, k), unique classical solutions were shown to exist globally. Recently, these results for non-symmetric kernels were extended in [24], in particular, moment boundedness assumptions for the uniqueness were replaced with milder conditions. For symmetric kernels, it was shown in [7] that the existence result can be generalized to kernels whose growth rate is bounded as K(j, k) ≤ C(j µ k v + j ν k µ ), with µ, ν ≤ 2, µ + v ≤ 3.…”

Entropic curvature and convergence to equilibrium for mean-field dynamics on discrete spaces

“…In this section, our method is perturbative, for which reason we restrict to the complete graph, that is p(x, y) = 1 for all x = y. In this case, the mean-field limit from the N -particle system was derived in Grosskinsky and Jatuviriyapornchai (2019), and the limit equation was investigated in Schlichting (2019). Since positive curvature is know in the case of independent particles on the complete graph (Erbar and Maas, 2012), we expect that for c(µ x , µ y ) = T +c(µ x , µ y ) with bounded c : P(X ) × P(X ) → [0, ∞), we should also obtain positive entropic curvature for the non-linear models when T is sufficiently large.…”