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.

arXiv comment34 pages, minor revised version accepted at the Journal of Nonlinear Science

References in corpus (17)

• Wealth Distributions in Models of Capital Exchange
• Condensation in the zero range process: stationary and dynamical properties
• Scaling Theory for Migration-Driven Aggregate Growth
• Exchange Driven Growth
• Explosive condensation in a mass transport model
• Gradient flow structure for McKean-Vlasov equations on discrete spaces
• Dynamics of condensation in the totally asymmetric inclusion process
• Asymptotic behavior of the generalized Becker-Döring equations for general initial data
• A non-local problem for the Fokker-Planck equation related to the Becker-Döring model
• Macroscopic limit of the Becker-Döring equation via gradient flows
• Derivation of mean-field equations for stochastic particle systems
• On a thermodynamically consistent modification of the Becker-Doering equations
• Trend to Equilibrium for the Becker-Döring Equations: An Analogue of Cercignani's Conjecture
• Coarsening dynamics of zero-range processes
• Explosive condensation in symmetric mass transport models
• Monotonicity and Condensation in Homogeneous Stochastic Particle Systems
• Mathematical Theory of Exchange-driven Growth

Cited by in corpus (3)

• Cosh gradient systems and tilting
• Self-similar behavior of the exchange-driven growth model with product kernel
• The role of zero-clusters in exchange-driven growth with and without input