双单体非线性 Becker-Döring 系统的渐近分析
Asymptotic Analysis of a bi-monomeric nonlinear Becker-D{ö}ring system
AI总结：
本文对双单体非线性 Becker-Döring 系统进行渐近分析，揭示持续后阻尼振荡的机制，刻画各动力学阶段，并给出周期、阻尼大小和振荡次数的定量近似，辅以数值模拟验证。
摘要：
为了对解聚实验中观察到的持续随后阻尼振荡提供机制性解释，文献 [DFMR] 提出了一种经典 Becker-Döring 系统的双单体变体。当所有反应速率均为常数时，方程如下：\n\begin{align*}\frac{dv}{dt} & =-vw+v\sum_{j=2}^{\infty}c_{j}, \qquad \frac{dw}{dt} =vw-w\sum_{j=1}^{\infty}c_{j}, \\\\ \frac{dc_{j}}{dt} & =J_{j-1}-J_{j}\\ \\ ,\\ \\ j\geq1\\ \\ ,\\ \\ \\ J_{j}=wc_{j}-vc_{j+1}\\ \\ ,\\ \\ j\geq1\\ \\ ,\\ J_{0}=0,\n\end{align*}\n其中 $v$ 和 $w$ 是两种不同的单体单元，$c_i$ 表示包含 $i$ 个单元的团簇浓度。我们详细研究了导致此类振荡的机制，并刻画了动力学的不同阶段：从初始的高幅振荡到逐渐阻尼，最终收敛到唯一的正稳态解。我们给出了主要感兴趣量的定量近似：振荡周期、阻尼大小（对应于能量损失）、表征每个阶段的振荡次数。我们通过数值模拟说明了这些结果，与理论结果一致，并提供了求解该系统的数值方法。
英文摘要：
To provide a mechanistic explanation of sustained {then} damped oscillations observed in a depolymerisation experiment, a bi-monomeric variant of the seminal Becker-Döring system has been proposed in \cite{DFMR}. When all reaction rates are constant, the equations are the following: \begin{align*}\frac{dv}{dt} & =-vw+v\sum_{j=2}^{\infty}c_{j}, \qquad \frac{dw}{dt} =vw-w\sum_{j=1}^{\infty}c_{j}, \\ \frac{dc_{j}}{dt} & =J_{j-1}-J_{j}\ \ ,\ \ j\geq1\ \ ,\ \ \ J_{j}=wc_{j}-vc_{j+1}\ \ ,\ \ j\geq1\ \ ,\ J_{0}=0, \end{align*} where $v$ and $w$ are two distinct unit species, and $c_i$ represents the concentration of clusters containing $i$ units. We study in detail the mechanisms leading to such oscillations and characterise the different phases of the dynamics, from the initial high-amplitude oscillations to the progressive damping leading to the convergence towards the unique positive stationary solution. We give quantitative approximations for the main quantities of interest: period of the oscillations, size of the damping (corresponding to a loss of energy), number of oscillations characterising each phase. We illustrate these results by numerical simulation, in line with the theoretical results, and provide numerical methods to solve the system.
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