Asymptotic Analysis of a bi-monomeric nonlinear Becker-D{ö}ring system
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909243146240000 |
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| author | Doumic, Marie Fellner, Klemens Mezache, Mathieu Velázquez, Juan J L |
| author_facet | Doumic, Marie Fellner, Klemens Mezache, Mathieu Velázquez, Juan J L |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_04320 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic Analysis of a bi-monomeric nonlinear Becker-D{ö}ring system Doumic, Marie Fellner, Klemens Mezache, Mathieu Velázquez, Juan J L Analysis of PDEs Dynamical Systems 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. |
| title | Asymptotic Analysis of a bi-monomeric nonlinear Becker-D{ö}ring system |
| topic | Analysis of PDEs Dynamical Systems |
| url | https://arxiv.org/abs/2407.04320 |