Abstract

We consider the one-dimensional coagulation–diffusion problem on a dynamical expanding linear lattice, in which the effect of the coagulation process is balanced by the dilatation of the distance between particles. Distances x(t) follow the general law ẋ (t) ∕ x (t) = α (1 + αt ∕ β) ^-1 with growth rate α and exponent β, describing both algebraic and exponential (β = ∞) growths. In the space continuous limit, the particle dynamics is known to be subdiffusive, with the diffusive length varying like t^1∕2−β for β < 1∕2, logarithmic for β = 1∕2, and reaching a finite value for all β > 1∕2. We interpret and characterize quantitatively this phenomenon as a second order phase transition between an absorbing state and a localized state where particles are not reactive. We furthermore investigate the case when space is discrete and use a generating function method to solve the time differential equation associated with the survival probability. This model is then compared with models of growth on geometrically constrained two-dimensional domains, and with the theory of fractional diffusion in the subdiffusive case. We found in particular a duality relation between the diffusive lengths in the inflating space and the fractional theory.

Graphical abstract

中文翻译：

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膨胀空间中有限的凝结-扩散动力学

摘要

我们考虑动态扩展线性晶格上的一维凝结-扩散问题，其中凝结过程的影响通过粒子之间距离的扩张得到平衡。距离x（t）遵循一般定律ẋ  （t）∕  x  （t）=  α  （1 +αt∕  β）  ^-1，具有增长率α和指数β，描述了代数和指数（β = ∞）增长。在空间连续极限中，粒子动力学被认为是亚扩散的，扩散长度像吨^{1 / 2- β}为β <1/2，对数为β = 1/2，到达所有的有限值β> 1∕2。我们将这种现象定量地解释和表征为吸收状态与颗粒不具有反应性的局部状态之间的二阶相变。我们还研究了空间是离散的情况，并使用生成函数方法来求解与生存概率相关的时间微分方程。然后将此模型与几何约束的二维域上的增长模型进行比较，并与亚扩散情况下的分数扩散理论进行比较。我们特别发现了膨胀空间中扩散长度与分数理论之间的对偶关系。

图形概要