Dynamics of condensation in the symmetric inclusion process
Dynamics of condensation in the symmetric inclusion process
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对称包含过程中的凝结动力学
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
K. Vafayi
中科院分区：
文献类型：
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作者：
S. Grosskinsky;F. Redig;K. Vafayi
包含过程是一种随机格子气体，它是研究得很好的排除过程的自然玻色子对应，与热传导模型和在种群遗传学中的应用有很强的联系。与零程过程类似，由于粒子之间的吸引相互作用，包合过程可以呈现凝聚转变。在这篇文章中，我们首先给出了这类模型凝析油形成动力学的严格结果。在强相互作用区域，即具有独立扩散速率$m=m_N\to 0$的情况下，研究了粒子总数为$N的有限集$S$上的对称包含过程.对于$Nm_N\to\Infty$情形，我们证明了在时间尺度上，$1/m_N$凝聚体产生于一般齐次初始条件，并精确地刻画了它们的极限动力学。在最简单的两个站点或一个完全连通的底层随机游走核的情况下，有一个凝聚作为连续时间随机游走跳过$S$。在非完全连通的情况下，在一个有趣的粗化过程中，几个凝析油可以共存并通过中间位置交换质量，这个过程由扩散运动和跳跃过程组成，直到形成单一的凝析油。我们的结果是基于一般的双尺度形式的生成器，具有快尺度的中性Wright-Fisher扩散和慢尺度的确定性运动。凝聚体的运动用确定性运动的生成元和与Wright Fisher扩散的吸收态相对应的调和投影来描述。
The inclusion process is a stochastic lattice gas, which is a natural bosonic counterpart of the well-studied exclusion process and has strong connections to models of heat conduction and applications in population genetics. Like the zero-range process, due to attractive interaction between the particles, the inclusion process can exhibit a condensation transition. In this paper we present first rigorous results on the dynamics of the condensate formation for this class of models. We study the symmetric inclusion process on a finite set $S$ with total number of particles $N$ in the regime of strong interaction, i.e. with independent diffusion rate $m=m_N \to 0$. For the case $Nm_N\to\infty$ we show that on the time scale $1/m_N$ condensates emerge from general homogeneous initial conditions, and we precisely characterize their limiting dynamics. In the simplest case of two sites or a fully connected underlying random walk kernel, there is a single condensate hopping over $S$ as a continuous-time random walk. In the non fully connected case several condensates can coexist and exchange mass via intermediate sites in an interesting coarsening process, which consists of a mixture of a diffusive motion and a jump process, until a single condensate is formed. Our result is based on a general two-scale form of the generator, with a fast-scale neutral Wright-Fisher diffusion and a slow-scale deterministic motion. The motion of the condensates is described in terms of the generator of the deterministic motion and the harmonic projection corresponding to the absorbing states of the Wright Fisher diffusion.