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arXiv 1401.4263cond-mat.stat-mech
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不稳定晶体生长的普适类

Universality classes for unstable crystal growth

Sofia Biagi, Chaouqi Misbah, Paolo Politi

备注：
15 pages, new section (VII.A) added, further considerations at the end of section VI
发表信息：
Phys. Rev. E 89, 62114 (2014)
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摘要：

普适性一直是平衡临界现象分类的关键概念，允许不同物理过程和模型之间的关联。然而，在处理非平衡问题时，普适性类的区别并不那么清楚，很少有例子，如相分离和动力学粗糙化，普适性允许在一般精神中对结果进行分类。在这里，我们专注于一个平衡的情况下，不稳定的晶体生长，躺在相序和图案形成之间。我们考虑一个良好建立的2+1维家庭的连续非线性方程的局部高度$h(\mathbf{x},t)$的晶体表面具有一般形式${\partial_t h(\mathbf{x},t)} = -\mathbf{\nabla}\cdot {[\mathbf{j}(\nabla h) + \mathbf{\nabla}(\nabla^2 h)]}$：$\mathbf{j}(\nabla h)$是一个任意的功能，这是线性的小$\nabla h$，其结构表示不稳定性，导致形成的平面尺寸$L$和高度$H$的类晶体结构。我们的任务是选择和计算可以作为临界指数的量，以及讨论什么与我们的普适性类的定义相关或不相关。这些目标是通过我们的模型的微扰，多尺度分析，导致相扩散方程的扩散系数封装所有相关的动力学信息。我们确定了两个临界指数：i）粗化指数$n$，控制图案的典型尺寸$L\sim t^n$的时间增加; ii）指数$β$，控制图案的典型斜率$M \sim t^β$的时间增加，其中$M\approx H/L$。我们的研究表明，只有两种不同的普适性类...

英文摘要：

Universality has been a key concept for the classification of equilibrium critical phenomena, allowing associations among different physical processes and models. When dealing with non-equilibrium problems, however, the distinction in universality classes is not as clear and few are the examples, as phase separation and kinetic roughening, for which universality has allowed to classify results in a general spirit. Here we focus on an out-of-equilibrium case, unstable crystal growth, lying in between phase ordering and pattern formation. We consider a well established 2+1 dimensional family of continuum nonlinear equations for the local height $h(\mathbf{x},t)$ of a crystal surface having the general form ${\partial_t h(\mathbf{x},t)} = -\mathbf{\nabla}\cdot {[\mathbf{j}(\nabla h) + \mathbf{\nabla}(\nabla^2 h)]}$: $\mathbf{j}(\nabla h)$ is an arbitrary function, which is linear for small $\nabla h$, and whose structure expresses instabilities which lead to the formation of pyramid-like structures of planar size $L$ and height $H$. Our task is the choice and calculation of the quantities that can operate as critical exponents, together with the discussion of what is relevant or not to the definition of our universality class. These aims are achieved by means of a perturbative, multiscale analysis of our model, leading to phase diffusion equations whose diffusion coefficients encapsulate all relevant informations on dynamics. We identify two critical exponents: i) the coarsening exponent, $n$, controlling the increase in time of the typical size of the pattern, $L\sim t^n$; ii) the exponent $β$, controlling the increase in time of the typical slope of the pattern, $M \sim t^β$ where $M\approx H/L$. Our study reveals that there are only two different universality classes...

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