Biagi Sofia, Misbah Chaouqi, Politi Paolo
Université Grenoble 1/CNRS, LIPhy UMR 5588, Grenoble, F-38401, France and Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche, Via Madonna del Piano 10, 50019 Sesto Fiorentino, Italy.
Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche, Via Madonna del Piano 10, 50019 Sesto Fiorentino, Italy and INFN Sezione di Firenze, via Giovanni Sansone 1, 50019 Sesto Fiorentino, Italy.
Phys Rev E Stat Nonlin Soft Matter Phys. 2014 Jun;89(6):062114. doi: 10.1103/PhysRevE.89.062114. Epub 2014 Jun 10.
Universality has been a key concept for the classification of equilibrium critical phenomena, allowing associations among different physical processes and models. When dealing with nonequilibrium problems, however, the distinction in universality classes is not as clear and few are the examples, such 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(x,t) of a crystal surface having the general form ∂_{t}h(x,t)=-∇·[j(∇h)+∇(∇^{2}h)]: j(∇h) is an arbitrary function, which is linear for small ∇h, and whose structure expresses instabilities which lead to the formation of pyramidlike 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 information 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∼t^{n}; (ii) the exponent β, controlling the increase in time of the typical slope of the pattern, M∼t^{β}, where M≈H/L. Our study reveals that there are only two different universality classes, according to the presence (n=1/3, β=0) or the absence (n=1/4, β>0) of faceting. The symmetry of the pattern, as well as the symmetry of the surface mass current j(∇h) and its precise functional form, is irrelevant. Our analysis seems to support the idea that also space dimensionality is irrelevant.
普遍性一直是平衡临界现象分类的关键概念，它使得不同的物理过程和模型之间能够建立联系。然而，在处理非平衡问题时，普遍性类别的区分并不那么清晰，而且像相分离和动力学粗糙化这样能用普遍性以一种通用方式对结果进行分类的例子很少。在这里，我们关注一个介于相序和图案形成之间的非平衡情况，即不稳定晶体生长。我们考虑一族成熟的二维连续非线性方程，用于描述晶体表面的局部高度(h(x,t))，其一般形式为(\partial_{t}h(x,t)=-\nabla\cdot[j(\nabla h)+\nabla(\nabla^{2}h)])：(j(\nabla h))是一个任意函数，在(\nabla h)较小时是线性的，其结构表示导致形成平面尺寸为(L)、高度为(H)的金字塔状结构的不稳定性。我们的任务是选择并计算能够作为临界指数的量，同时讨论哪些与我们普遍性类别的定义相关或不相关。这些目标是通过对我们的模型进行微扰多尺度分析来实现的，从而得到相扩散方程，其扩散系数封装了关于动力学的所有相关信息。我们确定了两个临界指数：（i）粗化指数(n)，它控制图案典型尺寸随时间的增长，(L\sim t^{n})；（ii）指数(\beta)，它控制图案典型斜率随时间的增长，(M\sim t^{\beta})，其中(M\approx H/L)。我们的研究表明，根据是否存在刻面（(n = \frac{1}{3}, \beta = 0)）或不存在刻面（(n = \frac{1}{4}, \beta > 0)），只有两种不同的普遍性类别。图案的对称性，以及表面质量流(j(\nabla h))的对称性及其精确函数形式，都是无关的。我们的分析似乎支持空间维度也无关的观点。
