Bedini Andrea, Owczarek Aleksander L, Prellberg Thomas
Department of Mathematics and Statistics, University of Melbourne, Melbourne 3010, Australia.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Jul;86(1 Pt 1):011123. doi: 10.1103/PhysRevE.86.011123. Epub 2012 Jul 20.
Trails (bond-avoiding walks) provide an alternative lattice model of polymers to self-avoiding walks, and adding self-interaction at multiply visited sites gives a model of polymer collapse. Recently a two-dimensional model (triangular lattice) where doubly and triply visited sites are given different weights was shown to display a rich phase diagram with first- and second-order collapse separated by a multicritical point. A kinetic growth process of trails (KGTs) was conjectured to map precisely to this multicritical point. Two types of low-temperature phases, a globule phase and a maximally dense phase, were encountered. Here we investigate the collapse properties of a similar extended model of interacting lattice trails on the simple cubic lattice with separate weights for doubly and triply visited sites. Again we find first- and second-order collapse transitions dependent on the relative sizes of the doubly and triply visited energies. However, we find no evidence of a low-temperature maximally dense phase with only the globular phase in existence. Intriguingly, when the ratio of the energies is precisely that which separates the first-order from the second-order regions anomalous finite-size scaling appears. At the finite-size location of the rounded transition clear evidence exists for a first-order transition that persists in the thermodynamic limit. This location moves as the length increases, with its limit apparently at the point that maps to a KGT. However, if one fixes the temperature to sit at exactly this KGT point, then only a critical point can be deduced from the data. The resolution of this apparent contradiction lies in the breaking of crossover scaling and the difference in the shift and transition width (crossover) exponents.
迹线（避免键合行走）为聚合物提供了一种替代自回避行走的晶格模型，并且在多次访问的位点添加自相互作用给出了聚合物塌缩的模型。最近，一个二维模型（三角晶格），其中对双次和三次访问的位点赋予不同权重，被证明展示了一个丰富的相图，其中一阶和二阶塌缩由一个多临界点分隔。推测一种迹线的动力学生长过程（KGTs）精确地映射到这个多临界点。遇到了两种低温相，一种球状相和一种最大密度相。在这里，我们研究了在简单立方晶格上类似的相互作用晶格迹线扩展模型的塌缩性质，对双次和三次访问的位点赋予单独的权重。我们再次发现一阶和二阶塌缩转变取决于双次和三次访问能量的相对大小。然而，我们没有发现仅存在球状相的低温最大密度相的证据。有趣的是，当能量比恰好是分隔一阶和二阶区域的那个值时，出现了反常的有限尺寸标度。在圆滑转变的有限尺寸位置，存在明显的一阶转变证据，该转变在热力学极限下持续存在。这个位置随着长度增加而移动，其极限显然在映射到KGT的点处。然而，如果将温度精确地固定在这个KGT点，那么从数据中只能推断出一个临界点。这个明显矛盾的解决在于交叉标度的破坏以及位移和转变宽度（交叉）指数的差异。
