摘要翻译：
我们在三维标量噪声模型(SNM)中考虑无序-有序相变。我们的结果与二维情况下的结果相似。当速度较小(v<=0.1)时，可以观察到连续的二级相变，附近粒子的扩散是各向同性的。随着粒子速度的增加，相变变为一级相变，扩散变为各向异性。后者的一级跃迁可能是由各向异性扩散和周期性边界条件相互作用引起的，导致解的对称破缺依赖于边界条件。
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英文标题：
《Phase transition in the scalar noise model of collective motion in three
  dimensions》
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作者：
Balazs Gonci, Mate Nagy, Tamas Vicsek
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最新提交年份：
2008
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分类信息：

一级分类：Physics        物理学
二级分类：Statistical Mechanics        统计力学
分类描述：Phase transitions, thermodynamics, field theory, non-equilibrium phenomena, renormalization group and scaling, integrable models, turbulence
相变，热力学，场论，非平衡现象，重整化群和标度，可积模型，湍流
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英文摘要：
  We consider disorder-order phase transitions in the three-dimensional version of the scalar noise model (SNM) of flocking. Our results are analogous to those found for the two-dimensional case. For small velocity (v <= 0.1) a continuous, second-order phase transition is observable, with the diffusion of nearby particles being isotropic. By increasing the particle velocities the phase transition changes to first order, and the diffusion becomes anisotropic. The first-order transition in the latter case is probably caused by the interplay between anisotropic diffusion and periodic boundary conditions, leading to a boundary condition dependent symmetry breaking of the solutions.
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PDF链接：
https://arxiv.org/pdf/801.0151