Universal scaling solution for a rigidity transition: Renormalization group flows near the upper critical dimension.

IF 2.4 3区 物理与天体物理 Q1 Mathematics
Stephen J Thornton, Itai Cohen, James P Sethna, Danilo B Liarte
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引用次数: 0

Abstract

Rigidity transitions induced by the formation of system-spanning disordered rigid clusters, like the jamming transition, can be well described in most physically relevant dimensions by mean-field theories. A dynamical mean-field theory commonly used to study these transitions, the coherent potential approximation (CPA), shows logarithmic corrections in two dimensions. By solving the theory in arbitrary dimensions and extracting the universal scaling predictions, we show that these logarithmic corrections are a symptom of an upper critical dimension d_{upper}=2, below which the critical exponents are modified. We recapitulate Ken Wilson's phenomenology of the (4-ε)-dimensional Ising model, but with the upper critical dimension reduced to 2. We interpret this using normal form theory as a transcritical bifurcation in the RG flows and extract the universal nonlinear coefficients to make explicit predictions for the behavior near two dimensions. This bifurcation is driven by a variable that is dangerously irrelevant in all dimensions d>2 which incorporates the physics of long-wavelength phonons and low-frequency elastic dissipation. We derive universal scaling functions from the CPA sufficient to predict all linear response in randomly diluted isotropic elastic systems in all dimensions.

刚性跃迁的通用标度解:重整化群在上临界维附近流动。
由系统跨越无序刚性团簇形成引起的刚性转变,如干扰转变,可以用平均场理论在大多数物理相关维度上很好地描述。通常用于研究这些跃迁的动态平均场理论相干势近似(CPA)在二维上显示对数修正。通过在任意维度上求解理论并提取通用尺度预测,我们证明了这些对数修正是上临界维d_{upper}=2的症状,在该临界维d_{upper}=2以下的临界指数被修改。我们概括了Ken Wilson关于(4-ε)维Ising模型的现象学,但将上临界维降为2维。我们使用范式理论将其解释为RG流中的跨临界分岔,并提取通用非线性系数以明确预测二维附近的行为。这种分岔是由一个变量驱动的,这个变量在所有维度上都是危险的不相关的,它包含了长波声子和低频弹性耗散的物理特性。我们从CPA中导出了通用尺度函数,足以预测随机稀释各向同性弹性系统在所有维度上的所有线性响应。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Physical review. E
Physical review. E 物理-物理:流体与等离子体
CiteScore
4.60
自引率
16.70%
发文量
0
审稿时长
3.3 months
期刊介绍: Physical Review E (PRE), broad and interdisciplinary in scope, focuses on collective phenomena of many-body systems, with statistical physics and nonlinear dynamics as the central themes of the journal. Physical Review E publishes recent developments in biological and soft matter physics including granular materials, colloids, complex fluids, liquid crystals, and polymers. The journal covers fluid dynamics and plasma physics and includes sections on computational and interdisciplinary physics, for example, complex networks.
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