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Condensed Matter > Statistical Mechanics
Title: Scaling theory of continuous symmetry breaking under advection
(Submitted on 3 Jan 2024 (v1), last revised 21 Jan 2024 (this version, v2))
Abstract: In this work, we discuss how the linear and non-linear advection terms modify the scaling behavior of the continuous symmetry breaking and stabilize the long-range order, even in $d=2$ far from equilibrium, by means of simple scaling arguments. For an example of the liner advection, we consider the $O(n)$ model in the steady shear. Our scaling analysis reveals that the model can undergo the continuous symmetry breaking even in $d=2$ and, moreover, predicts the upper critical dimension $d_{\rm up}=2$. These results are fully consistent with a recent numerical simulation of the $O(2)$ model, where the mean-field critical exponents are observed even in $d=2$. For an example of the non-linear advection, we consider the Toner-Tu hydrodynamic theory, which was introduced to explain polar-ordered flocks, such as the Vicsek model. Our simple scaling argument reproduces the previous results by the dynamical renormalization theory. Furthermore, we discuss the effects of the additional non-linear terms discovered by the recent re-analysis of the hydrodynamic equation. Our scaling argument predicts that the additional non-linear terms modify the scaling exponents and, in particular, recover the isotropic scaling reported in a previous numerical simulation of the Vicsek model. We discuss that the critical exponents predicted by the naive scaling theory become exact in $d=2$ by using a symmetry consideration and similar argument proposed by Toner and Tu.
Submission history
From: Harukuni Ikeda [view email][v1] Wed, 3 Jan 2024 08:14:18 GMT (8kb)
[v2] Sun, 21 Jan 2024 17:23:50 GMT (17kb)
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