Scaling theory of continuous symmetry breaking under advection
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909078146514944 |
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| author | Ikeda, Harukuni |
| author_facet | Ikeda, Harukuni |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_01603 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Scaling theory of continuous symmetry breaking under advection Ikeda, Harukuni Statistical Mechanics Soft Condensed Matter 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. |
| title | Scaling theory of continuous symmetry breaking under advection |
| topic | Statistical Mechanics Soft Condensed Matter |
| url | https://arxiv.org/abs/2401.01603 |