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In this work, we discuss how the linear and non-linear advection terms modifythe scaling behavior of the continuous symmetry breaking and stabilize thelong-range order, even in $d=2$ far from equilibrium, by means of simplescaling arguments. For an example of the liner advection, we consider the$O(n)$ model in the steady shear. Our scaling analysis reveals that the modelcan undergo the continuous symmetry breaking even in $d=2$ and, moreover,predicts the upper critical dimension $d_{\rm up}=2$. These results are fullyconsistent with a recent numerical simulation of the $O(2)$ model, where themean-field critical exponents are observed even in $d=2$. For an example of thenon-linear advection, we consider the Toner-Tu hydrodynamic theory, which wasintroduced to explain polar-ordered flocks, such as the Vicsek model. Oursimple scaling argument reproduces the previous results by the dynamicalrenormalization theory. Furthermore, we discuss the effects of the additionalnon-linear terms discovered by the recent re-analysis of the hydrodynamicequation. Our scaling argument predicts that the additional non-linear termsmodify the scaling exponents and, in particular, recover the isotropic scalingreported in a previous numerical simulation of the Vicsek model. We discussthat the critical exponents predicted by the naive scaling theory become exactin $d=2$ by using a symmetry consideration and similar argument proposed byToner and Tu.

Scaling theory of continuous symmetry breaking under advection