<p>The influence of a random environment on the dynamics of a fluctuating rough surface is investigated using the field theoretic renormalization group. The environment motion is modelled by the stochastic Navier–Stokes equation, which includes both a fluid in thermal equilibrium and a turbulent fluid. The surface is described by the generalized Pavlik’s stochastic equation. As a result of the renormalizability requirement, the model necessarily involves an infinite number of coupling constants. The one-loop counterterm is derived in an explicit closed form. The corresponding renormalization group equations demonstrate the existence of three two-dimensional surfaces of fixed points in the infinite-dimensional parameter space. If the surfaces contain IR attractive regions, the model allows for a large-scale, long-time scaling behaviour. For the first surface (advection is irrelevant) the critical dimensions of the height field <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2025_3410_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$\Delta _{h}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mi>h</mi> </msub> </math></EquationSource> </InlineEquation>, the response field <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2025_3410_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">$\Delta _{h'}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <msup> <mi>h</mi> <mo>′</mo> </msup> </msub> </math></EquationSource> </InlineEquation> and the frequency <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2025_3410_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$\Delta _{\omega }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mi>ω</mi> </msub> </math></EquationSource> </InlineEquation> are non-universal through the dependence on the effective couplings. For the other two surfaces (advection is relevant) these dimensions appear to be universal and are found exactly.</p>

Field Theoretic Renormalization Group in an Infinite-Dimensional Model of Random Surface Growth in Random Environment

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发表日期：

2025年2月14日

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N. V. Antonov,
A. A. Babakin,
N. M. Gulitskiy,
P. I. Kakin

摘要

The influence of a random environment on the dynamics of a fluctuating rough surface is investigated using the field theoretic renormalization group. The environment motion is modelled by the stochastic Navier–Stokes equation, which includes both a fluid in thermal equilibrium and a turbulent fluid. The surface is described by the generalized Pavlik’s stochastic equation. As a result of the renormalizability requirement, the model necessarily involves an infinite number of coupling constants. The one-loop counterterm is derived in an explicit closed form. The corresponding renormalization group equations demonstrate the existence of three two-dimensional surfaces of fixed points in the infinite-dimensional parameter space. If the surfaces contain IR attractive regions, the model allows for a large-scale, long-time scaling behaviour. For the first surface (advection is irrelevant) the critical dimensions of the height field $\Delta _{h}$ $Δ_{h}$ , the response field $\Delta _{h'}$ $Δ_{h^{'}}$ and the frequency $\Delta _{\omega }$ $Δ_{ω}$ are non-universal through the dependence on the effective couplings. For the other two surfaces (advection is relevant) these dimensions appear to be universal and are found exactly.