冰冻湖泊作为ΛCDM宇宙学及更广泛领域的类比模型
Freezing lakes as analogue models of $Λ$CDM cosmology and beyond
AI总结：
通过结合浮力驱动的热输运，将湖泊冰层生长与宇宙膨胀类比，推导出冰厚演化方程，其结构与弗里德曼方程相似，揭示非线性输运机制在经典边界问题中的作用。
摘要：
我们扩展了之前基于传导的类比，将湖泊冰层生长与宇宙学膨胀的类比扩展到考虑浮力驱动的热输运。通过将斯蒂芬问题重新公式化，结合传导和对流流量，得到冰厚s(t)的演化方程，其结构与宇宙学尺度因子a(t)的弗里德曼方程相似。除了重现辐射、物质和曲率行为外，我们引入了对流的简化描述，其中垂直积分的热量流到达移动的冰-水界面被建模为液层厚度的幂律函数，产生两个额外的有效贡献。第一个是常数项，直接类似于宇宙学常数，源于在几何约束下浮力驱动输运的持续性。第二个是s^{-1}项，源于移动冰边界与对流边界层的耦合。这一项反映了所采用的简化流量-高度假设，而非普遍物理预测。当以弗里德曼类宇宙学形式表达时，这一项对应于具有负能量密度和方程状态参数w=-2/3的流体。在宇宙学中，这一项可能与由奇异能量/物质构成的领域壁网络相关，但也可能源于宇宙学组分间的能量交换。总体而言，结果应被解释为演化方程之间的结构类比，展示了经典移动边界问题中的非线性输运机制如何在简化且可解析的框架内重现宇宙学中熟悉的尺度层级。
英文摘要：
We extend previous conduction-based analogies between ice growth in a lake and cosmological expansion by incorporating buoyancy-driven heat transport. Reformulating the Stefan problem with both conductive and convective fluxes yields an evolution equation for the ice thickness $s(t)$ that is structurally analogous to the Friedmann equations for the cosmological scale factor $a(t)$. Beyond reproducing radiation-, matter-, and curvature-like behaviors, we introduce a reduced description of convection in which the vertically integrated heat flux reaching the moving ice-water interface is modeled as a power-law function of the instantaneous liquid-layer thickness, generating two additional effective contributions. The first is a constant term, directly analogous to a cosmological constant, arising from the persistence of buoyancy-driven transport under geometric confinement. The second is a $s^{-1}$ contribution originating from the coupling between the moving ice boundary and the convective boundary layer. This term reflects the specific reduced flux-height Ansatz adopted, rather than a universal physical prediction. When expressed in Friedmann-like cosmological form, this term entails a fluid with negative energy density and equation-of-state parameter $w=-2/3$. In cosmology this term may be an effective one associated to a network of domain walls made of exotic energy/matter, but it might also arise from an energy exchange between cosmological components. Overall, the results should be interpreted as a structural analogy between evolution equations, showing how nonlinear transport mechanisms in a classical moving-boundary problem can reproduce the hierarchy of scaling terms familiar from cosmology within a reduced and analytically tractable framework.
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