Freezing Ice and Expanding Space ⚡ экспресс
The Stefan problem for ice growth has been reformulated to include convective heat transfer, yielding an evolution equation for ice thickness s(t) that is structurally analogous to the Friedmann equations for the scale factor of the Universe a(t). In addition to the familiar radiation, dust, and curvature terms, a simplified convection model — where the vertical heat flux to the ice-water boundary is set as a power law of the liquid layer thickness — produces two extra contributions. The constant term is analogous to the cosmological constant and arises from sustained convection in a confined geometry, while the s^{-1} term is linked to the interaction of the moving boundary with the convective boundary layer. In a cosmological interpretation, s^{-1} corresponds to a medium with negative energy density and an equation-of-state parameter w=−2/3, which could describe domain walls or energy exchange. The results highlight a structural analogy: nonlinear heat transfer in a classical problem reproduces the hierarchy of scale-dependent terms known from cosmology, in a form that is analytically tractable.
In winter, a lake freezes over—and this simple process turns out to be a living model of an expanding universe. Ice thickness grows according to the same mathematical law as the distances between distant galaxies.
The secret lies in the movement of water beneath the ice crust. When warm layers rise and cold ones sink, this circulation delivers heat to the growing ice. An extra term appears in the equations, behaving exactly like dark energy—a constant invisible force that causes galaxies to rush apart with acceleration. In the lake, this "energy" slows ice growth, while in space, it does the opposite—it pushes expansion forward.
Amazingly, the formula for this scenario was first written by physicist Josef Stefan back in 1891 while studying ice forming on ponds. Now that same equation underpins models that predict the fate of the entire cosmos.
🎯 Stefan's equation, derived for simple ice, is now even used to calculate the cooling of Mars' surface.