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For equal phase mass densities, equal sensible heat capacities per unit volume, and negligible thermal diffusion, a porous melting front driven by hot Darcy flux $U$ has speedHere $\Delta \varphi$ is the newly melted volume fraction. Mass conservation preserves $U$ on both sides: the liquid gained by melting fills the newly available pores. The enthalpy increase per unit bulk volume is $\rho c_p\Delta T+\rho L\Delta \varphi$, so its rate of production is balanced by incoming heat flux $\rho c_{p}U\Delta T$.

For a thermodynamically stable system, the entropy is concave in energy, soConsequently $\partial E/\partial T>0$: energy increases with temperature, and the corresponding heat capacity is positive.

Two thermal derivatives of a regulated Gaussian determinant produce this integral. For $0<D<4$, rescaling $p=\sqrt{t}q$ yields a divergent power $t^{D/2-2}$ and heat-capacity critical exponent $\alpha =(4-D)/2$. At four dimensions the singularity is logarithmic, with power exponent zero. Above four the leading heat capacity is a finite cutoff-dependent background; its leading bounded-power convention has exponent zero, although the background-subtracted singular Gaussian contribution retains the negative index $2-D/2$. In a stable quartic theory the mean-field ordered free energy proportional to $-t^2/u$ instead produces a heat-capacity jump. This is why a dangerously irrelevant coupling qualifies naive hyperscaling relation.

For $\mathcal{C}=C_0/\Delta C\gg 1$ and inverse Stefan number $\mathcal{S}=O(\mathcal{C})$, a stagnant mushy-layer model has nearly constant effective heat capacity, giving ${\Omega }^2{\theta }_t=\kappa {\theta }_{zz}$ with ${\Omega }^2=1+\mathcal{S}/\mathcal{C}$. With negligible liquid solutal diffusivity, roof value $\theta =-1$, positive liquid superheat ${\theta }_{\infty }$, and edge $h=2\lambda \sqrt{\kappa t}$, the mush profile is $\theta =-1+\mathrm{erf}(\Omega \eta )/\mathrm{erf}(\Omega \lambda )$. Matching heat flux requires $\Omega e^{{\lambda }^2}\mathrm{erfc}\lambda ={\theta }_{\infty }{e}^{{\Omega }^2{\lambda }^2}\mathrm{erf}(\Omega \lambda )$. The square on $\Omega$ in the exponential is essential. The porosity $\mathcal{C}/(\mathcal{C}-\theta )$ records the leading composition balance; the linear thermal profile is asymptotic, not exact at finite $\mathcal{C}$.

In units $G=\hslash =c=k_B=1$, a Schwarzschild black hole has Hawking temperature $T=1/(8\pi M)$, so its heat capacity is $dM/dT=-8\pi M^2$. It heats up as it loses mass, preventing stable canonical equilibrium with an unlimited thermal reservoir.