Heat Content on Self-Similar Fractals

• Heat content on self-similar fractals is the measure of short-time thermal energy diffusion governed by scale-dependent features like Minkowski content and fractal curvatures.
• It employs tube formulas, scaling zeta functions, and renewal equations to link complex dimensions with oscillatory heat kernel asymptotics.
• This framework aids in inferring fractal dimensions and modeling radiative heat transfer while integrating stochastic and numerical methods in fractal analysis.

Heat content on self-similar fractals refers to the short-time behavior of the total thermal energy present in a domain characterized by a boundary or geometry exhibiting exact or statistical self-similarity. In such settings, the asymptotics of heat diffusion are governed not by classical smooth invariants (like perimeter or surface area), but by scale-dependent geometric features—specifically, Minkowski content, complex fractal dimensions, and higher-order “fractal curvatures.” This subject connects fractal geometry, spectral theory, stochastic processes, and applied mathematical physics, leveraging tube formulas, scaling functional equations, and renewal-type arguments to analyze how intricate boundaries control the propagation of heat at small times.

1. Minkowski Measurability, Tube Formulas, and Heat Content

The paper of heat content on self-similar fractals is tightly linked to Minkowski measurability and tube formulas for fractal sets. The Minkowski content $\mathcal{M}(F)$ of a fractal $F \subset \mathbb{R}^d$ with Minkowski dimension $D$ is extracted from the asymptotic expansion of the volume of its $\varepsilon$-neighborhood:

$$\mathcal{M}(F) = \lim_{\varepsilon \to 0^+} V_F(\varepsilon)\, \varepsilon^{D-d}$$

where $V_F(\varepsilon)$ denotes the $d$-dimensional volume of the $\varepsilon$-neighborhood.

For self-similar fractals, the dichotomy between lattice and non-lattice types of the iterated function system (IFS) determines whether a well-defined Minkowski content—and by extension, a well-defined leading term in the heat content—exists. In non-lattice cases, the oscillatory cross-scale fluctuations average out, resulting in a unique Minkowski content; in lattice cases, persistent periodic logarithmic oscillations prevent a pointwise limit, so only a logarithmic average is meaningful (Deniz et al., 2010).

Mathematically, the tube formula, which expresses $V_F(\varepsilon)$ in terms of residues of a geometric zeta function,

$$V_\mathcal{T}^-(\varepsilon) = \sum_{\omega \in \mathcal{D}_\mathcal{T}} \operatorname{res}(\zeta_\mathcal{T}(s, \varepsilon), \omega),$$

parallels expansions for heat content, with complex dimensions ($\omega$) controlling oscillatory terms. The same framework extends to expansions for short-time heat content, with the leading-order behavior determined by the residue at $s = D$.

2. Scaling Laws, Zeta Functions, and Complex Dimensions

On self-similar fractal boundaries, scaling laws provide the analytic backbone for both geometric and heat content asymptotics. The principal object is the scaling zeta function:

$$\zeta_\Phi(s) = \frac{1}{1 - \sum_{\varphi \in \Phi} \lambda_\varphi^s}$$

where the $\lambda_\varphi$ are contraction ratios of the IFS maps. Its poles—the complex dimensions—arise as solutions to $1 = \sum_{\varphi \in \Phi} \lambda_\varphi^s$ and encode information about multi-scale oscillations in both geometric invariants and spectral quantities (Hoffer et al., 13 Aug 2025, Hoffer, 11 Jul 2025).

For heat content, the parabolic nature of the heat equation induces a functional equation of the form:

$$E(t) = \sum_{\varphi \in \Phi} E\left(\frac{t}{\lambda_\varphi^2}\right) + R(t)$$

where $R(t)$ accounts for non-disjoint overlaps of self-similar copies. Mellin transform techniques translate this recursion into a singular expansion for $E(t)$,

$$E(t) \sim \sum_{\omega \in \mathcal{D}_\Phi} r_\omega\, t^{(N-\omega)/2} + \text{error terms}$$

in which the $\omega$ run over complex dimensions—identical for tube volume and heat content due to the underlying scaling structure (Hoffer et al., 13 Aug 2025).

3. Heat Kernel Asymptotics, Renewal Equations, and Dimension Recovery

The spectral characteristics of the Laplacian on self-similar fractals yield heat kernel asymptotics that deviate sharply from the classical Weyl law. For the trace $K(t)$ of the heat kernel, one obtains leading sub-Gaussian behavior:

$$K(t) \sim c\, t^{-d_s/2} \left[1 + \alpha \cos\left(\frac{2\pi}{d_w \ln l} \ln t + \phi\right) + \cdots \right]$$

Here $d_s$ is the spectral dimension and $d_w$ the walk dimension, both of which can differ from the geometric (Hausdorff) dimension due to anomalous diffusion (Dunne, 2012).

In domains with statistically self-similar fractal boundaries, branching process techniques recover not only the Minkowski dimension from heat content scaling but also finer geometric fluctuations, expressed as regularly varying corrections (e.g., $E(s) \sim c\, s^\alpha e^{f(\log(1/s))}$). In almost sure expansions, random variables such as $N_\infty$—limits of martingales—appear as multiplicative factors modulating heat content (Charmoy, 2014).

Moreover, renewal equations serve as a mechanism to propagate asymptotics across scales, particularly in the treatment of fractal polyhedra or non-arithmetic scaling settings:

$$E(t) = \sum_{i=1}^N c_i E\left(\frac{t}{r_i^2}\right) + \phi(t)$$

with explicit solutions encoding contributions from multiple scales (Park et al., 2021, Berg et al., 2015).

4. Geometric and Physical Interpretation: Fractal Curvatures and Scaling Exponents

In domains where the boundary's Minkowski dimension $d$ exceeds $n-1$ (smooth case), the leading term in short-time heat content asymptotics shifts to

$$N(t) \sim \mathcal{M}^d(\partial\Omega, \Omega) (D_+ t)^{(n-d)/2}$$

with $\mathcal{M}^d$ the Minkowski content of the boundary (Rozanova-Pierrat et al., 5 Feb 2025). This “anomalous” scaling implies faster heat loss for boundaries with larger fractal dimension.

Beyond the leading term, “fractal curvature measures” enter as higher-order corrections: renormalized limits of Lipschitz–Killing curvature measures over parallel sets, rescaled by $r^{d-k}$. These measures encode the multi-scale geometry beyond mere size, allowing predictions about the impact of boundary roughness on thermal transport and the lower-order terms of the expansion.

In radiative settings, the fractal dimension controls collective heat transfer via universal scaling relations such as $\mathcal{G}_N \sim R_g^{D_f}$, with $R_g$ the gyration radius and $D_f$ the fractal dimension, establishing a direct connection between morphology and total heat loss (Nikbakht, 2017).

5. Stochastic, Network, and Numerical Models in Fractal Heat Diffusion

Analysis on self-similar fractals often exploits discrete network representations, where the fractal forms the boundary of a weighted graph (network). Here, discrete Laplacians (via conductance models or Markov chains) provide a framework for understanding heat content: random walks on the network converge to the fractal, enabling computation of Dirichlet forms and energy distributions (Pearse, 2011, Chen et al., 2016).

Iterative matrix and tensor decomposition methods (including tensor-train formats and Kronecker products) yield efficient numerical and analytical representations of high-dimensional self-similar sets. These frameworks are critical for extending heat content analysis to higher dimensions and for modeling non-Euclidean diffusion on fractal geometries (Gelß et al., 2018, Tzanov, 2015).

For post-critically finite (pcf) fractals, resolvent kernel decompositions and Phragmén–Lindelöf extension theorems enable sharp control over the spatial decay of the heat kernel and the resulting sub-Gaussian heat content estimates (Rogers, 2010). For certain Sierpinski carpet-like fractals lacking uniform scaling or connectivity, classical Dirichlet forms cannot exist, and sub-Gaussian estimates fail, producing fundamentally different heat content behavior (Cao et al., 2021).

6. Applications, Limitations, and Directions for Further Study

These methods and results unify spectral geometry, fractal analysis, and probabilistic techniques, illuminating how self-similar structure governs both stationary and anomalous thermal behaviors. Key applications include:

• Inferring Minkowski and spectral dimensions from heat content measurements;
• Quantifying the effect of nanoscale morphology on collective radiative heat transfer in fractal aggregates;
• Evaluating corrections due to fractal curvatures in thermal interface design;
• Connecting stochastic models (subordinate killed Brownian motion, martingale limits, random walks) to geometric features (Rozanova-Pierrat et al., 5 Feb 2025, Park et al., 2021).

Limitations arise for domains with complex overlap among self-similar pieces or lack of regular separation, where the “remainder” in scaling functional equations must be admissible for Mellin-based expansions to hold. The breakdown of classical diffusion behavior in certain “pathological” carpet-like fractals underscores the necessity of checking geometric and analytic assumptions before applying tube and scaling methods (Cao et al., 2021).

Current research is exploring extension to self-affine and stochastic fractals, more general Lévy processes, central limit corrections to heat content, explicit computation of residue-based coefficients from scaling zeta functions, and deeper connections between spectral and geometric invariants.

7. Summary Table: Main Features and Regimes

Feature	Non-Lattice IFS	Lattice IFS	Statistically Self-Similar
Minkowski Content	Exists, unique	Only as logarithmic avg	Exists a.s.
Heat Content Leading Term	Simple power law	Power law + log-periodic	Power law × random variable
Complex Dimensions	Single, non-periodic	Infinite, periodic line	Generic (non-lattice)
Fluctuations in Expansion	Suppressed	Present (oscillatory)	Modulated by martingale
Tube Formula via Zeta Function	Residue at $s = D$	Averaged residue	Leading term via branching
Spectral Heat Kernel Scaling	Sub-Gaussian, $t^{-d_s/2}$	Oscillations in $\ln t$	Sub-Gaussian, random prefac

This framework underscores that on self-similar fractals, the heat content can “hear” not only the dimension but also the fine oscillatory structure and curvature measures of the boundary, making these objects ideal laboratories for linking geometry, analysis, and stochastic models.