Most physical models on quasicrystals, as well as the related experimental results, exhibit fractal energy spectra. In order to have a deep insight on relevant thermodynamic implications of this feature, we have performed analytical and high precision numerical calculations of the specific heats $C_n^{\mathrm{band}}$ and $C_n^{\mathrm{disc}}$ associated with successive hierarchical approximations $(n=1,2,3,\dots )$ to bounded Cantor-set energy spectra (constructed with sets of continuous intervals for the banded case, and with discrete levels for the discretecase). Instructive anomalies are exhibited, namely (i) $C_n^{\mathrm{band}}\mathrm{}\left(T\right)\mathrm{}$ and $C_n^{\mathrm{disc}}\mathrm{}\left(T\right)\mathrm{}$ differ for all temperatures and finite $n$ (in particular, in units of $k_B,$ $C_n^{\mathrm{disc}}\mathrm{}\left(0\right)=0\mathrm{}$ whereas $C_n^{\mathrm{band}}\mathrm{}\left(0\right)=1)$, but, through an interesting nonuniform convergence, $C_{\infty }^{\mathrm{band}}{\mathrm{}\left(T\right)=C}_{\infty }^{\mathrm{disc}}\mathrm{}\left(T\right)\mathrm{}\equiv C_{\infty }\mathrm{}\left(T\right)\mathrm{}$ for all finite temperatures; (ii) in the $T\to 0$ limit, $C_{\infty }\mathrm{}\left(T\right)\mathrm{}$ exhibits an infinite number of small-amplitude oscillations symmetrically disposed precisely around the fractal dimensionality $d_f=\ln 2/\ln 3$; more precisely, $C_{\infty }\mathrm{}\left(T\right)\mathrm{}\sim C^{*}\mathrm{}\left(T\right)\mathrm{}$, where $C^{*}{\mathrm{}\left(T\right)=C}^{*}\mathrm{}\left(3T\right)={\sum }_{k=-\infty }^{\infty }\left[3^{k}T\cosh {(1/3\mathrm{}}^{k}T\right){]}^{-2}$ $=\ln 2/\ln 3+a\sin \left[2\pi \ln \mathrm{}\right(\mathrm{bT})/\ln 3]+\epsilon \mathrm{}\left(T\right)\mathrm{}$ with $a=1.27\mathrm{}\dots \times {10}^{-2}$, $b=1.97\mathrm{}\dots$ and $\epsilon \mathrm{}\left(T\right)<5\mathrm{}\times {10}^{-4}$ $\left(\forall \hspace{0.5em}T\right)$ $(T$ is measured in units of the outermost width of the Cantor set); (iii) in the $T\to \infty$ limit, $C_{\infty }\mathrm{}\left(T\right)\mathrm{}\sim {1/8T}^2$. In addition to this, we comment on a possible connection of this type of systems with the recently introduced nonextensive thermostatistics.

• Received 24 July 1997

DOI:https://doi.org/10.1103/PhysRevE.56.R4922