SPECIFIC-HEAT ANOMALIES ASSOCIATED WITH CANTOR-SET ENERGY-SPECTRA

Most physical models on quasicrystals, as well as the related experime ntal results, exhibit fractal energy spectra. In order to have a deep insight on relevant thermodynamic implications of this feature, we hav e performed analytical and high precision numerical calculations of th e specific heats C-n(band) and C-n(disc) associated with successive hi erarchical approximations (n=1,2,3,...) to bounded Canter-set energy s pectra (constructed with sets of continuous intervals for the banded c ase, and with discrete levels for the discrete case). Instructive anom alies are exhibited, namely (i) C-n(band)(T) and C-n(disc)(T) differ f or all temperatures and finite it (in particular, in units of k(B), C- n(disc)(0)=0 whereas C-n(band)(0)=1), but, through an interesting nonu niform convergence, nfinity(band)(T)=C-infinity(disc)(T)=C-infinity(T) for all finite temperatures; (ii) in the T-->0 limit, C-infinity(T) e xhibits an infinite number of small-amplitude oscillations symmetrical ly disposed precisely around the fractal dimensionality d(f)=ln2/ln3; more precisely, C-infinity(T)similar to C(T), where )(infinity)[3(k)T cosh(1/3(k)T)](-2)=ln2/ln3+asin[2 pi ln(bT)/ln3]+epsilon(T) with a=1.2 7...X10(-2), b=1.97... and epsilon(T)<5X10(-4)(For All T) (T is measur ed in units of the outermost width of the Canter set); (iii) in the T- ->infinity limit, C-infinity(T)similar to 1/8T(2). In addition to this , we comment on a possible connection of this type of systems with the recently introduced nonextensive thermostatistics.