Thermal properties of bodies in fractal and cantorian physics

Abstract

Abstract Fundamental laws describing the heat diffusion in fractal environment are discussed. It is shown that for the three-dimensional space the heat radiation process occur in structures with fractal dimension D ∈ 〈0,1), whereas in structures with D ∈ (1,3〉 heat conduction and convection have the upper hand (generally in the real gases). To describe the heat diffusion a new law has been formulated. Its validity is more general than the Plank’s radiation law based on the quantum heat diffusion theory. The energy density w = f ( K , D ), where K is the fractal measure and D is the fractal dimension exhibit typical dependency peaking with agreement with Planck’s radiation law and with the experimental data for the absolutely black body in the energy interval kT Kℏc . The positions of the energy density maximums (for fractal dimensions D m W - Function u ( A ) = A + W [− A exp(− A )], where A ≈ 1.9510 and u = hc / λ m kT m ≈ 1.5275. The agreement of the fractal model with the experimental outcomes is documented for the spectral characteristics of the Sun. The properties of stellar objects (black holes, relict radiation, etc.) and the elementary particles fields and interactions between them (quarks, leptons, mesons, baryons, bosons and their coupling constants) will be discussed with the help of the described mathematic apparatus in our further contributions. The general gas law for real gases in its more applicable form than the widely used laws (e.g. van der Waals, Berthelot, Kammerlingh–Onnes) has been also formulated. The energy density, which is in this case represented by the gas pressure p = f ( K , D ), can gain generally complex value and represents the behaviour of real (cohesive) gas in interval D ∈ (1,3〉. The gas behaves as the ideal one only for particular values of the fractal dimensions (the energy density is real-valued). Again, it is shown that above the critical temperature ( kT > Kℏc ) and for fractal dimension D m > 2.0269 the results are comparable to the kinetics theory of real (ideal) gas (van der Waals equation of state, compressibility factor, Boyle’s temperature). For the critical temperature ( Kℏc = kT r ) the compressibility factor gains Z = 1 (except for the ideal gas case D = 3) also for the fractal dimension D = 1/ ϕ = 1.618033989, where ϕ is the golden mean value of the El Naschie’s golden mean field theory. To determine the minimum it is also possible to employ the Lambert’s W − Function u ( A ) = A + W [− A exp(− A )], where A ≈ 0.6779 and u ≈ −0.7330. The thermal properties of fractal structures (thermal capacity, thermal conductivity, diffusivity) and additional parameters (enthalpy, entropy, etc.) will be defined using the mathematic apparatus in the future. Good agreement of the fractal model with experimental data is documented on the compressibility factor of various gases.