On the occurrence of oscillatory modulations in the power - law behavior of dynamic and kinetic processes in fractals

– The dynamic and kinetic behavior of processes occurring in fractals with spatial discrete scale invariance (DSI) is considered. Spatial DSI implies the existence of a fundamental scaling ratio (b1). We address time-dependent physical processes, which as a consequence of the time evolution develop a characteristic length of the form ξ ∝ t, where z is the dynamic exponent. So, we conjecture that the interplay between the physical process and the symmetry properties of the fractal leads to the occurrence of time DSI evidenced by soft log-periodic modulations of physical observables, with a fundamental time scaling ratio given by τ = b1. The conjecture is tested numerically for random walks, and representative systems of broad universality classes in the fields of irreversible and equilibrium critical phenomena. The understanding of the kinetic and dynamic behavior of the wide variety of phenomena and processes, taking place in disordered and low-dimensional media, is a challenging issue of increasing interdisciplinary interest [1,2]. The simplest and paradigmatic example is, most likely, the effort devoted to the study of random walks [3]. However, many other examples can also be quoted such as the dynamics of critical systems in condensed matter physics [2,4]; chemical reactions in catalysts, porous media, nanoand microcavities; annihilation reactions of a wide diversity ranging from magnetic monopoles in the early stages of the Universe to excitons in polymeric matrices [5]; epidemic propagation of diseases and forest fires in far fromequilibrium systems [1, 4, 6]; coarsening dynamics in many systems involving fluids, magnets, and eventually the formation of opinion in social systems [4]; etc. In this letter we address the subtle interplay between time-dependent processes and the symmetry properties of the underlying structure or substrate where the considered process actually takes place. For this purpose, fractal media that exhibit discrete scale invariance (DSI) [7] are considered. DSI is a weak kind of scale invariance such that an observable O(x), obeys the scaling law O(x) = μ(b)O(bx), (1) under the change x → bx. Here b is no longer an arbitrary real number, as in the case of continuous scale invariance, but it can only take specific discrete values of the form bn = (b1) , where b1 is a fundamental scaling ratio. If an observable O(x) satisfies equation (1) for an