Scale relativity theory for one-dimensional non-differentiable manifolds

Abstract

We discuss a rigorous foundation of the pure scale relativity theory for a one-dimensional space variable. We define several notions as ''representation'' of a continuous function, scale law and minimal resolution. We define precisely the meaning of a scale reference system and space reference system for non-differentiable one-dimensional manifolds. Ó

Fractal Transport Phenomena through the Scale Relativity Model

“…The non-differentiability, according with Cresson's mathematical procedures [29][30][31][32][33][34][35][36] and Nottale's physical principles [1-8, 16-19, 27, 28] implies the followings:…”

“…"In the framework of scale relativity, the physics is related to the behavior of the function during the "zoom" operation on the time resolution δt, here identified with the differential element dt ("substitution principle"), which is considered as an independent variable. The standard function F (t) is therefore replaced by a fractal function F (t, dt) (for details see [29][30][31][32][33][34][35][36]) explicitly dependent on the time resolution interval, whose derivative is undefined only at the unobservable limit dt → 0" [16,17]. As a consequence, this leads us to define the two derivatives of the fractal function as explicit functions of the two variables t and dt,…”

“…(iv) the differential of a fractal function F (t, dt) can be expressed as the sum of two differentials, one which is not scale-dependent, dF (t), and the other dependent on it, dF (t, dt), therefore [29][30][31][32][33][34][35][36] dF (t, dt) = dF (t) + dF (t, dt).…”

“…We call this procedure "an extension by differentiability" (Cresson's extension -for detail see [33][34][35][36]). Applying this operator to the "position vector" yields a complex speed…”

“…In the present paper the correspondence between the Nottale SR model [1-8, 16-19, 27, 28] and Cresson mathematical procedures [29][30][31][32][33][34][35][36][37][38][39] is analyzed. Thus, in Sect.…”