Implications of Non-Differentiable Entropy on a Space-Time Manifold

• Main Authors: Maricel Agop, Alina Gavriluţ, Gavril Ştefan, Bogdan Doroftei
• Format: Article
• Language: English
• Published: MDPI AG 2015-04-01
• Series: Entropy
• Subjects: non-differentiable entropy; fractal bit; space-time manifold; space-time scale relativity theory
• Online Access: http://www.mdpi.com/1099-4300/17/4/2184

Assuming that the motions of a complex system structural units take place on continuous, but non-differentiable curves of a space-time manifold, the scale relativity model with arbitrary constant fractal dimension (the hydrodynamic and wave function versions) is built. For non-differentiability through stochastic processes of the Markov type, the non-differentiable entropy concept on a space-time manifold in the hydrodynamic version and its correspondence with motion variables (energy, momentum, etc.) are established. Moreover, for the same non-differentiability type, through a scale resolution dependence of a fundamental length and wave function independence with respect to the proper time, a non-differentiable Klein–Gordon-type equation in the wave function version is obtained. For a phase-amplitude functional dependence on the wave function, the non-differentiable spontaneous symmetry breaking mechanism implies pattern generation in the form of Cooper non-differentiable-type pairs, while its non-differentiable topology implies some fractal logic elements (fractal bit, fractal gates, etc.).