Fractional Dynamics, Tiling Equilibrium states and Riemann's zeta function

TL;DR

This paper explores how fractality and singularities in space-time influence generalized dynamics, leading to connections with Riemann zeta functions and their zeros, and introduces a new discrete time concept with thermodynamic implications.

Contribution

It proposes a novel framework linking fractal space-time, singularities, and Riemann zeta functions, extending classical mechanics principles to complex geometries.

Findings

Fractality induces a new discrete time concept.

Singularities create horizons affecting equilibrium states.

Connections established between space-time singularities and Riemann zeta zeros.

Abstract

Il is argued that the generalisation of the mechanical principles to other variables than localisation, velocity and momentum leads to the laws of generalized dynamics under the condition of continuous and derivable space time. However, when the fractality arises, the mechanics principles may no more be extended especially because the time and space singularity appears on the boundary and creates curvature. There is no more equilibrium state, but only a horizon which might play a same role as equilibrium but does not close the problem - especially the problem of the invariance of the energy - which requires two complementary factors: a first one related to the closure in the dimensional space, and the second to scan dissymmetry stemming from the default of tilling the space time. A new discreet time arises from fractality. It leads irreversible thermodynamic properties. Space and time singularities lead to the relation between the above mentioned problematic and the Riemann zeta functions as well as its zeros.