Non-Linear Friction-Type Behaviors in Liquid Crystals in a Multifractal Paradigm of Motion

Abstract

Fractal analysis of liquid crystals is usually and mainly focused on their intrinsic (structural) properties, and less on their dynamic (functional) properties. In the present paper, a new model, based on a multifractal structural-functional synergy for describing liquid crystals dynamics, is proposed. Then, liquid crystal entities dynamics will be explained in relation with the Scale Relativity Theory through continuous and non-differentiable (multifractal) curves, described with Schrödinger’s theory and the hydrodynamic theory. Examining the hidden symmetry of the friction tensor at different scale resolutions, in the Madelung framework, synchronization modes among the liquid crystals entities are found through SL(2R) transformations groups. As such, at various scale resolutions, Riemannian manifolds with hyperbolic metrics, generated by means of direction parallel transport (in the Levi-Civita sense) can be defined. Different non-stationary friction characteristics can thus be highlighted at various scale resolutions, using harmonic mappings from the ordinary space to the hyperbolic one. Of these, the cellular and channel-type ones become fundamental.

Keywords

• liquid crystals
• fractal analysis
• multifractal friction tensor
• scale relativity theory