A new model of the transport phenomena in nanostructures, considering that the motion of the particles takes place on continuous but non-differentiable curves is build. It results that the synchronization of the movements at different scales (fractal scale, differential scale etc.) gives conductive type properties to the "matter," while the absence of synchronization induces properties of convective type. These behaviors (conductive or convective) at nano-time scales, are illustrated through numerical simulations of a plasma generated by laser ablation.

Considering that the particle movement takes place on fractal curves, the mathematical and physical aspects in fractal space-time theory are analyzed. In such context, the harmonic oscillator problem implies that the microscopic-macroscopic scale transition could be associated with an evolution scenario towards chaos. The splitting of the plasma plume, generated by laser ablation, into two patterns, has been successfully reproduced through a numerical simulation using the fractal hydrodynamic model. For the free time-dependent particle in a fractal space-time, the uniform movement is naturally obtained by a specific mechanism of vacuum polarization.

Implications of the fractal potential in the system dynamics using an extended scale relativity model assuming the fractal character of the particle movements, are established. So, in the dissipative approximation of the model it is shown that the fractal potential comes from the non-differentiability of the space-time, i.e. by means of imaginary part of a complex speed field. In the dispersive approximation of the same model, the fractalization of the differential part of the complex speed field induces a normalized fractal potential which controls through coherence the system dynamics. In such context the type I superconductivity results: the temperature dependences of the superconducting parameter, the accumulator effect etc.

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