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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.

Periodic current bursts observed in the dynamic current-voltage characteristic of a probe in the presence of a plasma fireball in dynamic state were modeled in the frame of the scale relativity model, based on both the fractal space-time concept and the generalization of Einstein’s principle of relativity to scale transformations. The bursts appear in the probe characteristic when a certain relation exists between the fireball dynamics frequency and the frequency of the probe voltage sweep. The double layer dynamics is described by a set of time-dependent Schrödinger-type equations and the self-structuring is given by means of the negative differential resistance. The obtained experimental and theoretical results are proven to be in very good agreement.

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.