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Fractal Ohm’s law and the Drude model in fractal time
Journal article   Peer reviewed

Fractal Ohm’s law and the Drude model in fractal time

Alireza Khalili Golmankhaneh, Airton Deppman, Rawid Banchuin and Palle E.T. Jørgensen
Physica. B, Condensed matter, Vol.731, 418501
06/01/2026
DOI: 10.1016/j.physb.2026.418501

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Abstract

Classical electrical conduction theories, including Ohm’s law and the Drude model, assume charge transport in smooth Euclidean time. However, many disordered and glassy materials exhibit anomalous conduction and relaxation that classical models cannot fully explain. In this work, we present a generalized electrical conduction framework based on fractal calculus, where time evolves on a fractal set with dimension 0<α≤1. Using local fractal derivatives and the associated staircase function, we derive a fractal equation of motion for charge carriers and obtain explicit formulas for drift velocity, current density, and conductivity. The resulting conductivity follows a power-law dependence on the relaxation time, reflecting temporal fractality. The classical Drude result is recovered in the Euclidean limit α=1. The model also produces stretched-exponential relaxation, giving a unified description of steady and transient transport in complex media. •Fractal-time framework for electrical conduction is introduced.•Ohm’s law is reformulated using local fractal derivatives.•Drude model is extended to fractal time dynamics.•Conductivity shows power-law dependence on relaxation time.
 Anomalous electrical conduction Fractal calculus Fractal Ohm’s law Fractal time Glassy and disordered materials Stretched-exponential relaxation

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