Choosing Fractional Derivatives in Physics Through Equivalence, Mechanism, and Measurement

This review develops an operator-selection framework in which equivalence, physical mechanism, mathematical domain, and independently measurable observables are examined before a fractional derivative is used. The principal result is a systematic operator passport and an equivalence atlas that distinguish exact representation identities, restrictions induced by initial data, coordinate conjugacies, and limiting relations. The order–asymmetry representation is used for full-line translation-invariant operators, while kernel, regularization, variable or distributed order, geometry, and boundary realization provide independent classification criteria. This review covers the Riemann–Liouville, Gerasimov–Caputo, Grünwald–Letnikov, Marchaud, Weyl, Riesz–Feller, Hadamard, Hilfer, Prabhakar, tempered, variable-order, distributed-order, general-kernel, fractional-Laplacian, conformable, and nonsingular-kernel constructions. Four physical settings—viscoelastic and dielectric relaxation, anomalous transport, attenuating waves, and bounded-domain nonlocality—connect operator differences with response functions, trajectory statistics, dispersion, and spectra. A published single-particle-tracking comparison illustrates that the same anomalous mean-square-displacement scaling does not uniquely identify its stochastic mechanism. Mathematical limitations of nonsingular kernels and a phenomenological absorbing-contact transport construction are made explicit. The resulting decision protocol separates mathematically equivalent representations from physically testable competing models.