Until about 10 years ago, expressions involving fractional derivatives and integrals were pretty much restricted to the realm of mathematics. But over the past decade, many physicists have discovered that a number of systems—particularly those exhibiting anomalously slow diffusion, or subdiffusion—are usefully described by fractional calculus. Those systems include charge transport in amorphous semiconductors, the spread of contaminants in underground water, relaxation in polymer systems, and tracer dynamics in polymer networks and in arrays of convection rolls.

Fractional diffusion equations generalize Fick’s second law and the Fokker–Planck equation by taking into account memory effects such as the stretching of polymers under external fields and the occupation of deep traps by charge carriers in amorphous semiconductors. Such generalized diffusion equations allow physicists to describe complex systems with anomalous behavior in much the same way as simpler systems.

Physicists are all familiar with the high-school calculus that introduces students to derivatives of...

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