FRACTAL TIME IN CONDENSED MATTER

Temporal scaling laws involving noninteger exponents have appeared in the physics of many systems, including charge transport in xerographic films (1-4), electron-hole recombination in amorphous materials (5-7) and other heterogeneous kinetics (8), dielectric, magnetic, and mechanical relaxation in glassy materials (9-12), and in the time-dependent reactivity of radiation-induced chemistry in frozen liquids (13). While at first quite puzzling, all of these complex phenomena have been explained using a concept called fractal time to describe the transport of charges and defects in these systems. Fractals are usually considered to be self-similar geo­ metric objects in space with features on an infinite number of scales. Fractal time describes highly intermittent self-similar temporal behavior that does not possess a characteristic time scale. If the average time of an event were finite, then this would provide a time scale. Thus for fractal time, the average time for an event must be infinity. The event can be the time a charge is trapped at a site in an amorphous solid, the time the phase in a Josephson junction rotates in the same direction, the time a fluid particle spends in a given vortex, etc. In 1963, Berger & Mandelbrot (14) first used the concept of fractal time to characterize transmission errors in telephone networks, and Mandelbrot coined the term in his 1977 book (15) to describe the Scher-Montroll (1) model of transport in amorphous media. We are able to trace the scaling inherent in fractal time back to an idea introduced by Nicolas Bernoulli