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Active systems of synthetic motors operating out of equilibrium on micro- and nano-scales have unique properties that can be exploited in applications in a number of different fields, and the descriptions of the mechanisms by which they operate present challenges for theory and simulation. In this chapter we focus on synthetic chemically-propelled motors without moving parts that operate by a diffusiophoretic mechanism, where self-generated concentration gradients are responsible for motor motion. Both macroscopic continuum treatments and microscopic models of the dynamics are described and used to investigate motor dynamics. While different motor geometries are discussed, Janus motors with catalytic and non-catalytic portions are used to illustrate the phenomena. Single motor motion, the collective dynamics of many motors, and the dynamics of motors in crowed media are described.

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