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Diffusion of small particles is omnipresent in a plentiful number ofprocesses occurring in Nature. As such, it is widely studied and exerted inalmost all branches of sciences. It constitutes such a broad and often rathercomplex subject of exploration that we opt here to narrow down our survey forthe case of the diffusion coefficient for a Brownian particle which can bemodeled in the framework of Langevin dynamics. Our main focus will center onthe temperature dependence of the diffusion coefficient for several fundamentalmodels of diverse physical systems. Starting out with diffusion in equilibriumfor which the Einstein theory holds we consider a number of physical situationsaway from free Brownian motion and end with surveying nonequilibrium diffusionfor a time-periodically driven Brownian particle dwelling randomly in aperiodic potential. For this latter situation the diffusion coefficientexhibits an intriguingly non-monotonic dependence on temperature.

Diffusion coefficient of a Brownian particle in equilibrium and nonequilibrium: Einstein model and beyond