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We know that the diffusion equation can be viewed as the continuum limit of a discreet unbiased random walk.

My question is the following:

Is diffusion (or an unbiased random walk) a non-equilibrium phenomenon?

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The question is far from being trivial. Basically, one has to distinguish between a macroscopic diffusion as the effect of a starting macroscopic gradient of density or concentration in a condensed phase and microscopic diffusive motion of individual atoms/molecules.

By combining the continuity equation and Fick's law, one easily gets a time-dependent diffusion equation for the macroscopic density/concentration whose solution is time-dependent. Therefore we are in the presence of a non-equilibrium process.

However, in an equilibrium system, fluctuations are always present. It is the content of the so-called fluctuation-dissipation theorem that the time response of a system perturbated by an external weak perturbation is indistinguishable from the decay of an equilibrium fluctuation. Moreover, the atomic motion of a selected particle in an equilibrium system can be well approximated by a suitable model for diffusive behavior (the Brownian motion is a good model for fluid phases). From this microscopic point of view, diffusion is an equilibrium property.

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The answer depends a bit on the type of equilibrium you are talking about. Having said that, if there is net diffusion happening, then the density or distribution of particles you are considering is probably changing with time time and if things are changing in time, they probably aren't in equilibrium

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