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\begin{enumerate}  \item Why do the swimmers float on top?  \item For now consider the simple case of a single inert particle in a (fixed)  concentration gradient. Say it moves towards the source of the chemical. It will follow some mean trajectory, but with plenty of scatter. This is an out-of-equilibrium process. \begin{enumerate}  \item Understand why it moves (diffusiophoresis): how fast and in what direction on average?   \item What are the statistics of this motion -- how much variance is there in the motion?   \item  Which part of this setup  is out of equilibrium? \item What does the scatter in trajectories tell us about the system? Anything useful? Use linear-response theory (is this justified? Why?).  \end{enumerate}