Measurements on soft condensed matter and biological systems often track low dimensional physical observables projected from the full system phase space as a function of time. Most prominent examples thereof are single-particle tracking an various forms of single-molecule spectroscopy. In most cases the projection renders the observed dynamics non-Markovian and the memory has intriguing manifestations.
Moreover, in many such measurements the duration of the observation is shorter than the respective ergodic time. In order to describe such systems theoretically one is required to study functionals of projected dynamics along finite trajectories.
We will review our recent theoretical results on the dynamics of projected observables with emphasis on the physical meaning of memory and its manifestations, the emergence of a broken time-translation invariance, and surprising asymmetries in relaxation from temperature quenches.