Least Rattling Feedback from Strong Time-scale Separation

In most interacting many-body systems associated with some "emergent phenomena," we can identify sub-groups of degrees of freedom that relax on dramatically different time-scales. Time-scale separation of this kind is particularly helpful in nonequilibrium systems where only the fast variables are subjected to external driving; in such a case, it may be shown through elimination of fast variables that the slow coordinates effectively experience a thermal bath of spatially-varying temperature. In this work, we investigate how such a temperature landscape arises according to how the slow variables affect the character of the driven quasi-steady-state reached by the fast variables. Brownian motion in the presence of spatial temperature gradients is known to lead to the accumulation of probability density in low temperature regions. Here, we focus on the implications of attraction to low effective temperature for the long-term evolution of slow variables. After quantitatively deriving the temperature landscape for a general class of overdamped systems using a path integral technique, we then illustrate in a simple dynamical system how the attraction to low effective temperature has a fine-tuning effect on the slow variable, selecting configurations that bring about exceptionally low force fluctuation in the fast-variable steady-state. We furthermore demonstrate that a particularly strong effect of this kind can take place when the slow variable is tuned to bring about orderly, integrable motion in the fast dynamics that avoids thermalizing energy absorbed from the drive. We thus point to a potentially general feedback mechanism in multi-time-scale active systems, that leads to the exploration of slow variable space, as if in search of fine-tuning for a "least rattling" response in the fast coordinates.