Abstract
In most interacting many-body systems associated with some "emergent phenomena," we can identify subgroups 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 paper, we investigate how such a temperature landscape arises according to how the slow variables affect the character of the driven quasisteady 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.
Highlights
A broad range of many-body nonequilibrium systems have in common that different degrees of freedom within them undergo motion on two well-separated time scales, and that the faster degrees of freedom are the only ones directly subject to external driving
In many systems one can usefully identify coarse-grained variables describing global features of the many-body dynamics, which may relax more slowly than the coordinates of individual particles. Such order parameters might be thought of as a set of slowly varying constraints on the driven fast dynamics, as, for example, in [2]. It is possible in principle for the particular configuration of a set of slow variables to have a significant influence on the specific nonequilibrium steady state reached by the fast variables
In general, a feedback loop can arise in which the slow variables first establish the features of the fast steady state, and the statistics of this steady state in turn determine the stochastic dynamics of the resulting local motion
Summary
A broad range of many-body nonequilibrium systems have in common that different degrees of freedom within them undergo motion on two well-separated time scales, and that the faster degrees of freedom are the only ones directly subject to external driving. There is extensive literature studying the conditions and effects of deviations from this basic result, which are generally termed “anomalous diffusion”— see, e.g., [5] Within this context, the “slow” degrees of freedom lack their own dynamics, and are considered only as probes of the fast bath. III, we will carry out a numerical analysis of the kicked rotor on a cart—a time-scale separated, damped, driven dynamical system that is ideally suited for demonstrating the predictive power of the leastrattling framework Will this analysis draw clear connections to methods of equilibrium statistical physics and show how they generalize in such a nonequilibrium scenario, but it will underline how least rattling helps to explain the nontrivial relationship between dissipation rate and local kinetic stability in driven systems
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