Least-rattling feedback from strong time-scale separation

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.

Figure 1

(a) Schematic of the kicked-rotor-on-a-cart toy model. (b) Numerical realizations of typical cart trajectories over time for driving force below ($K=3$: regular/ordered regime) and above ($K=8$: chaotic regime) the ordering transition (at $K_c\sim 5$), along with samples of the corresponding fast $(\theta ,v)$ dynamics. (c, d, e) Average force, fluctuations, and dissipation rates in the cart dynamics, measured from trajectories as in panel (b), for the various values of $K$, along with analytical predictions (in black) from Eq. (6).

Figure 2

Histograms (gray) of the steady-state cart positions in simulation with (a) constant $K\left(x\right)=K$ and in a potential landscape $U\left(x\right)$ plotted in blue (the black curve gives the expected Boltzmann distribution). (b) $U\left(x\right)=\text{const}$ and $K\left(x\right)$ landscape in red [the black curve shows the $1/T_{\text{eff}}\left(x\right)$ solution of the Fokker-Planck equation]. (c) $U\left(x\right)$ as in panel (a) and the $K\left(x\right)$ step function [the analytical prediction plotted in black is given by Eq. (8)]. (d) $U\left(x\right)=\text{const}$ and $K\left(x\right)$ as in panel (b), but shifted down to dip below the critical $K_c\sim 5$ value—dotted red line [the black curve again shows $1/T_{\text{eff}}\left(x\right)$ outside of the ordered region]—this shows that probability gets localized at the two transition points at long times.

Figure 3

Typical cart trajectories in linear $K\left(x\right)$ landscape $\left[U\right(x)=\text{const}]$ all starting from one point, along with their mean (purple) and median (brown). The black curve shows the analytical prediction for the median, while the mean is expected to be constant at small times. Inset: The regularization transition at $K_c\sim 5$, where effective temperature drops abruptly to zero, and the median departs from the smooth decay. The $x$ axis is labeled in units of $K$.

Figure 4

Simulated steady-state cart-position distributions for the shown $U\left(x\right)$ (blue) and $K\left(x\right)$ (red) landscapes ($x$ is periodic). These result in a pumped steady-state current (block arrows), with the typical barrier-crossing trajectories shown at the bottom. Unlike in all the above simulations, thermal bath temperature $T_0={10}^{-4}>0$ was taken in these to smooth out the critical behavior. Straddling the critical point with $K\left(x\right)$ in panel (b) produces a tenfold larger (and reversed) current, even for smaller absolute variation in $K\left(x\right)$.

Figure 5

Different realizations of kicked-rotor trajectories starting from the same initial conditions in the regular and chaotic driven phases, as well as the freely diffusing undriven case (for reference). While the steady state in the chaotic phase looks like large random fluctuations about a constant mean (thus giving a high $T_{\text{eff}}$), in the regular phase, the drive serves to synchronize all trajectories, thus yielding small fluctuations about an oscillating mean in the steady state (and so a low $T_{\text{eff}}$).