Least-rattling feedback from strong time-scale separation

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.

Low rattling: A predictive principle for self-organization in active collectives

“…While the covariance matrix reflects the amplitude of local fluctuations, in estimating effective diffusivity we are instead interested in a measure of their disorder. This follows from the observation that high-amplitude ordered oscillations do not contribute to the rate of stochastic diffusion (10). We suggest that the degree of disorder of fluctuations may be captured by the entropy of the distribution of ṽq vectors, which is how we define "rattling" R(q).…”

Driven probe under harmonic confinement in a colloidal bath

“…This introduces memory effects in the probe dynamics, with an explicit dependence on the microscopic details of the surrounding bath particles. At variance with some previous works [35][36][37], our approach does not assume a slow relaxation of the tracer dynamics compared to the bath. Our approach relies on the fact that the probe is linearly coupled to the density fluctuations φ k , which are Gaussian fields with first order dynamics.…”

Systematic model reduction captures the dynamics of extrinsic noise in biochemical subnetworks

“…Note that the reduced dynamics ( 18) is valid for any noise level ε, i.e., all the intrinsic noise terms are still present in contrast to the projected equation ( 13) that relates to conditional averages over the intrinsic fluctuations. The overall effective noise in the 3GA on the subnetwork is given by χ(t) in (18). Similar to the random force, we can split this as χ(t) = χ 0 (t) + χ 1 (t), where χ 0 (t) is the noise of the linearized dynamics (12), while χ 1 (t) stems from the nonlinear corrections; hence, it contains δx s -dependent terms as follows:…”

“…The first intermediate approximation we explore is "linear noise," where in the 3GA-reduced dynamics (18), we keep the nonlinear memory [i.e., the integrals containing M ss (t−t ′ ) and M ss,s 3GA (t, t ′ , t ′′ )] but we replace the nonlinear (hence multiplicative) and colored effective noise χ(t) = χ 0 (t) + χ 1 (t) with its linear (hence additive), but still time-correlated, part χ 0 (t) from (12). We refer to this approximation as 3GA-χ 0 .…”

“…Importantly, our results do not rely on any timescale separation, while projection methods and more recently path integrals have so far been applied primarily to coarsegraining by elimination of fast bulk variables. [16][17][18][19] As a second novel contribution, we propose and analyze approximations of the extrinsic noise terms that help us evaluate them more expediently and elucidate the relative contribution of different noise properties such as color, non-linearity, and multiplicativity of the source.…”