Self-organization is frequently observed in active collectives as varied as ant rafts and molecular motor assemblies. General principles describing self-organization away from equilibrium have been challenging to identify. We offer a unifying framework that models the behavior of complex systems as largely random while capturing their configuration-dependent response to external forcing. This allows derivation of a Boltzmann-like principle for understanding and manipulating driven self-organization. We validate our predictions experimentally, with the use of shape-changing robotic active matter, and outline a methodology for controlling collective behavior. Our findings highlight how emergent order depends sensitively on the matching between external patterns of forcing and internal dynamical response properties, pointing toward future approaches for the design and control of active particle mixtures and metamaterials.

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.

Complex systems with many degrees of freedom are typically intractable, but some of their behaviors may admit simpler effective descriptions. The question of when such effective descriptions are possible remains open. The paradigmatic approach where such "emergent simplicity" can be understood in detail is the renormalization group (RG). Here, we show that for general systems, without the self-similarity symmetry required by the RG construction, the RG flow of model parameters is replaced by a more general flow of the Fisher Information Metric on the model manifold. We demonstrate that the systems traditionally studied with RG comprise special cases where this metric flow can be induced by a parameter flow, keeping the global geometry of the model-manifold fixed. In general, however, the geometry may deform, and metric flow cannot be reduced to a parameter flow -though this could be achieved at the cost of augmenting the manifold by one new parameter, as we discuss. We hope that our framework can clarify how ideas from RG may apply in a broader class of complex systems.