Self-Organized Resonance during Search of a Diverse Chemical Space

Recent studies of active matter have stimulated interest in the driven self-assembly of complex structures. Phenomenological modeling of particular examples has yielded insight, but general thermodynamic principles unifying the rich diversity of behaviors observed have been elusive. Here, we study the stochastic search of a toy chemical space by a collection of reacting Brownian particles subject to periodic forcing. We observe the emergence of an adaptive resonance in the system matched to the drive frequency, and show that the increased work absorption by these resonant structures is key to their stabilization. Our findings are consistent with a recently proposed thermodynamic mechanism for far-from-equilibrium self-organization.

Figure 1

(a) Twenty particles experience viscous drag as they move in one dimension. The only interaction between any two particles is a bonding reaction that follows two-state Arrhenius kinetics, as shown in (b), whereby a bond forms or breaks at a Poissonian rate determined by the distance $\mathrm{\Delta }x$ between them. A bonding energy $ϵ$ controls the tendency to form bonds at random. (c) In the absence of any drive, the particles tend to form a random graph at equilibrium (left). When a single particle (yellow) is subjected to an oscillatory external force, vibrations in the network change the rates of formation and breakage of certain bonds, and new structures are sampled in a nonequilibrium steady state. A color gradient superimposed on the bonds indicates minimal connectivity distance from the driven particle (with reddest being the closest). Particles are depicted with sizes proportional to their number of bonded neighbors in the graph. Produced using the ForceAtlas2 algorithm in Gephi.

Figure 2

(a) The normal mode spectrum $\mathcal{P}(\omega )$ is sampled for an equilibrium ensemble of spring networks (blue) and exhibits a stereotypical haystack shape. When driven at frequency ${\omega }_d=1.2$, the normal mode spectrum rearranges (red) in two noticeable respects: First, the average number of springs decreases; indeed, an equilibrium normal mode spectrum computed for a value of $ϵ$ that leads to the same average number of bonds (orange) has a very similar shape. However, in the nonequilibrium spectrum, there is an additional peak at the chosen drive frequency. (b) The drive induces a peak in the nonequilibrium normal mode spectrum at any chosen drive frequency within the haystack region of the spectrum. (c) The average forcing magnitude distributed to normal modes of a given frequency is plotted for different drive frequencies. The diagonal line indicates that modes with greater resonant response to the drive frequency tend to be more strongly forced. Eigendirection signs for the normal mode basis have been chosen so that the corresponding driving force components $F_0^{(i)}$ are all positive.

Figure 3

(a) Scatter plot of rates of dissipation $\dot{Q}$ for spring networks at adjacent iteration steps along a driven Markov microtrajectory of the network graph. States with atypically high values of $\dot{Q}$ tend to jump to states that retain this property, indicating that this subset of spring networks is kinetically stabilized by driving. (b) Mean dissipation rate $\dot{Q}$ over time of the ensemble of random graphs, initialized in the equilibrium and driven with an oscillatory drive. Across a wide range of drive frequencies ${\omega }_d$, the dissipation rate $\dot{Q}$ increases monotonically from its value in the initial equilibrium distribution (c) Distributions of stochastic entropy $\mathrm{\Delta }S=\ln (r_{\mathrm{fwd}}/r_{\mathrm{rev}})$ for transitions in the driven steady state. Markov transitions in spring network connectivity that lead to an increase in $\dot{Q}$ (red) show a significantly greater tendency toward positive $\mathrm{\Delta }S$ than those that lead to a decrease, and are thus more irreversible. (d) Distributions of transient work $\mathrm{\Delta }W$ for transitions in the driven steady state. This quantity measures the extra work performed by the drive during dissipative relaxation after formation or breakage of a bond. Transitions that lead to an increase in $\dot{Q}$ typically absorb and dissipate more work from the drive relative to those that lead to a decrease in $\dot{Q}$.