Active Matter under Control: Insights from Response Theory

Active constituents burn fuel to sustain individual motion, giving rise to collective effects that are not seen in systems at thermal equilibrium, such as phase separation with purely repulsive interactions. There is a great potential in harnessing the striking phenomenology of active matter to build novel controllable and responsive materials that surpass passive ones. Yet, we currently lack a systematic roadmap to predict the protocols driving active systems between different states in a way that is thermodynamically optimal. Equilibrium thermodynamics is an inadequate foundation to this end, due to the dissipation rate arising from the constant fuel consumption in active matter. Here, we derive and implement a versatile framework for the thermodynamic control of active matter. Combining recent developments in stochastic thermodynamics and response theory, our approach shows how to find the optimal control for either continuous- or discrete-state active systems operating out of equilibrium. Our results open the door to designing novel active materials that are not only built to stabilize specific nonequilibrium collective states but are also optimized to switch between different states at minimum dissipation.

Popular Summary

Active matter is a class of physical systems where each individual unit constantly converts a given source of energy into sustained dynamics, resulting in collective behavior that is very different from systems at thermal equilibrium. Efficient control of active systems opens the door to machines and materials that can perform novel functions. But their out-of-equilibrium nature presents major difficulties to existing control frameworks that are built for equilibrium systems. Here, we present a systematic optimal control framework that takes into account the out-of-equilibrium nature of active matter.

Our new framework leverages recent developments in stochastic thermodynamics and response theory, coupled with a standard calculus of variations, to build a generic recipe that finds the optimal protocol for driving a continuous-state or discrete-state active system between two different states. The framework requires computing only steady-state averages and response functions, based on correlation functions in the unperturbed state, for which we provide explicit formulas.

We apply our framework to one- and many-body scenarios, revealing a general feature arising from the control of active systems: In contrast to systems at thermal equilibrium, where the protocol achieving the least dissipation is always the slowest one, the optimal protocol has instead a finite duration that achieves the best trade-off between the dissipation stemming from the external perturbation and that coming from internal activity.

Our control framework reveals new insights into the thermodynamics of active systems and lays down a road map for the optimal control of a broad array of active systems, taking one step closer to active-matter technology.

Figure 1

Illustration of the heat dissipated for a control protocol with external perturbation in finite time. In passive systems, the protocol achieving the least dissipation is always the slowest one, as expected from standard thermodynamics. In active systems, the optimal protocol instead has a finite duration $t_p^{*}$, which achieves the best trade-off between the dissipation stemming from the external perturbation and that coming from internal activity.

Figure 2

Optimal thermodynamic control of an active particle in a one-dimensional trap. (a) Illustration of an active particle (blue) in a harmonic trap. The protocol consists in driving the trap strength from ${\alpha }_0$ to ${\alpha }_1$. (b) Control speed $\dot{\alpha }$ [Eq. (14)] against trap strength $\alpha$ for different $t_p=\{0.1\tau ,1\tau ,5\tau ,10\tau ,20\tau ,100\tau \}$. (c) Protocols for $t_p=\{0.1\tau ,\dots ,100\tau \}$. (d) Comparing the scaled heat, where $T=D/\mu$, against protocol duration as approximated in the response framework (blue) [Eq. (8)] and the exact expression (orange) [Eq. (43)]. For both curves, we use the protocols shown in (c). Parameters: $\mu =1$, $D=1$, $D_1=0.01$, $\tau =1$, ${\alpha }_0=1$, and ${\alpha }_1=5$.

Figure 3

Optimal protocol duration $t_p^{*}$ as a function of the active noise amplitude $D_1$, as predicted by the scaling relation Eq. (17) in conjunction with Eq. (46). Parameters: ${\alpha }_0/{\alpha }_1=1/5$, $\mu =\tau =D=1$.

Figure 4

Simulation snapshots of the repulsive active Brownian particles showing two distinct phases determined by the particle size $\alpha$. (a) Small size of particles ($\alpha v/D=2\times {10}^3$) resulting in a homogeneous phase (HOM). (b) Larger size of particles ($\alpha v/D=4\times {10}^3$) leading to MIPS.

Figure 5

Optimal thermodynamic control of a many-body system consisting of repulsive active Brownian particles in the HOM (left) and MIPS (right) cases. (a),(b) Derived protocols for driving the particle size $\alpha$ as a function of time. (c),(d) Control speed $\dot{\alpha }$ against control parameter $\alpha$. (e),(f) Scaled heat against protocol duration, computed using protocols shown in (a),(b).