Active matter represents a class of nonequilibrium systems that constantly dissipate energy to produce directed motion. Controlling active matter to achieve a target state holds great potential for advancements in synthetic molecular motors, targeted drug delivery, and adaptive smart materials. However, the inherently nonequilibrium nature of active matter poses a significant challenge in achieving optimal control with minimal energy cost. In this work, we extend the concept of thermodynamic geometry, originally developed to provide geometric representations of energy cost in passive systems, to active systems. We propose a systematic geometric framework for minimizing energy cost in active matter with interparticle interactions. Specifically, we derive a cost metric that defines a Riemannian manifold for control parameters, enabling the application of powerful geometric tools to design optimal control protocols. The geometric perspective reveals that, unlike in passive systems, minimizing energy cost in active systems entails a universal trade-off scaling relation, leading to an optimal transportation speed in the geometric space that intriguingly coincides with the self-propulsion speed of an active Brownian particle. This insight enriches the broader concept of thermodynamic geometry. Furthermore, the derived scaling relation suggests an optimal protocol duration that aligns with the general expectation proposed by Davis et al. [Phys. Rev. X 14, 011012 (2024)] for active matter. We illustrate the utility of this approach by optimizing the performance of an active monothermal engine within the geometric framework.

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• Nonequilibrium & irreversible thermodynamics
• Stochastic thermodynamics
• Heat engines
• Fokker–Planck equation