Shortcuts to state transitions for active matter

Abstract: Shortcut schemes can accelerate quasi-static processes in passive systems by adding auxiliary controls to realize swift transitions between equilibrium states. In active systems, however, inherently directed motion driven by free energy consumption continually drives the system away from equilibrium. In this work, we develop a shortcut framework to realize swift state transitions for active systems operating in the weak activity regime. An auxiliary potential is introduced to guide the system along a predefined distribution path, allowing it to reach the target state within a finite time. Considering unavoidable energy cost in such a finite-time process, we derive a thermodynamic metric from the dissipative work to induce a Riemann manifold on the space spanned by the control parameters. The optimal protocol with minimum dissipative work is then identical to the geodesic path in the geometric space. We demonstrate this framework by considering active systems confined in an external harmonic trap and interacting via two distinct internal potentials, respectively: an attractive harmonic coupling and a repulsive pairwise Gaussian-core coupling. The strengths of both the external trap and the internal interactions are controllable. For the latter case, since the auxiliary potential can not be derived precisely, we adopt a variational method to obtain an approximate auxiliary control. Compared to linear protocols, the geodesic protocols can effectively reduce dissipation.

• The paper develops an analytical framework extending geometric control to active matter by deriving finite-time protocols for state transitions in AOUP systems.
• It demonstrates both exact and variational methods for constructing auxiliary potentials and computing the Riemannian metric that governs dissipative work.
• Results for harmonic and Gaussian-core models highlight optimal control landscapes with significant implications for experimental applications in active systems.

Shortcuts to State Transitions in Active Matter: Analytical and Computational Framework

Overview

This paper develops a formal framework for constructing finite-time protocols that optimally drive active matter systems between specified nonequilibrium distributions. It extends geometric control theory, previously established for passive systems, to the context of active Ornstein-Uhlenbeck particles (AOUPs). The work systematically derives evolution equations for the probability distribution under AOUP dynamics (using the Fox approximation), constructs auxiliary control potentials to realize target state transitions, and demonstrates that the associated mean input work—including dissipation—admits a geometric (Riemannian) form up to second order in active persistence time. The formalism is analytically tractable for certain models and also supports variational-sampling-based implementation for more complex, interacting systems.

AOUP Dynamics and Probability Evolution

The foundational model is a many-body AOUP system governed by

$$\dot{r}_i^\alpha = F_i^\alpha(\mathbf{r}) + \xi_i^\alpha(t) + \eta_i^\alpha(t)$$

where $F$ is a conservative force, $\xi$ is thermal white noise, and $\eta$ is colored noise representing self-propulsion. The authors derive a closed-form evolution (Fokker-Planck-like) equation for the probability distribution $\rho(\mathbf{r}, t)$ based on the Fox approximation, which systematically accounts for the non-Markovian colored noise:

$$\frac{\partial \rho}{\partial t} = -\nabla_i \cdot \mathbf{J}_i$$

with explicit current terms that depend nontrivially on the active noise amplitude and correlation time, including a spatially nonlinear inverse operator $(\mathbf{I} + \tau_p \nabla\nabla U)^{-1}$. This highlights how active forces break detailed balance and induce non-gradient contributions to the probability current.

Geometric Structure of Dissipation

The primary objective is to design "shortcuts"—external driving protocols, possibly through time-dependent parameters $\boldsymbol{\lambda}(t)$—that steer the system between initial and final distributions $\rho_i$ and $\rho_f$ within finite time $F$0. Drawing from stochastic thermodynamics, the mean input (and dissipative) work is expressed in terms of $F$1 and the probability current $F$2, revealing boundary and bulk (dissipative) contributions:

$F$3

Critically, by constraining the dynamics to evolve along a pre-selected path of probability distributions, the dissipative term becomes a quadratic form over the protocol velocities $F$4, with the coefficient being a positive semidefinite Riemannian metric $F$5. Explicit formulas are derived for $F$6 to first and second order in the active persistence time $F$7; the latter introduces higher-order derivative and cross-correlation terms, reflecting active non-Markovianity.

Strong claim: The dissipative cost of optimal state transition protocols for AOUPs is completely characterized, to second order in $F$8, by a computable Riemannian metric on the space of control parameters.

Auxiliary Potentials and Inverse Engineering

To enforce the desired evolutionary path, the paper employs an inverse engineering approach analogous to counterdiabatic driving. For any $F$9 on the path, the required time-dependent "auxiliary" potential is found by solving a partial differential equation (PDE) for $\xi$0.

The first-order (in $\xi$1) case enjoys linearity and admits explicit solutions for simple potentials. For higher-order expansions and for general non-quadratic many-body interactions, the authors recast the PDE as a variational minimization over parametrized functionals that can be efficiently estimated using path sampling and Monte Carlo methods.

Analytical Results for Model Systems

Harmonic Coupling

For particles in a harmonic trap combined with harmonic pairwise interactions (quadratic total potential), the authors derive exact analytical expressions for both the auxiliary potential and the thermodynamic metric. The metric tensor components (including off-diagonal) are computed in closed form as functions of system parameters, number of particles, and activity. The corresponding protocol geodesics—i.e., minimum dissipation paths—are obtained analytically as nonlinear ODEs for the control parameters, with explicit solution formulas.

Key technical result: In the harmonic case, the optimal protocols and cost metric reduce to tractable expressions involving standard linear algebra (traces, inverses) and the Sherman-Morrison formula.

Gaussian-core Pair Interactions

For interacting systems with non-quadratic potentials (e.g., Gaussian-core repulsion), the PDE for the auxiliary potential is not analytically solvable in high dimensions. The authors develop a variational (functional) formulation and employ low-order polynomial ansatzes to reduce the problem to finite-dimensional optimization, which can be efficiently solved by sampling with respect to the reference (target) distribution. This enables practical computation of the thermodynamic metric and geodesic paths even for strongly correlated active systems.

Implication: The framework supports both analytical and scalable numerical implementation, extending optimal protocol construction to the many-body, nonlinear regime.

Theoretical and Practical Implications

The paper extends stochastic geometric control theory to the full AOUP class, explicitly incorporating the effects of activity-induced colored noise up to leading nonlinear corrections. The results reveal that activity renormalizes both the driving cost and optimal protocols in a precise way—by modifying the metric on control space via higher-order, spatially nonlocal correlations. On the practical side, the analytical possibilities for harmonic and quadratic models enable closed-form predictions, while the variational approach facilitates optimal control in complex active matter assemblies, where explicit solutions are out of reach.

Key implications include:

• Nonequilibrium control: Provides a rigorous method to design finite-time protocols that minimize dissipation in active systems, relevant for micromanipulation, optimal transport, and synthetic microswimmer control.
• Thermodynamic geometry: Establishes that the full family of AOUP control problems has a Riemannian geometric structure, justifying the use of path/geodesic optimization for dissipation minimization.
• Role of activity: Demonstrates that active fluctuations fundamentally reshape optimal control landscapes in both analytical and computationally feasible ways.
• Extension to sampling-based algorithms: The variational functional prescription is directly compatible with high-dimensional Monte Carlo—enabling its use for high-$\xi$2 particle systems with arbitrary control landscapes.

Future Directions

Potential avenues for further theoretical development include:

• Incorporating constraints (such as limited control resources) and additional physical effects (e.g., non-conservative forces and hydrodynamic interactions).
• Extending to other non-equilibrium active matter models beyond AOUPs (e.g., run-and-tumble, chiral active fluids).
• Applying the geometric toolbox to design and optimization problems in real experimental active systems, such as synthetic colloidal assemblies and living matter.

Conclusion

The paper provides a technically detailed and comprehensive method for constructing optimal state-transition protocols in active matter, generalizing the geometric framework of passive stochastic control to AOUPs. Both analytical and variational solutions are established, and the practical utility for nontrivial many-body systems is demonstrated. These contributions represent a robust analytical and computational foundation for future exploration and exploitation of nonequilibrium control in active matter systems.