Time-correlation functions of stochastic three-sphere micromachines.

Non-equilibrium statistical mechanics provides the foundational framework for understanding how microscopic systems convert energy into directed motion within fluctuating environments. These systems frequently operate where thermal noise is significant, necessitating a stochastic approach to their internal dynamics. Prior research has shown that the violation of the fluctuation-dissipation theorem is a hallmark of active matter and non-equilibrium engines. In the context of microscale robotics, researchers have developed various models to simulate the behavior of small-scale swimmers in viscous fluids. These models often utilize simplified geometries, such as three spheres connected by adjustable linkages, to study the emergence of locomotion. While the role of non-conservative forces is well-documented, the specific statistical signatures that distinguish different driving mechanisms remain poorly understood. This absence of evidence motivated a detailed comparative analysis of the time-correlation functions in odd and thermal micromachine models.

This investigation evaluates the steady-state statistical properties of two distinct stochastic three-sphere micromachines to identify their underlying physical differences. The investigation focuses on comparing the odd micromachine, driven by non-conservative elastic forces, with the thermal micromachine, driven by temperature gradients. By calculating the time-correlation functions of the internal degrees of freedom, the researchers aim to quantify the extent of time-reversal symmetry breaking. The analysis seeks to decompose these correlation functions into symmetric and antisymmetric components to reveal the influence of odd elasticity. Another objective is the derivation of the entropy production rate, which serves as a measure of the system's distance from equilibrium. The study also aims to establish Green-Kubo relations that connect microscopic fluctuations to macroscopic transport properties. Finally, the researchers intend to demonstrate how internal heat flow facilitates directional locomotion when hydrodynamic interactions are present.

The researchers utilize a theoretical model of a three-sphere micromachine where spheres are linked by harmonic springs in a linear or triangular configuration. To analyze the dynamics, the team formulates Langevin equations that describe the stochastic evolution of the spring extensions under the influence of thermal noise. The mathematical framework specifically addresses the steady-state regime where the system's statistical properties are time-independent. For the odd micromachine, the researchers introduce an odd elasticity matrix that represents non-conservative interactions between the springs. In contrast, the thermal micromachine model incorporates distinct local temperatures for each sphere, creating a non-isothermal environment. The team performs a rigorous decomposition of the resulting time-correlation functions into symmetric and antisymmetric parts to isolate the effects of the driving forces. Analytical expressions for the entropy production rate are derived by considering the log-ratio of path probabilities under time-reversal.

The analysis reveals that the cross-correlation between the two spring extensions possesses a non-zero antisymmetric part in both micromachine models. This antisymmetric component serves as a direct mathematical indicator of the broken time-reversal symmetry within the stochastic system. In the case of the odd micromachine, the researchers found that this antisymmetric part is strictly proportional to the value of the odd elastic constant. For the thermal micromachine, the antisymmetric correlation scales linearly with the temperature difference maintained between the two edge spheres. The study successfully obtained the Green-Kubo relations, providing a link between the steady-state fluctuations and the response of the micromachines. A significant finding is the derivation of an effective odd elastic constant for the thermal model, which depends on the internal temperature gradient. This effective elasticity drives an internal heat flow that is essential for achieving directional locomotion in the presence of hydrodynamic interactions.

The results establish a clear correspondence between the non-conservative forces of odd elasticity and the thermal gradients found in non-isothermal systems. These findings imply that thermal micromachines can be effectively described using the language of odd mechanics, simplifying the analysis of complex heat-driven engines. The study confirms that the presence of an antisymmetric correlation part is a universal feature of micromachines that break time-reversal symmetry. By quantifying the entropy production rate, the researchers provide a thermodynamic basis for evaluating the efficiency of microscale locomotion. The identified mechanism for directional movement highlights the importance of coupling internal heat flow with hydrodynamic interactions in viscous fluids. These insights offer a robust theoretical foundation for the development of autonomous synthetic swimmers that operate in fluctuating environments. The researchers suggest that the Green-Kubo relations derived here will be instrumental in future efforts to engineer micromachines with optimized transport properties.