Time-correlation functions of stochastic three-sphere micromachines

We discuss and compare the statistical properties of two stochastic three-sphere micromachines, i.e., odd micromachine and thermal micromachine. We calculate the steady state time-correlation functions for these micromachines and decompose them into symmetric and antisymmetric parts. In both models, the cross-correlation between the two spring extensions has an antisymmetric part, which is a direct consequence of the broken time-reversal symmetry. For the odd micromachine, the antisymmetric part of the correlation function is proportional to the odd elasticity, whereas it is proportional to the temperature difference between the two edge spheres for the thermal micromachine. The entropy production rate and the Green-Kubo relations for the two micromachines are also obtained. Comparing the results of the two models, we argue an effective odd elastic constant of the thermal micromachine. We find that it is proportional to the temperature difference among the spheres, which causes an internal heat flow and leads to directional locomotion in the presence of hydrodynamic interactions.