Nonreciprocality of a micromachine driven by a catalytic chemical reaction

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.

Irreversibility of stochastic state transitions in Langevin systems with odd elasticity

Active microscopic objects, such as an enzyme molecule, are modeled by the Langevin system with the odd elasticity, in which energy injection from the substrate to the enzyme is described by the antisymmetric part of the elastic matrix. By applying the Onsager-Machlup integral and large deviation theory to the Langevin system with odd elasticity, we can calculate the cumulant generating function of the irreversibility of the state transition. For an N-component system, we obtain a formal expression of the cumulant generating function and demonstrate that the oddness λ, which quantifies the antisymmetric part of the elastic matrix, leads to higher-order cumulants that do not appear in a passive elastic system. To demonstrate the effect of the oddness under the concrete parameter, we analyze the simplest two-component system and obtain the optimal transition path and cumulant generating function. The cumulants obtained from expansion of the cumulant generating function increase monotonically with the oddness. This implies that the oddness causes the uncertainty of stochastic state transitions.

Thermodynamic Bound on the Asymmetry of Cross-Correlations

The principle of microscopic reversibility says that, in equilibrium, two-time cross-correlations are symmetric under the exchange of observables. Thus, the asymmetry of cross-correlations is a fundamental, measurable, and often-used statistical signature of deviation from equilibrium. Here we find a simple and universal inequality that bounds the magnitude of asymmetry by the cycle affinity, i.e., the strength of thermodynamic driving. Our result applies to a large class of systems and all state observables, and it suggests a fundamental thermodynamic cost for various nonequilibrium functions quantified by the asymmetry. It also provides a powerful tool to infer affinity from measured cross-correlations, in a different and complementary way to the thermodynamic uncertainty relations. As an application, we prove a thermodynamic bound on the coherence of noisy oscillations, which was previously conjectured by Barato and Seifert [Phys. Rev. E 95, 062409 (2017)PRESCM2470-004510.1103/PhysRevE.95.062409]. We also derive a thermodynamic bound on directed information flow in a biochemical signal transduction model.

Time-correlation functions for odd Langevin systems

We investigate the statistical properties of fluctuations in active systems that are governed by non-symmetric responses. Both an underdamped Langevin system with an odd resistance tensor and an overdamped Langevin system with an odd elastic tensor are studied. For a system in thermal equilibrium, the time-correlation functions should satisfy time-reversal symmetry and the anti-symmetric parts of the correlation functions should vanish. For the odd Langevin systems, however, we find that the anti-symmetric parts of the time-correlation functions can exist and that they are proportional to either the odd resistance coefficient or the odd elastic constant. This means that the time-reversal invariance of the correlation functions is broken due to the presence of odd responses in active systems. Using the short-time asymptotic expressions of the time-correlation functions, one can estimate an odd elastic constant of an active material such as an enzyme or a motor protein.

Self-organized swimming with odd elasticity

We theoretically investigate self-oscillating waves of an active material, which were recently introduced as a nonsymmetric part of the elastic moduli, termed odd elasticity. Using Purcell's three-link swimmer model, we reveal that an odd-elastic filament at low Reynolds number can swim in a self-organized manner and that the time-periodic dynamics are characterized by a stable limit cycle generated by elastohydrodynamic interactions. Also, we consider a noisy shape gait and derive a swimming formula for a general elastic material in the Stokes regime with its elasticity modulus being represented by a nonsymmetric matrix, demonstrating that the odd elasticity produces biased net locomotion from random noise.