Space-time symmetry of brownian motors controlled by a dichotomous process
Studying the symmetry of different systems allows one to formulate a variety of conclusions regarding their properties without knowing the detailed information about a particular system. The models of Brownian motors (ratchets) relate the spatial-temporal dependence of the potential energy of Brownian particles and the resulting ratchet effect. In this article, we use the symmetry description of ratchet systems to study the mechanisms of the influence of the spatial and (or) temporal asymmetry of a nanosystem on the average velocity of Brownian motors of the two main types - with fluctuating periodic potential energy and fluctuating force (the so-called pulsating and forced (or tilted) ratchets).
We consider the overdamped motion of a Brownian particle in an unbiased force field, which is described by a periodic stepwise coordinate function, that undergoes dichotomous changes with time. The use of symmetry transformations and symmetry properties of Brownian motors made it possible to obtain compact analytical representations of the average velocity of pulsating and forced ratchets as a function of the parameters of the spatial and temporal asymmetries of the force fields. The fundamental difference between the dependences of the average velocity of the Brownian motors of these two types on the asymmetry parameters is revealed. For the pulsating Brownian motors, the directed motion of nanoparticles is absent in a spatially symmetric system regardless of the presence of temporal asymmetry; the motion appears in the presence of spatial asymmetry, and it is possible to obtain the motor stopping points by tuning the temporal asymmetry parameter of the control process. For the forced Brownian motors, it is the temporal asymmetry that permits the ratchet effect in spatially symmetric systems. For this reason, stopping points as a result of competition of spatial and temporal asymmetries are easier to realize precisely for the motors with fluctuating forces (forced ratchets).
The results presented in this article were obtained by analyzing only general symmetry properties, without involving hidden symmetries of ratchet systems, therefore the results are also valid for inertial dynamics.
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