Directional fidelity of nanoscale motors and particles is limited by the second law of thermodynamics via a universal equality

Directional motion of nanoscale motors and driven particles in an isothermal environment costs a finite amount of energy despite zero work as decreed by the 2nd law, but quantifying this general limit remains difficult. Here we derive a universal equality linking directional fidelity of an arbitrary nanoscale object to the least possible energy driving it. The fidelity-energy equality depends on the environmental temperature alone; any lower energy would violate the 2nd law in a thought experiment. Real experimental proof for the equality comes from force-induced motion of biological nanomotors by three independent groups for translational as well as rotational motion. Interestingly, the natural self-propelled motion of a biological nanomotor (F1-ATPase) known to have nearly 100% energy efficiency evidently pays the 2nd-law decreed least energy cost for direction production.
Comment: 20 pages, 3 figures