  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.
