Optimal control of overdamped systems

Nonequilibrium physics encompasses a broad range of natural and synthetic small-scale systems. Optimizing transitions of such systems will be crucial for the development of nanoscale technologies and may reveal the physical principles underlying biological processes at the molecular level. Recent work has demonstrated that when a thermodynamic system is driven away from equilibrium then the space of controllable parameters has a Riemannian geometry induced by a generalized inverse diffusion tensor. We derive a simple, compact expression for the inverse diffusion tensor that depends solely on equilibrium information for a broad class of potentials. We use this formula to compute the minimal dissipation for two model systems relevant to small-scale information processing and biological molecular motors. In the first model, we optimally erase a single classical bit of information modeled by an overdamped particle in a smooth double-well potential. In the second model, we find the minimal dissipation of a simple molecular motor model coupled to an optical trap. In both models, we find that the minimal dissipation for the optimal protocol of duration $\tau$ is proportional to $1/\tau$, as expected, though the dissipation for the erasure model takes a different form than what we found previously for a similar system.

Figure 1

Continuous erasure protocol. The left-hand well of the double-well potential merges with the right and the central barrier lowers simultaneously as $\lambda$ decreases from 2 to 0.

Figure 2

Dependence of the leading order term of ${\langle ▵s^{\text{tot}}\rangle }_{{\mathbf{\Lambda }}_{\text{opt}}}$ on $\alpha$ for the erasure model given by Eq. (49). Note the saturation at 2 as $\alpha$ becomes very large.

Figure 3

Dependence of minimal dissipation for ratchet model on (scaled) tilt $f$ where $\alpha =1,\epsilon =0.1$, and $y_0\left(0\right)=0$ and $y_0\left(\tau \right)=2$. Note that the minimal dissipation is a periodic function of $f$.