How nanomechanical systems can minimize dissipation

Information processing machines at the nanoscales are unavoidably affected by thermal fluctuations. Efficient design requires understanding how nanomachines can operate at minimal energy dissipation. Here we focus on mechanical systems controlled by smoothly varying potential forces. We show that optimal control equations come about in a natural way if the energy cost to manipulate the potential is taken into account. When such a cost becomes negligible, an optimal control strategy can be constructed by transparent geometrical methods which recover the solution of optimal mass transport equations in the overdamped limit. Our equations are equivalent to hierarchies of kinetic equations of a form well known in the theory of dilute gases. From our results, optimal strategies for energy efficient nanosystems may be devised by established techniques from kinetic theory.

Figure 1

Means for Gaussian boundary conditions with parameters $\tau =1$, $m=1$, $\beta =1$, $t_{\mathrm{f}}=1$, $k=1$. The initial and final means of $\mathbf{x}\equiv [\mathbf{q},\mathbf{p}]$ are $(0,0)$ and $(\sqrt{2},0)$, respectively. The parameter $g$ varies logarithmically from $1.28\times {10}^{-1}$ (green/light) to $1.25\times {10}^{-4}$ (blue/dark). The arrows indicate the behavior for decreasing $g$.

Figure 2

Covariances for Gaussian boundary conditions with parameters as in Fig. 1. The initial and final covariance matrices of $\mathbf{x}\equiv [\mathbf{q},\mathbf{p}]$ are given by $\left(\begin{array}{cc}1\hfill & 0\\ 0\hfill & 1\end{array}\right)$ and $\left(\begin{array}{cc}1.7\hfill & 0\\ 0\hfill & 1\end{array}\right)$, respectively.