Geometry of thermodynamic control

A deeper understanding of nonequilibrium phenomena is needed to reveal the principles governing natural and synthetic molecular machines. Recent work has shown that when a thermodynamic system is driven from equilibrium then, in the linear response regime, the space of controllable parameters has a Riemannian geometry induced by a generalized friction tensor. We exploit this geometric insight to construct closed-form expressions for minimal-dissipation protocols for a particle diffusing in a one-dimensional harmonic potential, where the spring constant, inverse temperature, and trap location are adjusted simultaneously. These optimal protocols are geodesics on the Riemannian manifold and reveal that this simple model has a surprisingly rich geometry. We test these optimal protocols via a numerical implementation of the Fokker-Planck equation and demonstrate that the friction tensor arises naturally from a first-order expansion in temporal derivatives of the control parameters, without appealing directly to linear response theory.

Figure 1

(a) Our model system. A particle (black dot) diffusing in a harmonic potential with adjustable spring constant $k$, position $y_0$, and inverse temperature $\beta =\frac{1}{k_{\mathrm{B}}T}$ (indicated by thermometers). (b) Representative optimal protocols (orange and red curves) plotted for two of the three control parameters, $k$ and $\beta$. An optimal protocol (e.g., red curve) results in the minimum dissipation for any path taking the system from one particular state (black square) to another (black triangle) in a fixed amount of time. (c) A change of variables $\{\beta ,k\}\to \{z,x\}$ [Eq. (34)] reveals that our model system has an underlying structure described by hyperbolic geometry, represented here as the Poincaré half-plane, in which geodesics form half circles (orange curve) or vertical lines (red line). (d) A piece of the $(z,x)$ manifold may be isometrically embedded as a saddle in ${\mathbb{R}}^3$. The distortions in each of these two optimal paths as shown in panels (b) and (c) reflect the curvature of this manifold.

Figure 2

Optimal protocols differ substantially from linear interpolation (red dashed lines). Blue solid curves represent geodesics of the inverse diffusion tensor and are thus optimal protocols for transitioning the system from one state to another in a fixed amount of time. Blue dots indicate points separated by equal times along each of the eight optimal paths shown.

Figure 3

Geodesics describe protocols that outperform naive straight line paths in parameter space. A geodesic between two fixed points in the $(\beta ,k)$ plane (black) and several comparison protocols are pictured above. The comparison protocols were generated via a linear interpolation between the constant speed straight line (pink) and the geodesic. The tick marks represent points separated by equal times. The solid pink dots correspond to the constant speed parametrization of the line, whereas the open red circles correspond to the optimal parametrization along this straight path. The ratio of excess work to that of the geodesic protocol is: $6.12$ (pink circle), $4.37$ (red open circle), $3.67$ (cyan downward-pointing triangle), $1.38$ (orange upward-pointing triangle), $1.00$ [black star (geodesic)], $2.86$ (magenta square), and $6.29$ (green circle). These ratios are plotted in the inset figure along with a graph of the ratio as a function of the interpolating parameter (light gray curve). All excess work values were calculated using the Fokker-Planck system Eq. (56). Here, $A={10}^{-2},B={10}^{-3},M=1$, placing the system within the near-equilibrium regime and ensuring accuracy of the inverse diffusion tensor approximation.