Optimal control in nonequilibrium systems: Dynamic Riemannian geometry of the Ising model

A general understanding of optimal control in nonequilibrium systems would illuminate the operational principles of biological and artificial nanoscale machines. Recent work has shown that a system driven out of equilibrium by a linear response protocol is endowed with a Riemannian metric related to generalized susceptibilities, and that geodesics on this manifold are the nonequilibrium control protocols with the lowest achievable dissipation. While this elegant mathematical framework has inspired numerous studies of exactly solvable systems, no description of the thermodynamic geometry yet exists when the metric cannot be derived analytically. Herein, we numerically construct the dynamic metric of the two-dimensional Ising model in order to study optimal protocols for reversing the net magnetization.

Figure 1

Minimum dissipation, finite-time protocols for reversing the magnetization of the 2D Ising model with initial and final conditions below the critical temperature $t_{\mathrm{C}}\approx 2.269$. The outermost protocol is unconstrained, whereas the inner two protocols have a constraint on the maximum temperature. We control the external field $h$ and the spin-spin coupling constant $J$ as a function of time. Initially, the protocols ramp up the external field followed by a temperature increase as the field is turned off. Low dissipation protocols circumscribe the critical region to avoid large spatial and temporal correlations near the second-order phase transition. The first-order phase transition ($h=0, t<t_{\mathrm{C}}$) is shown as a dashed line ending at the critical point.

Figure 2

(a) Caloric, (b) magnetocaloric, and (c) magnetic friction coefficients of the 2D Ising model, as defined by Eq. (2), plotted in the magnetic field ($h$), temperature $T=1/\beta J$ plane. Both relaxation times and static correlations diverge at the critical point which gives rise to the cusp in each of these plots. The friction coefficients are the matrix elements of a Riemannian metric with the property that geodesics minimize the average excess work that a protocol exercises over the system.

Figure 3

The black arrows show tangents vectors of geodesic paths passing through $h=0, t=4.6$, where the optimal protocol plotted in Fig. 1 crosses the supercritical line. Optimal protocols follow these geodesic flows. Starting below the critical temperature, geodesics flow towards high field and low temperatures before raising the temperature and subsequently reducing the field. Contours show $log\mathrm{Tr}\zeta .$