Global thermodynamic manifold for conservative control of stochastic systems

TL;DR

This work introduces a global thermodynamic manifold framework with a full-control friction tensor to describe slow-driving dissipation in stochastic systems under conservative control. It shows how partial-control friction tensors arise as inherited metrics on submanifolds, and derives two practical decompositions: a Drazin-inverse operator form and a spectral form, linking dissipation to relaxation modes. The linear-response excess work is expressed either as a sum over relaxation modes or via mode projections onto control directions, yielding design principles that drive dynamics orthogonally to slow modes to reduce dissipation. The approach is demonstrated through three illustrative examples (two-state, harmonic trap, and four-spin systems), where computation of minimum-work protocols is facilitated and the role of relaxation spectra clarified. The results offer scalable pathways for optimizing finite-time protocols in mesoscopic and complex systems, with potential extensions to larger systems and higher-order corrections.

Abstract

Optimal control of stochastic systems plays a central role in nonequilibrium physics, with applications in the study of biological molecular motors and the design of single-molecule experiments. While exact analytic solutions to optimization problems are rare, under slow driving conditions, the problem can be reformulated geometrically solely in terms of equilibrium properties. In this framework, minimum-work protocols are geodesics on a thermodynamic manifold whose metric is a generalized friction tensor. Here, we introduce a new foundation for this friction-tensor formalism for conservatively driven systems. Under complete control of the potential energy, a global thermodynamic manifold (on which points are identified with instantaneous energy landscapes) has as its metric a full-control friction tensor. Arbitrary partial-control friction tensors arise naturally as inherited metrics on submanifolds of this global manifold. Leveraging a simple mathematical relationship between system dynamics and the geometry of the global manifold, we derive new expressions for the friction tensor that offer powerful tools for interpretation and computation of friction tensors and minimum-work protocols. Our results elucidate a connection between relaxation and dissipation in slowly driven systems and suggest optimization heuristics. We demonstrate the utility of these developments in three illustrative examples.

Figures (8)

• Figure 1: Convergence of the exact excess work $\beta\langle\mathcal{W}_{\text{ex}}\rangle$ (solid red curve) to the linear-response approximation $\beta\langle\mathcal{W}_{\text{ex}}\rangle^{(\text{LR})}$ (dashed teal line) as the protocol duration $\tau$ becomes large, for the $(h,J_{\text{NN}}, J_{\text{NNN}}, K, L)$ protocol described in Sec. \ref{['ssec:spinSystems']}. As $\tau \to 0$, the excess work asymptotically approaches the relative entropy $D(\pi_0 \| \pi_{\tau})$ between the initial and final equilibrium distributions blaberStepsMinimizeDissipation2021 (horizontal dashed black line).
• Figure 2: Partial-control manifolds $(\tilde{\mathcal{M}}, \tilde{\zeta})$ (blue) coordinatized by a set of control parameters $\bm{u}$ are submanifolds of the global thermodynamic manifold $(\mathcal{M},\zeta)$, with $\tilde{\zeta}$ inherited from $\zeta$ by Eq. \ref{['eq:inheritedMetric']}. The restriction $V = V(\bm{u})$ of conservative control defines the immersion $\mathcal{F}$\ref{['eq:immersion']}. The pushforward $\mathcal{F}_*$---that maps vectors on $\tilde{\mathcal{M}}$ to vectors on $\mathcal{M}$ (red arrows)---is related to the conjugate forces $\bm{f}$\ref{['eq:conjforcesPushforwardDisc']}.
• Figure 3: Schematic of the geometric condition on the completeness of a set of control parameters $\bm{u}$. A control parameter set $\bm{u}$ is complete, i.e., able to fully determine state energies up to a constant offset when the space $\Delta_f$ of average conjugate forces $\left\langle \bm{f} \right\rangle$ forms an $(n-1)$-simplex in $\mathbb{R}^{n-1}$ (red triangle, right), since this allows a bijective transformation between $\Delta_{\bm{f}}$ and the probability simplex $\Delta_{\bm{p}}$ (green triangle, left).
• Figure 4: Model systems studied in Sec. \ref{['sec:examples']}: (a) two-state system with transition rates $w_{01}$ and $w_{10}$ (left), one instantiation of which is a quantum dot interacting with a metallic reservoir espositoFiniteTimeThermodynamics2010 (right); (b) overdamped diffusion $X_t$ of a particle in a harmonic trap $V(x,t)$ parameterized by tunable stiffness $\kappa(t)$ and center $x_0(t)$; (c) four-spin system with single spin-flip dynamics, giving rise to a state graph that is isomorphic to a hypercube (left). Control-parameter sets are (possibly constrained) collections of $k$-spin couplings ($k = 1,2,3,4$) drawn from 15 parameters of the cluster expansion \ref{['eq:clusterEnergy']}, e.g., the 5-parameter set $(h, J_{\text{NN}}, J_{\text{NNN}}, K, L)$ where $h = h_0=h_1=h_2=h_3$, $J_{\text{NN}}=J_{01}=J_{02}=J_{13}=J_{23}$, $J_{\text{NNN}}=J_{03}=J_{12}$, and $K=K_{012}=K_{013}=K_{023}=K_{123}$ (right).
• Figure 5: (a) LR excess work for LR optimal protocols for the four-spin system computed for 200 different control sets with between $1$ and $15$ control parameters. The protocols featured in Fig. \ref{['fig:threeProtocols']} and Fig. \ref{['fig:eigPlot']} are labeled. (b) Two-dimensional histogram showing the number of independently varying parameters against the number of available control parameters (e.g., 18 of the calculated 5-parameter optimal protocols had only 4 independently varying parameters).