Steps minimize dissipation in rapidly driven stochastic systems

Abstract

Micro-and nano-scale systems driven by rapid changes in control parameters (control protocols) dissipate significant energy. In the fast-protocol limit, we find that protocols that minimize dissipation at fixed duration are universally given by a two-step process, jumping to and from a point that balances jump size with fast relaxation. Jump protocols could be exploited by molecular machines or thermodynamic computing to improve energetic efficiency, and implemented in nonequilibrium free-energy estimation to improve accuracy.

Limited-control optimal protocols arbitrarily far from equilibrium

“…Figure A.1 gives the corresponding numerical solutions for these cases. Comparing the numerically obtained protocol with the analytic solution(27), which is plotted in lighter-colored solid lines for each t f , we see a very a close match, with typical root-mean-squared error value 10 −5 , never exceeding 1.8 × 10 −3 (we get larger RMS error values for for larger t f , as the time step ∆t ≈ t f /N is generally larger). The closeness between the numerical and analytic solutions suggests a sufficiently fine discretization of space and time.…”

“…(a, b) illustrate optimal protocols for the trap stiffness, across various finite protocol duration values t f . We see that for short times t f 1, the optimal protocol asymptotes to the fast protocol as given in [27], whereas for long times t f 1, the optimal protocol asymptotes to the slow protocol as given in [16]. We observe discontinuous jumps at t = 0 and t = t f in our numerically calculated optimal protocols, which is often the case for optimal protocols [4,45,5].…”

“…We see that the optimal protocol outperforms all other protocols, with the fast and slow protocols asymptoting in performance to the optimal protocol in their respective small-and large-t f limits. The form and performance of these optimal protocols are qualitatively similar to those for the harmonic oscillator control case [4], which we have illustrated in Also plotted are the fast [27] and slow [16] protocols, which have been derived to be optimal for the smalland large-t f limits, respectively. We see that the numerically solved optimal protocol asymptotes to these protocols in the respective t f limits.…”

“…, and generally is not optimal in any regime. The fast protocol, also known as the shorttime efficient protocol (STEP) as developed in [27], is optimal for small-t f limit, and involves a step to an intermediate value λ STEP for the duration of the protocol. The slow protocol first derived in [16], also known as the near-equilibrium protocol, is optimal for large t f , and is obtained by considering the thermodynamic geometry of protocol parameter space induced by the friction tensor ξ(λ), from the linear response of excess work from changes in λ(t).…”

“…Only for the simplest case of a Brownian particle in a harmonic potential has the fully nonequilibrium optimal protocol been analytically solved and studied [4,12,22,23]. For arbitrary potentials, limited control optimal protocol approximations for the slow near-equilibrium t f 1 [16,24,25,26] and the fast t f 1 [27] regimes have been derived, but these approximations generally are optimal only within the specified limits. Very recently, gradient methods have been devised to calculate fully non-equilibrium optimal protocols through sampling many stochastic trajectories [28,29,30].…”