Optimized finite-time work protocols for the Higgs RNA model with external force

The Higgs RNA model with an added term for a coupling to an external force is studied in regard to finite-time force-driving protocols with a minimal-work requirement. In this paper, RNA sequences which at low temperature exhibit hairpins are considered, which are often cited as typical template systems in stochastic thermodynamics. The optimized work protocols for this glassy many-particle system are determined numerically using the parallel tempering method. The protocols show distinct jumps at the beginning and end, which have been observed for single-particle systems and are proven to be optimal in the fast protocol limit generally. Optimality seems to be achieved by staying close to the equilibrium unfolding transition point, in agreement with experimental and theoretical observations. The change of work distributions, compared to those resulting from a naive linear driving protocol, are discussed generally and in terms of free energy estimation as well as the effect of optimized protocols on rare work process starting conditions.

Figure 1

Optimized protocols $\lambda$ as a function of time $t$, which display distinct jumps at the beginning and end, for three sequences (curves with symbols). The linear protocol (blue full line) and the constant excess power protocols (broken lines without symbols) are also shown for reference. The arrow on the vertical axis indicates the critical value ${\lambda }_c$ of the folding-unfolding transition. Left: $T=0.3$; right: $T=1.0$.

Figure 2

Protocols $\lambda$ as a function of mean-free length $\overline{n}$. The equilibrium curves are also shown for reference. Top: Optimized protocols. The initial jumps are close to the equilibrium unfolding transition point. Bottom: Linear protocols for comparison. Left: $T=0.3$; right: $T=1.0$.

Figure 3

Mean system observables as a function of time $t$ for $T=0.3$. The trajectories for the optimized protocols have an almost linear behavior. Left: mean-free length $\overline{n}$; right: mean number of paired bases $\overline{\left|\mathcal{S}\right|}$.

Figure 4

Work distributions for the linear and optimized protocols. Top: hairpin sequence; middle: continuous sequence; bottom: asymmetric sequence. ${10}^5$ work processes are used to estimate the histograms, except for the optimized protocol at $T=0.3$ for the hairpin sequence, where $4\times {10}^5$ processes are generated. For reference, the work distribution for the constant excess power protocol is shown in the asymmetric sequence plots. Lines are a guide to the eye only.

Figure 5

Shade-coded work distributions conditioned to the initial free length $n\left({\mathcal{S}}_0\right)$ at $T=0.3$. A darker shade corresponds to a higher probability. For each initial free length 5500 work processes are sampled. Left: linear protocol; right: optimized protocol. Top: hairpin sequence; middle: continuous sequence; bottom: asymmetric sequence.