Similarity of extremely rare nonequilibrium processes to equilibrium processes

For system coupled to heat baths, typical nonequilibrated processes, e.g., induced by varying an external parameter without waiting for equilibration in between, are very different from the corresponding equilibrium infinitely slow processes. Nevertheless, there are connections between equilibrium and nonequilibrated behaviors, e.g., the theorems of Jarzynski and Crooks, which relate the distribution $P\left(W\right)$ of nonequilibrium work to the free energy differences $\mathrm{\Delta }F$. Here we study the naturally arising question, whether those relevant but rare trajectories, which exhibit these work values, show a higher degree of similarity to equilibrium. For convenience, we have chosen a simple model of RNA secondary structures (or single-stranded DNA), here modeling a medium-size hairpin structure, under influence of a varying external force. This allows us to measure the work $W$ during the resulting fast unfolding and refolding processes within Monte Carlo simulations, i.e., in nonequilibrium. Also we sample numerically efficiently directly in exact equilibrium, for comparison. Using a sophisticated large-deviation algorithm, we are able to measure work distributions with high precision down to probabilities as small as ${10}^{-46}$, enabling us to verify the Crooks and Jarzynski theorems. Furthermore, we analyze force-extension curves and the configurations of the secondary structures during unfolding and refolding for typical equilibrium processes and nonequilibrated processes. We find that the nonequilibrated processes where the work values are close to those which are most relevant for applying Crooks and Jarzynski theorems, respectively, but which occur with exponential small probabilities, are most and quite similar to the equilibrium processes.

Figure 1

An example for a RNA secondary structure with one globule and a line indicating the extension $n\left(\mathcal{S}\right)$ of the folded RNA. Circles denote bases, thick black lines links between consecutive bases, and thin blue lines hydrogen bonds between complementary bases.

Figure 2

Exemplary equilibrium secondary structures at $T=1$ for different forces. Top: $f=0$. Middle: $f=0.805$. Bottom: $f=2$. Drawn with the VARNA package [59].

Figure 3

Plain and mirrored work distributions for $T=1$ and 16 sweeps of the forward and reverse process, respectively. They intersect near $W=\mathrm{\Delta }F$, which is the exact value and indicated by the vertical line. The inset shows the same plot but with the distribution for the reverse process (cross symbols) rescaled as $P_{\mathrm{rev}}(-W)exp(-(\mathrm{\Delta }F-W)/T)$, according to the Crooks equation, yielding a good agreement with $P\left(W\right)$.

Figure 4

Plain and mirrored work distributions for $T=0.3$ and 16 sweeps of the forward and reverse process, respectively. They intersect near $W=\mathrm{\Delta }F$, which is the exact value and indicated by the vertical line. The inset shows the same plot but with the distribution for the reverse process (cross symbols) rescaled as $P_{\mathrm{rev}}(-W)exp(-(\mathrm{\Delta }F-W)/T)$, according to the Crooks equation, yielding a good agreement with $P\left(W\right)$.

Figure 5

Jarzynski integrand of the forward process for eight sweeps at $T=1$. Inset: same for the reverse process, in which the maximum is not reached. Error bars are smaller than symbol sizes.

Figure 6

Average nonequilibrated overlap profiles $\overline{\sigma }\left(f\right)$ for some sample processes at $T=1$ and 16 sweeps, with mentioned nonequilibrium work values $W$. The average is performed over 100 equilibrium trajectories while keeping the same nonequilibrium one. The solid line is the average overlap between 2500 pairs of independently sampled equilibrium structures for every force value. Top row: forward process for typical values of $W$ (left) and for $W\approx \mathrm{\Delta }F$ (right). Bottom row: the same for the reverse process. Error bars are smaller than symbol size.

Figure 7

Integrated difference $I^{\sigma }$ between equilibrium and nonequilibrium overlap profiles at $T=1$, for forward (left) and backward (right) processes. For 16 sweeps the data is only partially shown, for better visibility. The horizontal line indicates $I_0$, the value of $I^{\sigma }$ in equilibrium. Vertical lines indicate work values at (from left to right) the maximum $W_{\mathrm{J}}^{*}$ of the Jarzynski integrand, the free energy difference $\mathrm{\Delta }F$, the maximum $W_f^{\mathrm{typ}}$ of the forward process work distribution, the negative free energy difference $-\mathrm{\Delta }F$, and the maximum point $W_r^{\mathrm{typ}}$ of the reverse process work distribution. At eight sweeps there are a total of 90 947 RNA MC runs for the forward and 44 136 for the reverse process. At 16 sweeps there are a total of 50 763 RNA MC runs for the forward and 25 214 for the reverse process.

Figure 8

Top left: mean FECs, in equilibrium as well as in nonequilibrium for typical forward processes and for work values near $\mathrm{\Delta }F$, for two different numbers $n_{\mathrm{MC}}$ of sweeps at $T=1$. The nonequilibrated FECs that were averaged are selected from a work range $\pm 1$ around the specified values yielding at least 443 or more contributing RNA MC runs. Top right: samples of such single FECs in equilibrium. Bottom left: single, i.e., nonaveraged, samples of nonequilibrated FECs with $n_{\mathrm{MC}}=8$ for $W$ near $\mathrm{\Delta }F$. Bottom right: single, i.e., nonaveraged, samples of typical nonequilibrated FECs, i.e., where $W\gg \mathrm{\Delta }F$, with $n_{\mathrm{MC}}=8$. The solid line represents always the mean equilibrium FEC; see Eq. (5).

Figure 9

Top left: mean FECs, in equilibrium as well as in nonequilibrium for typical reverse processes and for work values near $\mathrm{\Delta }F$, for two different numbers of $n_{\mathrm{MC}}$ of sweeps at $T=1$. The nonequilibrated FECs that were averaged are selected from a work range $\pm 1$ around the specified values yielding at least 164 or more contributing RNA MC runs. Top right: samples of such single FECs in equilibrium. Bottom left: samples of nonequilibrated reverse FECs with $n_{\mathrm{MC}}=8$ for $W$ near $\mathrm{\Delta }F$. Bottom right: samples of typical nonequilibrated reverse FECs, i.e., where $W\gg \mathrm{\Delta }F$, with $n_{\mathrm{MC}}=8$. The solid line represents always the mean equilibrium FEC as shown already in Fig. 8.

Figure 10

Integrated extension difference $I^n$ between averaged equilibrium and nonequilibrium at $T=1$. For eight sweeps the entire work range is plotted, where for 16 sweeps only a range around the minimum is shown, for better visibility. $I_0$, represented by a horizontal line, is the averaged value of $I^n$ when comparing equilibrium FECs to the averaged equilibrium FEC. The left curves represent the forward, the right ones the reverse process. Vertical lines indicate work values at (from left to right) the maximum $W_J^{*}$ of the Jarzynski integrand, the free energy difference $\mathrm{\Delta }F$, the maximum $W_f^{\mathrm{typ}}$ of the forward process work distribution, the negative free energy difference $-\mathrm{\Delta }F$, and the maximum point $W_r^{\mathrm{typ}}$ of the reverse process work distribution. At eight sweeps there are a total of 90 947 RNA MC runs for the forward and 44 136 for the reverse process. At 16 sweeps there are a total of 50 763 RNA MC runs for the forward and 25 214 for the reverse process.