Simulation of extremely rare ultra-fast non-equilibrium processes close to equilibrium

Work theorems like Crooks equation allow to obtain equilibrium quantities from non-equilibrium processes. A standard setup is starting a system in contact to a heat bath in equilibrium, then executing a process by changing some external parameter which leads to performing work, and ending in some non-equilibrium configuration. The distribution P(W) of the work allows to extract the free energy difference \Delta F between equilibrium starting state and the imaginary final equilibrium state, which would be obtained if one waited long enough after the process has been finished. The region of P(W) which is most relevant to obtain \Delta F is where  W is about  \Delta F.  Nevertheless, if the investigated system is not too small, P(W) will be tiny, like 10^{-15} or smaller (decreasing with increasing system size) being located in the rare-event tail. Thus, when studying such processes by simulation, one needs to use large-deviation approaches applied to the dynamic evolution of the respective model.

Here we will investigate the question how similar  non-equilibrium processes are to the equilibrium ones beyond comparing a scalar number like the work. Still, we study this question as a function of the measured work W.  For that purpose we investigate numerically the unfolding and refolding of RNA secondary structures under influence of an external force f. Fortunately, for this model the equilibrium behavior can be accessed exactly by dynamic programming algorithms allowing to sample equilibrium unfolding and folding processes. We compare between equilibrium and non-equilibrium dynamics by means of force-extension curves n(f) and overlap profiles \sigma(f). Our results indicate that indeed the extreme low-probability trajectories which exhibit W near \Delta F, and thus contribute most to the determination of \Delta F via Crooks equation, are most similar to the equilibrium trajectories.