Length of Time’s Arrow

An unresolved problem in physics is how the thermodynamic arrow of time arises from an underlying time reversible dynamics. We contribute to this issue by developing a measure of time-symmetry breaking, and by using the work fluctuation relations, we determine the time asymmetry of recent single molecule RNA unfolding experiments. We define time asymmetry as the Jensen-Shannon divergence between trajectory probability distributions of an experiment and its time-reversed conjugate. Among other interesting properties, the length of time’s arrow bounds the average dissipation and determines the difficulty of accurately estimating free energy differences in nonequilibrium experiments.

Figure 1

In this Letter, we discuss the definition and measurement of time asymmetry in microscopic systems. (a) As a concrete example, we analyze the time asymmetry of a single molecule experiment in which an RNA molecule is attached between two beads [29]. One bead is captured in an optical laser trap that can measure the applied force on the bead. The other bead is fixed to a piezoelectric actuator. The controllable parameter $\lambda$ is the distance between the fixed bead and the center of the laser trap. For the forward protocol, the RNA hairpin is initially in thermal equilibrium in the folded state with extension $\lambda (a)$. The extension is then increased to $\lambda (b)$, unfolding the RNA. In the conjugate, time-reversed protocol, the RNA is initially in thermal equilibrium in the unfolded state, and the extension is lowered from $\lambda (b)$ back to $\lambda (a)$, allowing the RNA to refold [29, 30, 31, 32]. (b) Histograms of work measurements for folding and unfolding an RNA hairpin at three different rates. Observations are binned into integers centered at $1k_{B}T$ intervals. Note that Eq. (6) predicts that the folding and unfolding work distributions cross at the free energy change.

Figure 2

The squared length of time’s arrow $A$ [Eq. (1)] versus the hysteresis $h$ [Eq. (10)], the dissipation averaged across a conjugate pair of forward and reversed experiments. The experiment is explained in Fig. 1. We equalize the number of data points between conjugate experiments, estimate the free energy from the data, obtain error bars by applying a Bayesian bootstrap [33], and apply a correction for experimental errors, as described in 14. The slower the experiment is performed, the closer to thermodynamic reversibility, the lower the dissipation, and the lower the time asymmetry. The slowest experiments are known to contain the greatest experimental error [14], which may explain the deviation of the slowest data from the linear response trend.