How does a system respond when driven away from thermal equilibrium?

I t is widely appreciated that our understanding of nonequilibrium phenomena has not kept pace with its equilibrium counterpart. In recent years, however, consideration of the above question, posed at the microscopic level of statistical mechanics, has yielded some intriguing theoretical results distinguished by two common features. First, the results remain valid far from equilibrium, that is, even if the system is disturbed violently from its initial equilibrium state. Second, they incorporate information about the history of the system over some span of time; effectively, these results are statistical predictions about what we would see if we could watch a movie of the system filmed at the atomic level, rather than predictions about individual snapshots. To date, this work has been theoretical, although it has been supplemented with numerical simulations. However, in the current issue of PNAS, Hummer and Szabo (1) show how to combine these theoretical advances with single-molecule manipulation experiments, so as to extract useful equilibrium information from nonequilibrium laboratory data. What these authors propose amounts to a distinctive method of deducing the equilibrium mechanical properties of individual molecules. The scenario, roughly, is the following. Imagine a molecule, perhaps a linear polymer, that can be stretched like a tiny rubber band by tugging at one end by using micromanipulation technology such as atomic force microscopy or optical tweezers. Suppose we want to determine the equilibrium tension of this molecule as a function of its elongation or extension at a given temperature. We therefore stretch it, pulling out one end of the molecule at some constant speed while simultaneously measuring the restoring force. From these data, we can construct a plot of force vs. extension, as shown in figure 2 Inset of ref. 1. However, this information might not be what we are after; if we stretch the molecule too fast, we drive it out of equilibrium, resulting in hysteresis. The measured force will then tend to overestimate the equilibrium tension. (The same happens with macroscopic rubber bands, which heat up and become more tense when stretched rapidly.) One solution to this problem is to pull very slowly, allowing the molecule to maintain a gradually changing state of equilibrium. Hummer and Szabo’s surprising alternative (1) involves numerous ‘‘rapid’’ pulling experiments, rather than a single slow pull. The authors provide a prescription for combining the data from these repeated experiments, so that what ultimately emerges is the equilibrium tension as function of elongation, even if the molecule was driven away from equilibrium during the pulling process! Moreover, they make a solid case—by using simulations as well as analysis of published micromanipulation data—that their method is experimentally feasible. Hummer and Szabo (1) anchor their proposal in rigorous analysis, invoking the Feynman–Kac theorem for stochastic processes. Although this analysis is important for anyone wishing to gain a true appreciation of the theoretical ideas behind the work, the essence of those ideas can be conveyed without delving into the technical details of the derivation. In what follows, I will present a cartoon version of the experiments proposed in ref. 1 and use this model to illustrate the underlying principles. Imagine the chain-like, one-dimensional ‘‘molecule’’ shown in Fig. 1. The three beads depict atoms, the springs represent the forces between them. One end of the chain is tethered to a wall (or some other appropriately immovable object), while the other is attached to a handle that we are able to grasp and move at will. The variable l, denoting the distance between the wall and the handle, is viewed as an external parameter. The vector x 5 (x1, x2, x3) denotes the microscopic configuration of the molecule, specifying the position of each atom. Let us furthermore imagine that this molecule is immersed in a thermal environment at temperature T, so that, if we fix the value of l, the molecule will relax to a state of equilibrium. We now adopt a statistical attitude, imagining infinitely many copies or realizations of the system. We will then make statements about the response of this ensemble to a rapid change in l, assuming an initial state of equilibrium. Effectively, we will be discussing the statistics of a collection of microscopic histories, each representing one possible scenario for the evolution of the molecule over a fixed time interval during which the handle is drawn out from one position to another. In equilibrium, with the handle held at some distance lA from the wall, the configurations of the molecule are distributed according to the Boltzmann–Gibbs formula: