# What matters to me, and why?

Students at my college asked every Tuesday. They gathered in a white, windowed room near the center of campus. “We serve,” read advertisements, “soup, bread, and food for thought.” One professor or visitor would discuss human rights, family,  religion, or another pepper in the chili of life.

I joined occasionally. I listened by the window, in the circle of chairs that ringed the speaker. Then I ventured from college into physics.

The questions “What matters to you, and why?” have chased me through physics. I ask experimentalists and theorists, professors and students: Why do you do science? Which papers catch your eye? Why have you devoted to quantum information more years than many spouses devote to marriages?

One physicist answered with another question. Chris Jarzynski works as a professor at the University of Maryland. He studies statistical mechanics—how particles typically act and how often particles act atypically; how materials shine, how gases push back when we compress them, and more.

“How,” Chris asked, “should we quantify precision?”

Chris had in mind nonequilibrium fluctuation theoremsOut-of-equilibrium systems have large-scale properties, like temperature, that change significantly.1 Examples include white-bean soup cooling at a “What matters” lunch. The soup’s temperature drops to room temperature as the system approaches equilibrium.

Nonequilibrium. Tasty, tasty nonequilibrium.

Some out-of-equilibrium systems obey fluctuation theorems. Fluctuation theorems are equations derived in statistical mechanics. Imagine a DNA molecule floating in a watery solution. Water molecules buffet the strand, which twitches. But the strand’s shape doesn’t change much. The DNA is in equilibrium.

You can grab the strand’s ends and stretch them apart. The strand will leave equilibrium as its length changes. Imagine pulling the strand to some predetermined length. You’ll have exerted energy.

How much? The amount will vary if you repeat the experiment. Why? This trial began with the DNA curled this way; that trial began with the DNA curled that way. During this trial, the water batters the molecule more; during that trial, less. These discrepancies block us from predicting how much energy you’ll exert. But suppose you pick a number W. We can form predictions about the probability that you’ll have to exert an amount W of energy.

How do we predict? Using nonequilibrium fluctuation theorems.

Fluctuation theorems matter to me, as Quantum Frontiers regulars know. Why? Because I’ve written enough fluctuation-theorem articles to test even a statistical mechanic’s patience. More seriously, why do fluctuation theorems matter to me?

Fluctuation theorems fill a gap in the theory of statistical mechanics. Fluctuation theorems relate nonequilibrium processes (like the cooling of soup) to equilibrium systems (like room-temperature soup). Physicists can model equilibrium. But we know little about nonequilibrium. Fluctuation theorems bridge from the known (equilibrium) to the unknown (nonequilibrium).

Experiments take place out of equilibrium. (Stretching a DNA molecule changes the molecule’s length.) So we can measure properties of nonequilibrium processes. We can’t directly measure properties of equilibrium processes, which we can’t perform experimentally. But we can measure an equilibrium property indirectly: We perform nonequilibrium experiments, then plug our data into fluctuation theorems.

Which equilibrium property can we infer about? A free-energy difference, denoted by ΔF. Every equilibrated system (every room-temperature soup) has a free energy F. F represents the energy that the system can exert, such as the energy available to stretch a DNA molecule. Imagine subtracting one system’s free energy, F1, from another system’s free energy, F2. The subtraction yields a free-energy difference, ΔF = F2 – F1. We can infer the value of a ΔF from experiments.

How should we evaluate those experiments? Which experiments can we trust, and which need repeating?

Those questions mattered little to me, before I met Chris Jarzynski. Bridging equilibrium with nonequilibrium mattered to me, and bridging theory with experiment. Not experimental nitty-gritty.

I deserved a dunking in white-bean soup.

Suppose you performed infinitely many trials—stretched a DNA molecule infinitely many times. In each trial, you measured the energy exerted. You processed your data, then substituted into a fluctuation theorem. You could infer the exact value of ΔF.

But we can’t perform infinitely many trials. Imprecision mars our inference about ΔF. How does the imprecision relate to the number of trials performed?2

Chris and I adopted an information-theoretic approach. We quantified precision with a parameter $\delta$. Suppose you want to estimate ΔF with some precision. How many trials should you expect to need to perform? We bounded the number $N_\delta$ of trials, using an entropy. The bound tightens an earlier estimate of Chris’s. If you perform $N_\delta$ trials, you can estimate ΔF with a percent error that we estimated. We illustrated our results by modeling a gas.

I’d never appreciated the texture and richness of precision. But richness precision has: A few decimal places distinguish Albert Einstein’s general theory of relativity from Isaac Newton’s 17th-century mechanics. Particle physicists calculate constants of nature to many decimal places. Such a calculation earned a nod on physicist Julian Schwinger’s headstone. Precision serves as the bread and soup of much physics. I’d sniffed the importance of precision, but not tasted it, until questioned by Chris Jarzynski.

The questioning continues. My college has discontinued its “What matters” series. But I ask scientist after scientist—thoughtful human being after thoughtful human being—“What matters to you, and why?” Asking, listening, reading, calculating, and self-regulating sharpen my answers those questions. My answers often squish beneath the bread knife in my cutlery drawer of criticism. Thank goodness that repeating trials can reduce our errors.

1Or large-scale properties that will change. Imagine connecting the ends of a charged battery with a wire. Charge will flow from terminal to terminal, producing a current. You can measure, every minute, how quickly charge is flowing: You can measure how much current is flowing. The current won’t change much, for a while. But the current will die off as the battery nears depletion. A large-scale property (the current) appears constant but will change. Such a capacity to change characterizes nonequilibrium steady states (NESSes). NESSes form our second example of nonequilibrium states. Many-body localization forms a third, quantum example.

2Readers might object that scientists have tools for quantifying imprecision. Why not apply those tools? Because ΔF equals a logarithm, which is nonlinear. Other authors’ proposals appear in references 1-13 of our paper. Charlie Bennett addressed a related problem with his “acceptance ratio.” (Bennett also blogged about evil on Quantum Frontiers last month.)