The Universal Law That Aims Time’s Arrow

Pour milk in coffee, and the eddies and tendrils of white soon fade to brown. In half an hour, the drink cools to room temperature. Left for days, the liquid evaporates. After centuries, the cup will disintegrate, and billions of years later, the entire planet, sun and solar system will disperse. Throughout the universe, all matter and energy is diffusing out of hot spots like coffee and stars, ultimately destined (after trillions of years) to spread uniformly through space. In other words, the same future awaits coffee and the cosmos.

This gradual spreading of matter and energy, called “thermalization,” aims the arrow of time. But the fact that time’s arrow is irreversible, so that hot coffee cools down but never spontaneously heats up, isn’t written into the underlying laws that govern the motion of the molecules in the coffee. Rather, thermalization is a statistical outcome: The coffee’s heat is far more likely to spread into the air than the cold air molecules are to concentrate energy into the coffee, just as shuffling a new deck of cards randomizes the cards’ order, and repeat shuffles will practically never re-sort them by suit and rank. Once coffee, cup and air reach thermal equilibrium, no more energy flows between them, and no further change occurs. Thus thermal equilibrium on a cosmic scale is dubbed the “heat death of the universe.”

But while it’s easy to see where thermalization leads (to tepid coffee and eventual heat death), it’s less obvious how the process begins. “If you start far from equilibrium, like in the early universe, how does the arrow of time emerge, starting from first principles?” said Jürgen Berges, a theoretical physicist at Heidelberg University in Germany who has studied this problem for more than a decade.

Over the last few years, Berges and a network of colleagues have uncovered a surprising answer. The researchers have discovered simple, so-called “universal” laws governing the initial stages of change in a variety of systems consisting of many particles that are far from thermal equilibrium. Their calculations indicate that these systems — examples include the hottest plasma ever produced on Earth and the coldest gas, and perhaps also the field of energy that theoretically filled the universe in its first split second — begin to evolve in time in a way described by the same handful of universal numbers, no matter what the systems consist of.

The findings suggest that the initial stages of thermalization play out in a way that’s very different from what comes later. In particular, far-from-equilibrium systems exhibit fractal-like behavior, which means they look very much the same at different spatial and temporal scales. Their properties are shifted only by a so-called “scaling exponent” — and scientists are discovering that these exponents are often simple numbers like $latex \frac{1}{2}$ and $latex -\frac{1}{3}$. For example, particles’ speeds at one instant can be rescaled, according to the scaling exponent, to give the distribution of speeds at any time later or earlier. All kinds of quantum systems in various extreme starting conditions seem to fall into this fractal-like pattern, exhibiting universal scaling for a period of time before transitioning to standard thermalization.

“I find this work exciting because it pulls out a unifying principle that we can use to understand large classes of far-from-equilibrium systems,” said Nicole Yunger Halpern, a quantum physicist at Harvard University who is not involved in the work. “These studies offer hope that we can describe even these very messy, complicated systems with simple patterns.”

Berges is widely seen as leading the theoretical effort, with a series of seminal papers since 2008 elucidating the physics of universal scaling. He and a co-author took another step this spring in a paper in Physical Review Letters that explored “prescaling,” the ramp-up to universal scaling. A group led by Thomas Gasenzer of Heidelberg also investigated prescaling in a PRL paper in May, offering a deeper look at the onset of the fractal-like behavior.

Some researchers are now exploring far-from-equilibrium dynamics in the lab, as others dig into the origins of the universal numbers. Experts say universal scaling is also helping to address deep conceptual questions about how quantum systems are able to thermalize at all.

There’s “chaotic progress on various fronts,” said Zoran Hadzibabic of the University of Cambridge. He and his team are studying universal scaling in a hot gas of potassium-39 atoms by suddenly dialing up the atoms’ interaction strength, then letting them evolve.

Energy Cascades

When Berges began studying far-from-equilibrium dynamics, he wanted to understand the extreme conditions at the beginning of the universe when the particles that now populate the cosmos originated.

These conditions would have occurred right after “cosmic inflation” — the explosive expansion of space thought by many cosmologists to have jump-started the Big Bang. Inflation would have blasted away any existing particles, leaving only the uniform energy of space itself: a perfectly smooth, dense, oscillating field of energy known as a “condensate.” Berges modeled this condensate in 2008 with collaborators Alexander Rothkopf and Jonas Schmidt, and they discovered that the first stages of its evolution should have exhibited fractal-like universal scaling. “You find that when this big condensate decayed into the particles that we observe today, that this process can be very elegantly described by a few numbers,” he said.

To understand what this universal scaling phenomenon looks like, consider a vivid historical precursor of the recent discoveries. In 1941, the Russian mathematician Andrey Kolmogorov described the way energy “cascades” through turbulent fluids. When you’re stirring coffee, for instance, you create a vortex on a large spatial scale. Kolmogorov realized that this vortex will spontaneously generate smaller eddies, which spawn still smaller eddies. As you stir the coffee, the energy you inject into the system cascades down the spatial scales into smaller and smaller eddies, with the rate of the transfer of energy described by a universal exponential decay factor of $latex -\frac{5}{3}$, which Kolmogorov deduced from the fluid’s dimensions.

Kolmogorov’s “$latex -\frac{5}{3}$ law” always seemed mysterious, even as it served as a cornerstone of turbulence research. But now physicists have been finding essentially the same cascading, fractal-like universal scaling phenomenon in far-from-equilibrium dynamics. According to Berges, energy cascades probably arise in both contexts because they are the most efficient way to distribute energy across scales. We instinctively know this. “If you want to distribute your sugar in your coffee, you stir it,” Berges said — as opposed to shaking it. “You know that’s the most efficient way to redistribute energy.”

There’s one key difference between the universal scaling phenomenon in far-from-equilibrium systems and the fractal eddies in a turbulent fluid: In the fluid case, Kolmogorov’s law describes energy cascading across spatial dimensions. In the new work, researchers see far-from-equilibrium systems undergoing fractal-like universal scaling across both time and space.

Take the birth of the universe. After cosmic inflation, the hypothetical oscillating, space-filling condensate would have quickly transformed into a dense field of quantum particles all moving with the same characteristic speed. Berges and his colleagues conjecture that these far-from-equilibrium particles then exhibited fractal scaling governed by universal scaling exponents as they began the thermal evolution of the universe.