Fractal Patterns Offer Clues to the Universe's Origin

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.

Jürgen Berges, a professor of physics at Heidelberg University, is a leader in the effort to understand universality in far-from-equilibrium dynamics. Philip Benjamin
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 ½ and -⅓. 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.”

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 -5/3.

Berges is widely seen as leading the theoretical effort, with a series of seminal papers since 2008 elucidating the physics of universal scaling. His 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.