CORE CONCEPTS Core Concept: Ergodic theory plays a key role in multiple fields Steven Ashley Science Writer

Statistical mechanics is a powerful set of professor Tom Ward, reached a key milestone mathematical tools that uses probability the- in the early 1930s when American mathema- ory to bridge the enormous gap between the tician George D. Birkhoff and Austrian-Hun- unknowable behaviors of individual atoms garian (and later, American) mathematician and molecules and those of large aggregate sys- and physicist John von Neumann separately tems of them—a volume of gas, for example. reconsidered and reformulated Boltzmann’ser- Fundamental to statistical mechanics is godic hypothesis, leading to the pointwise and ergodic theory, which offers a mathematical mean ergodic theories, respectively (see ref. 1). means to study the long-term average behavior These results consider a dynamical sys- of complex systems, such as the behavior of tem—whetheranidealgasorothercomplex molecules in a gas or the interactions of vi- systems—in which some transformation func- brating atoms in a crystal. The landmark con- tion maps the phase state of the system into cepts and methods of ergodic theory continue its state one unit of time later. “Given a mea- to play an important role in statistical mechan- sure-preserving system, a probability space ics, physics, mathematics, and other fields. that is acted on by the transformation in Ergodicity was first introduced by the a way that models physical conservation laws, Austrian physicist Ludwig Boltzmann laid Austrian physicist Ludwig Boltzmann in the what properties might it have?” asks Ward, 1870s, following on the originator of statisti- who is managing editor of the journal Ergodic the foundation for modern-day ergodic the- cal mechanics, physicist James Clark Max- Theory and Dynamical Systems.Themeasure ory. Image courtesy of Goethe University well. Boltzmann coined the word ergodic— on a mathematical space assigns weights to Frankfurt. combining two Greek words: ἔργo¸ν (ergon: parts of the space. “ ” ὁδός “ ” “ ” — “ work )and (odos: path or way ) Measure theory provides a precise formu- of the total phase-space volume; they are ’ to describe his hypothesis. Although some lation, saying that while a system doesn tlit- proportional in size,” Simányi says. It formulations of his hypothesis were not com- erally pass through every point in the phase was the first time that someone had rigor- pletely accurate, his ideas seeded an impor- space, it may come arbitrarily close to every ously proved that a dynamical system that ” “ ’ tant set of principles and tools. point, he explains. That is, there snofa- seemed related to a real physical situation Boltzmann, working with gases, suggested vored part of the space.” This formulation was ergodic. that the spatial average values giving rise to gives rise to a notion of ergodicity weak Ergodic theory continues to have wide macroscopic features also arose as averages enough to hold for many systems and strong impact in statistical physics, number theory, over time of observable quantities that could enough to have significant consequences. probability theory, functional analysis, and be calculated from microscopic states. He After decades of relative quiescence, efforts other fields. One prominent example is the realized that this might hold if a system passed to devise ergodicity models underwent a re- Green–Tao theorem. In 2004, Ben Green of through all of the possible microscopic states. vival in the 1960s, notably including work by the University of Oxford and Terrence Tao This “strong” form of Boltzmann’sergodic Yakov Sinai, a mathematician now at Prince- of University of California, Los Angeles, hypothesis was, however, later seen to be too ton University, Simányi says. In 1963, Sinai building on the work of American-Israeli strong in practice on topological and measure introduced the idea of dynamical billiards, or mathematician Hillel Furstenberg, fused theoretic grounds, says Nándor Simányi, pro- “Sinai Billiards,” in which an idealized “hard” methods from analytic number theory and fessor of mathematics at the University of particle bounces around inside a square ergodic theory to prove a “folklore” conjec- Alabama at Birmingham. “Ergodicity was boundary without loss of energy. Inside the ture dating back to the 1770s that the se- loosely defined. It was an assumption made square is a circular wall, off of which the par- quence of prime numbers contains arbi- about the time-evolution of a dynamical sys- ticle bounces. He proved that for most initial trarily long sequences of numbers in which tem that worked, but the idea that a system trajectories of the ball, this system is ergodic— the difference between consecutive terms is goes through every state, that a system goes that is, over infinite time, the time average of aconstant(forexample,3,5,7issuchase- through every point in the phase space, is an observed quantity will equal its average quence of length three). Tao went on to earn impossible; mathematically, it’snonsense,” over the space. For any part of the space, the 2006 Fields Medal. he says. “It was, however, satisfactory for a the proportion of time the system spends physical understanding of systems.” in that region will be proportional to its size. “ 1 Moore CC (2015) Ergodic theorem, ergodic theory, The path toward greater theoretical rigor, If the system spends 3% of the time in a and statistical mechanics. Proc Natl Acad Sci USA explains University of Warwick mathematics region of its phase space, that region is 3% 112:1907–1911.

1914 | PNAS | February 17, 2015 | vol. 112 | no. 7 www.pnas.org/cgi/doi/10.1073/pnas.1500429112 Downloaded by guest on October 1, 2021