Ashutosh Jogalekar has an interesting article, The Fermi-Pasta-Ulam-Tsingou problem: A foray into the beautifully simple and the simply beautiful (3 Quarks Daily, Jan 20, 2020) about an important foundational result in non-linear dynamics. From the conclusion:
Fermi’s sense of having made a “little discovery” has to be one of the great understatements of 20th century physics. The results that he, Ulam, Pasta and Tsingou obtained went beyond harmonic systems and the MANIAC. Until then there had been two revolutions in 20th century physics that changed our view of the universe – the theory of relativity and quantum mechanics. The third revolution was quieter and started with French mathematician Henri Poincare who studied non-linear problems at the beginning of the century. It kicked into high gear in the 1960s and 70s but still evolved under the radar, partly because it spanned several different fields and did not have the flashy reputation that the then-popular fields of cosmology and particle physics had. The field went by several names, including “non-linear dynamics”, but the one we are most familiar with is chaos theory.
As James Gleick who gets the credit for popularizing the field in his 1987 book says, “Where chaos begins, classical science stops.” Classical science was the science of pen and pencil and linear systems. Chaos was the science of computers and non-linear systems. Fermi, Ulam, Pasta and Tsingou’s 1955 paper left little reverberations, but in hindsight it is seminal and signals the beginning of studies of chaotic systems in their most essential form. Not only did it bring non-linear physics which also happens to be the physics of real world problems to the forefront, but it signaled a new way of doing science by computer, a paradigm that is the forerunner of modeling and simulation in fields as varied as climatology, ecology, chemistry and nuclear studies. Gleick does not mention the report in his book, and he begins the story of chaos with Edward Lorenz’s famous meteorology experiment in 1963 where Lorenz discovered the basic characteristic of chaotic systems – acute sensitivity to initial conditions. His work led to the iconic figure of the Lorenz attractor where a system seems to hover in a complicated and yet simple way around one or two basins of attraction. But the 1955 Los Alamos work got there first. Fermi and his colleagues certainly demonstrated the pull of physical systems toward certain favored behavior, but the graphs also showed how dramatically the behavior would change if the coefficients for the quadratic and other non-linear terms were changed. The paper is beautiful. It is beautiful because it is simple.
It is also beautiful because it points to another, potentially profound ramification of the universe that could extend from the non-living to the living. The behavior that the system demonstrated was non-ergodic or quasiergodic. In simple terms, an ergodic system is one which visits all its states given enough time. A non-ergodic system is one which will gravitate toward certain states at the expense of others. This was certainly something Fermi and the others observed. Another system that as far as we know is non-ergodic is biological evolution. It is non-ergodic because of historical contingency which plays a crucial role in natural selection. At least on earth, we know that the human species evolved only once, and so did many other species. In fact the world of butterflies, bats, humans and whales bears some eerie resemblances to the chaotic world of pendulums and vibrating strings. Just like these seemingly simple systems, biological systems demonstrate a bewitching mix of the simple and the complex. Evolution seems to descend on the same body plans for instance, fashioning bilateral symmetry and aerodynamic shapes from the same abstract designs, but it does not produce the final product twice. Given enough time, would evolution be ergodic and visit the same state multiple times? We don’t know the answer to this question, and finding life elsewhere in the universe would certainly shed light on the problem, but the Fermi-Pasta-Ulam-Tsingou problem points to the non-ergodic behavior exhibited by complex systems that arise from simple rules. Biological evolution with its own simple rules of random variation, natural selection and neutral drift may well be a Fermi-Pasta-Ulam-Tsingou problem waiting to be unraveled.
I'd like to add some observations:
1. It seems to me that quantum mechanics and relativity are focused on explanatory principles whereas non-linear dynamics tends more toward description, description of a wide variety of phenomena. Moreover quantum mechanics and relativity are most strongly operative in different domains, the microscopic and macroscopic respectively.
2. Back in the 1970s and 1980s Ilya Prigogine observed that living organisms are relatively large objects operating in the macroscopic domain, but the internal processes of individual cells are in touch with the microscopic quantum domain. So life exists in the overlap between those two domains.
3. And then we've got computation. In many cases there are various computational paths from the initial state to the completion of the computation. As a simple example, when adding a group of numbers, the order of the numbers doesn't matter; the sum will be the same in each case. In the case of non-linear systems successive states in the computation 'mirror' successive states in the system being modeled so the temporal evolution of the computation is intrinsic to the model rather than extrinsic.