Erik Van Aken in Aeon Magazine: A slight shift in Cleopatra’s beauty, and the Roman Empire unravels. You miss your train, and an unexpected encounter changes the course of your life. A butterfly alights from a tree in Michoacán, triggering a hurricane halfway across the globe. These scenarios exemplify the essence of ‘chaos’, a term scientists coined during the middle of the 20th century, to describe how small events in complex systems can have vast, unpredictable consequences.
Beyond these anecdotes, I want to tell you the story of chaos and answer the question: ‘Can the simple flutter of a butterfly’s wings truly trigger a distant hurricane?’ To uncover the layers of this question, we must first journey into the classical world of Newtonian physics. What we uncover is fascinating – the Universe, from the grand scale of empires to the intimate moments of daily life, operates within a framework where chaos and order are not opposites but intricately connected forces.
In his bestselling book Chaos: Making a New Science (1987), James Gleick observes that 20th-century science will be remembered for three things: relativity, quantum mechanics (QM), and chaos. These theories are distinctive because they shift our understanding of classical physics toward a more complex, mysterious and unpredictable world.
Classical physics, which reached its pinnacle in the work of Isaac Newton, painted a universe ruled by determinism and order. It was a world akin to a perfectly designed machine, where each action, like the fall of a domino, inevitably triggered a predictable effect. This absolute predictability – a world where understanding the present means knowing the future – became the essence of Newtonian mechanics.
lassical physics not only presented an orderly universe among Newton’s followers, but it also instilled a profound sense of mastery over the natural world. Newton’s discoveries fostered the belief that the Universe, previously shrouded in mystery, was now laid bare, sparking an unprecedented optimism in the power of science. Armed with Newton’s laws and revolutionary mathematics, leading thinkers felt they had finally unlocked the secrets of reality.
In this atmosphere of scientific triumph, Alexander Pope, the great poet of the Enlightenment, wrote a fitting epitaph for Newton that captured the monumental impact of his contribution:
Nature and Nature’s laws lay hid in night.
God said, Let Newton be! and all was light.
Not everyone was excited. In his beautiful work Lamia (1820), John Keats poignantly expressed concern over the loss of mystery and wonder in the face of empirical scrutiny:
Do not all charms fly
At the mere touch of cold philosophy?
There was an awful rainbow once in heaven:
We know her woof, her texture; she is given
In the dull catalogue of common things.
Philosophy will clip an Angel’s wings,
Conquer all mysteries by rule and line,
Empty the haunted air, and gnomed mine –
Unweave a rainbow, as it erewhile made
The tender-person’d Lamia melt into a shade.
The ‘cold philosophy’ of classical physics seemed to ‘unweave a rainbow’, stripping the natural world of its enchantment and mystery. Keats resented the process of scientific rationalisation, which could ‘clip an Angel’s wings’ and reduce the world’s wonders to simple entries in ‘the dull catalogue of common things’.
And yet, the 20th century witnessed a dramatic shift with the emergence of relativity, which redefines our understanding of space and time; QM, which revolutionised our understanding of the subatomic world; and chaos theory. The orderly and predictable world of Newtonian physics, the dream of a mechanical universe ready to unveil her innermost workings, was, happily or not, something of an illusion. In the 20th century, science revealed a far more intricate, less predictable and, indeed, chaotic universe.
Like the other two pillars Gleick identified, chaos theory challenges our understanding of classical physics. However, unlike QM and relativity, chaos theory operates within a Newtonian framework – it assumes a deterministic reality governed by specific laws. Yet chaos theory reveals a beguiling level of unpredictability, particularly at a macroscopic level.
The unpredictability revealed by chaos theory, seemingly at odds with a deterministic worldview, arises from the complex nature of nonlinear systems.
In dynamical systems, behaviour changes over time. The concept of determinism implies that future states are precisely determined by current conditions, without any randomness or chance involved. However, when dynamical systems exhibit nonlinearity, their behaviour becomes more complex and less predictable. This complexity arises from a disproportionate relationship between input or cause and output or effect.
Consider a simple faucet. At low pressure, water flows in a smooth, or laminar, pattern. As pressure increases, the flow remains steady but broadens slightly. At one critical point, however, marked by no more than a tiny pressure change, we see a ‘phase transition’ – the orderly flow suddenly becomes turbulent, exemplifying chaos: the sensitivity of nonlinear systems like fluids to minor changes, leading to unpredictable outcomes.
Think about the movement of a small pebble rolling down a mountain slope. Tiny variations in its starting point, uneven terrain, soil density, even wind direction can drastically alter its path and final position. For instance, imagine we drop a pebble at a specific location and it comes to rest in another location. Imagine we run a simple experiment, dropping the pebble one millimetre away from where we dropped it in the first place. If the pebble’s movement is slightly altered by external factors like wind, hitting a patch of highly dense soil or a large rock, its speed could increase dramatically, ultimately stopping in an unexpected location 5,000 mm away from where it landed in the first drop.
A parallel in celestial mechanics is the so-called three-body problem, with three bodies in space like the recent Netflix series. Consider two bodies in space: Earth and the Moon. Newtonian mechanics allows us to predict the orbital motions of these two bodies perfectly. Yet, when we add a third body, the Sun, we discover a level of complexity that defies Newtonian predictability. The gravitational interactions among these three bodies create a dynamic, nonlinear system where slight variations in initial conditions, for example, minor variations in the distances or velocities of any one body, can lead to vastly different outcomes; the long-term positions of the three bodies become practically impossible to predict.
In broader mathematical and scientific terms, ‘chaos’ refers to systems that appear random yet are inherently deterministic. Take the example of a roulette wheel, commonly perceived as a game of chance. While we might assume the outcome is purely random, the underlying mechanics of the roulette wheel, including the motion, friction and the force of the spin, adhere to deterministic physical laws. The true source of unpredictability stems from its extreme sensitivity to initial conditions: how forcefully the ball is dropped, the speed at which the wheel spins, subtle vibrations from environmental factors like an air conditioner, and even the movement of patrons around the table. These factors, often unnoticed, can significantly influence the outcome of each spin. Chaos theory teaches us that even seemingly insignificant variations in initial conditions – a fraction of a millimetre difference in the ball’s drop point – can lead to disproportionately large effects.
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