initial conditions

In a rubber band stretched in the lab, three different elasticity mechanisms operate simultaneously. At small strains, Hooke's law prevails; at intermediate strains, entropic chain elasticity; at large strains, strain-induced crystallization.

Imagine a ball in an unfamiliar hall: two guests start from opposite corners. Their movements are not a smooth waltz but chaotic improvisation in search of rare treasures. Contrary to intuition, the outcome hinges not on tempo but on the geometry of the floor.

A space tug with an ion engine accelerates slowly, but it sips fuel. To get from one orbit to another, you have to solve a tricky problem: how long to fire the engine and where to point it. Usually, the goal is to find the fastest transfer for a given spacecraft mass.

In the scorching heart of a dying star, an invisible stream is born — a neutrino gas. The particles race through matter almost without stopping, but when their density is enormous, they start to feel each other. Old calculations described this as a jostle in a crowd: each shove changes direction, but no connection arises between passersby.

The universe is expanding, and its history features two puzzling episodes that don't fit into a simple picture. The first is a brief period in its youth when everything flew apart faster than expected, as if someone had floored the gas pedal. The second is today's acceleration, as if the same pedal were pressed again, but more gently.

A time crystal resembles an eternal clock: its atoms spontaneously perform rhythmic movements. In ordinary clocks, a pendulum is driven by a spring or battery, but here the motion arises without an energy supply — the system simply returns to its initial state at regular intervals. This idea was proposed by Frank Wilczek in 2012.

If you scatter seeds haphazardly, the seedlings will tangle into chaos. But if you lay them out in neat rows, the field will grow in an orderly fashion. A similar principle works with bacteria. Colonies of elongated cells usually churn with disordered flows, but scientists have found a way to impose order from the very start.

In a pot of boiling water, steam bubbles rise and pop. In the quantum world of a superfluid, bubbles behave differently: they can collide and give birth to vortices — tiny swirling flows, like whirlpools in a river. Scientists modeled such a liquid using a holographic approach.

Magnetic fields on the surface of the Sun appear as dark spots and bright faculae. But they are born deep inside, where flows of hot gas act like a giant dynamo. The question is whether sunspots on the surface can predict what will happen inside the star years from now. The authors applied a mathematical theorem similar to the rule linking the area of a shape to its boundary.

At the Large Hadron Collider, not only nuclei but also individual protons collide. Even in such small systems, collective behavior resembling the flow of a liquid is observed. But the proton is a quantum object; its state is described by a wave function, not by temperature and pressure.

An amoeba turns left. For a long time, this was considered the result of a rigid chain of causes, like dominoes. But at the core of matter, there's a gap for freedom. The 'Agent Choice via Quantum Flux' model shows: each decision corresponds not to one, but to many quantum histories.

Toss a handful of sand on the floor: there are few large grains, many medium ones, and even more tiny ones. This skew, where small events are common and large ones are rare, is described by a power law. It shows up in city sizes, earthquake magnitudes, and citation counts of scientific papers.

In everyday life, time flows on its own, like a river past its banks. But in quantum gravity, things are different: time does not flow; it emerges from relations between objects, like a pattern in a carpet that is only visible when viewed from a certain spot.

A thin stream of water on a shower glass gathers into winding rivulets that quiver and shift before your eyes. Such a liquid film flowing down a vertical surface is a classic example of chaos in nature: it all flows downward, sure, but the surface lives its own unpredictable life.

Intelligent action begins with a plan. Just as an architect needs blueprints to evaluate options, any agent with will needs to repeatedly copy data about reality.

An orchestra without a conductor plays a complex piece. The listener hears the overall music but wants to understand who leads the melody and who merely echoes it in unison.

In the Pacific Ocean, every few years the water off the coast of Peru warms up, and fishermen call it El Niño. Sometimes the warming is mild, sometimes so strong that it changes weather across the entire planet. Forecasting a strong El Niño six months to a year ahead is a task at the edge of possibility: weather is forgotten within two weeks, while climate remembers for decades.