equation of state function

Two black holes, caught in a dance, spiral closer and closer until they merge into one. This dance is not silent: it shakes spacetime itself, sending gravitational waves rippling across the Universe. To understand what we hear in these waves, we need a precise mathematical model of the dance.

Traditionally, machine learning algorithms operate in a static environment, like water flowing along a pre-cut channel without changing its shape. The new method flips this paradigm: it lets an agent simultaneously master actions and deliberately modify the very environment it inhabits.

Photographing hummingbird wings with an ordinary camera gives a blurry smudge. For a sharp picture, you take many rapid shots from different angles and piece them together: that's how a quantum 'snapshot' is made. In the lab, scientists 'feel' the vibrations of a tiny mirror with a laser beam.

Eight hundred participants, firms and workers, enter the market. Each holds a strict preference list in mind. The challenge: pair up so that no one wants a do-over. In mathematics, this is called a game theory with a ban on repeat alliances, and its roots trace back to the work of John von Neumann and Oskar Morgenstern.

A sugar cube dropped into tea dissolves, and soon the entire liquid is uniformly sweet. This path to equilibrium is so familiar that it seems obvious.

In a football stadium, the crowd's emotions arise from the chatter of individual fans. If mood could be both joyful and sorrowful at once, you'd have a quantum superposition. But the murmur of thousands of voices instantly forces that blend into a single definite state.

A pot of water on the stove eventually settles down: the temperature evens out, the liquid stops boiling. Physicists call this thermal equilibrium — a state where energy is distributed in the most probable way.

On the surface of a tiny sphere live two invisible fields, like two kinds of ripples on a soap bubble. These fields don't just oscillate on their own: they constantly influence each other, like musicians playing in unison. In a new study, physicists derived exact equations that describe how this ripple changes over time, heats up, and descends into chaos.

Imagine a spring being compressed with tremendous force. The harder you press, the denser it gets, but at some point the compression stops being uniform — some parts give way more easily, others are stiffer. That's roughly how the matter inside a neutron star behaves — an object born in the flames of a supernova.

Hot tea cools down, and its temperature is the same throughout the cup. But in many systems (for example, in the rarefied gas between stars or in plasma) temperature is not uniform: in one place particles are faster, in another slower. Physicists call such systems "superstatistical": it's as if the tea consisted of many small regions, each with its own temperature.

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.

Water can be ice, liquid, or steam. Physicists call these states phases. In theory, each phase has its own measure — a way to describe how likely it is. But not every measure you can imagine occurs in nature. Some phases are real, while others are just mathematical illusions. Scientists have found a simple criterion that separates reality from mirage.

A pot on a stove receives heat, but not work: the gas under the lid just heats up. But what if the stove is not ordinary, but quantum? Then heat and work can arrive simultaneously, like from an engine that also heats. In a new study, exactly this case is described: a classical system (say, a pendulum or a small mirror) bathes in a quantum environment.

A hot coffee cools by radiating invisible light. To maintain temperature, it must receive exactly as much light back. This silent trade of photons is what we feel as steady warmth.

In every handful of fertile soil, there are bacteria that do something no factory can: they take nitrogen straight from the air and turn it into fertilizer. The secret lies in the enzyme nitrogenase, and its working core, a cluster of iron and molybdenum atoms, looks like a tiny factory floor. Chemists have long tried to understand how this factory works from the inside.

The idea of a expanding universe, proposed by Georges Lemaître, finds an unexpected analogy. Space behaves like a piston in a cylinder: dark energy pushes it, much like steam. The relationship between pressure and volume in this 'engine' is strikingly similar to that of a steam boiler.

Heat flow usually evens everything out: hot things cool down, cold things warm up, and in the end you are left with a uniform warm swamp. The same holds in the quantum world: the environment, whether a crystal lattice or a gas of photons, tends to destroy the fragile entanglement between particles.