Boltzmann equation

Water in a river, flowing around a stone, changes its course: some streams press against the surface, others detach and swirl downstream. A classical computer calculates such flows by dividing the fluid into millions of tiny parcels and tracking each one.

A wave packet is like a traveler who either loses steam and spreads out along the way, holds a steady pace, or gathers into a fist. In a 50-meter flume with water 5 meters deep, scientists tracked how a bell-shaped group of waves behaves when its height is changed but not its width.

When a fluid flows around an obstacle, it spawns vortices and shock waves. Describing their behavior means solving a nonlinear equation in which the flow velocity depends on itself. Classical supercomputers spend days on this, while quantum computers have barely been used: their native language is linear operations, but fluids are inherently nonlinear.

A courtroom interpreter is distracted by every rustle: the creak of a chair, a cough in the back row, the rustle of papers. By the end of the day, they are translating not what the witness said, but what they heard through the noise. Rare translation errors are not the result of a single loud shout, but of the accumulation of many small disturbances.

A cluster of glowing dots drifts through the darkness, now compressing into islands, now spreading out in waves. These are not fish or birds: this is a mathematical model of active matter, where each particle spends its own energy on movement, and also is born, dies, and competes with neighbors for space.

On a steep mountain slope, a single falling rock sometimes triggers an avalanche, and sometimes gets stuck. It all depends on the state of the slope.

A fraction of a second after the big bang, the universe was a searing hot soup of particles and antiparticles. By all rights, they should have annihilated each other completely, leaving only emptiness. But for some unknown reason, one "extra" matter particle survived for every billion pairs. It was from these surviving scraps that galaxies, stars, and people eventually formed.

A fraction of a second after the big bang, the universe was a searing hot soup of particles and antiparticles. By all rights, they should have annihilated each other completely, leaving only emptiness. But for some unknown reason, one "extra" matter particle survived for every billion pairs. It was from these surviving scraps that galaxies, stars, and people eventually formed.

Cosmic ripples begin with tiny clumps. 380,000 years after the Big Bang, the universe was nearly homogeneous — density fluctuations were only one part in 100,000. But gravity turned these weak ripples into galaxies, clusters, and giant voids.

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.

A billiard ball strikes another, and the sum of their velocities before and after the collision obeys strict rules. The same happens in the microscopic world: energy, momentum, and charge cannot simply vanish.

Biological complexity has long eluded us: you can count genes, cells too, but how do you grasp the organization of the proteome itself — a multi-thousand ensemble of proteins woven from carbon, oxygen, and hydrogen?

Textbooks divide particles into two camps: bosons love to crowd into the same state, fermions absolutely refuse. But nature allows a third option. Anyons are quasiparticles whose behavior lies somewhere in between. They can 'remember' how they were exchanged and obey a nonstandard exclusion principle.

You can think of a political system as a vast galaxy — a gravitationally bound system of stars, gas, and dust. In this analogy, parties are the stars, radiating the light of public activity, while their supporters are dark matter, the invisible substance that emits no electromagnetic radiation but whose mass holds the whole structure…

An empty glass on the table seems reliable. But if there's a microcrack in the bottom, one day it will break — not with a crash, but quietly, spreading in a web of lines. So too with the vacuum: it may not be the true vacuum but a false one — temporarily stable, yet ready to transition to a deeper, real state.

Picture a pot of boiling soup, but every grain in it is not just a point but a tiny top spinning around its own axis. That's the world of spin-1 particles. For example, some short-lived mesons born in accelerator collisions.

A lump of sugar drops into hot tea, a spoon stirs a few times, and the sugar disappears. But how exactly does it spread through the churning liquid? Physicists have peered into this process with unprecedented detail: they simulated turbulence on a grid of 8192³ points. If you were to split every millimeter of the cup into a hundred cells, you'd get a similar resolution.

Tea in a cup cools quietly on the table. But if you zoom past at near-light speed, the thermometer will read higher — even though no one heated the tea. Calculations confirm: the needle will creep up. A century ago, physicists split into three camps.

An eternal Universe where the measure of disorder (энтропия) grows is, for most of its time, a homogeneous void. In it, complex structures occasionally assemble by chance, such as fleeting consciousnesses — ‘мозги Больцмана’.

A busy intersection without traffic lights. Cars drive, brake, and change lanes, and from the outside, it looks like complete chaos. But if you watch for a while, you can notice a rhythm: the flow thickens and thins. In physics, this rhythm of disorder is called entropy: it shows how far a system is from calm equilibrium, like water in a boiling kettle.

A coffee whirlpool can span two spots in a cup — a blur of swirling liquid. Quantum particles behave similarly: a spacetime whirlpool where they exist in two places at once. But gravity acts as the spoon that stirs the fabric of space and time.

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.

A stone falls into a pond, and ripples spread across the water. They reflect off the shore, overlap, and die out. All this motion unfolds in space. But what if the pond itself changed its depth or density for an instant — not at one point, but everywhere at once? A temporal boundary is exactly such a jump in properties over time, not space.

If you pour sand into a glass, it settles in a flat layer. But instead of sand, take the tiniest particles of gas: in a gravitational field they don't just fall: each particle races along its own trajectory, collides with neighbors, and all this commotion creates pressure.

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.

A stream of electrons enters plasma like a jet of water hitting a calm pond. The jet disturbs the surface, giving rise to waves that grow quickly and mix the water. In plasma, this process is called the Buneman instability: the electron beam excites oscillations that then scatter its energy.

An ultracold cloud of atoms trapped in laser light can 'breathe': it rhythmically contracts and expands, much like lungs. Physicists have long been able to observe this breathing, but only now have they realized that its amplitude is not just a pretty picture: it is a precise measure of hidden energy jitters inside the cloud.

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.

A neural network learning something new is like a traveler in mountainous terrain. Each piece of learned knowledge is a depression in the landscape into which the traveler rolls and gets stuck. To escape and reach a new basin, energy is needed. But as the traveler gains experience, the walls of their pit become steeper, and each subsequent climb requires more effort.

Water in a cup cools down in minutes. Gas in a room spreads evenly in seconds. But if you trap atoms in a tube as thin as a laser beam and give them a good kick, they will take several seconds to "cool" toward equilibrium — an eternity by atomic standards.

When water boils in a kettle, waves of steam ripple through the kitchen. A similar process happens in space, in plasma — gas so hot that electrons are stripped from atoms. Around black holes and neutron stars, this plasma isn't just hot: its particles zip around at nearly the speed of light.

A black hole shrouded in dark matter is like a whirlpool in invisible water. This substance reveals itself only through gravity.

The lymphatic system is like a city's network of canals, through which water slowly moves, carrying away waste from tissues. But it has no single pump-heart. Instead, along the vessels are placed small station-locks — lymphangions. Each such lock squeezes itself like a fist and pushes fluid onward. How they coordinate their work has long remained a mystery.

In winter, a lake freezes over, and this simple process turns out to be a living model of an expanding universe. Ice thickness grows according to the same mathematical law as the distances between distant galaxies. The secret lies in the movement of water beneath the ice crust.

In the 19th century, James Clerk Maxwell imagined a tiny demon. It knows the speed of every gas molecule and opens a door: fast ones to the left, slow ones to the right. This creates a temperature difference without expending energy, seemingly violating the laws of physics.

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.

A swimmer in a pool of syrup moves differently than in clear water: each stroke is harder, but you can push off more strongly. Roughly the same goes for single-celled organisms — tiny rowers that propel themselves through fluid using flagella.

Bacteria in a drop of water jostle and swim: each on its own, but together they create patterns. With weak motion, the fluid churns like soup on a high flame: vortices collide, and the flow direction changes every second. This is called bacterial turbulence.