stochastic processes

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.

As you slowly turn the tuning knob of an old radio, the sound doesn't fade in gradually from the static: it bursts forth in a sudden jump, the moment the antenna hits resonance with the wave. Genes in a living cell behave much the same way.

A light switch clicks, and the light comes on. But imagine a switch that, with the same press, can either turn on the lamp or leave the room dark. It also remembers what it did last time, and sometimes it seems to flip a coin to decide whether to light up. This kind of behavior was found in a crystal of black phosphorus — a semiconductor just a few atomic layers thick.

A snow-covered mountain slope looks like a chaotic jumble. But measure the roughness and you find that the height variations follow strict mathematics. Stochastic modeling shows these random processes aren't entirely random. They produce universal patterns, whether in snow, the flame front of a wildfire, or the growth of a bacterial colony.

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.

Water droplets shoot out of a nozzle one after another, like cyclists on a narrow track. The faster ones catch up with the slower ones and merge into larger drops. This simple mechanism is called ballistic aggregation, and until now it had only been observed on cosmic scales: this is how dust particles stick together to form planets.

A kitchen mixer runs at a single speed, but start jabbing the button and the mixture whips up differently. Chemical reactions behave much the same way when the rates of their individual steps are not constant but pulse over time.

Imagine a cookbook that gets rewritten from generation to generation. One day, the borscht recipe receives a secret ingredient from a pastry chef, and from that moment, the taste remembers not only ancestors but also a neighbor. In the Standard Model of evolution, genes behave the same way.

For a long time, biologists argued: if a herd of animals stably maintains a mix of aggressive and peaceful individuals, does each one alternately change behavior, or do some always fight while others always yield? In classical equations, these options are indistinguishable.

The cells of an organism are like twin chefs sharing a kitchen. They have the same cookbook (DNA), but when copying the recipe to a cheat sheet and assembling the dish, random errors occur. This unpredictability, entropy, means that in identical cells, protein levels are sometimes high, sometimes low.

Evolution flows like water. In a wide river, changes are smooth and slow, while a mountain stream churns with every stone. Population size and mutation rate set the channel: broad and calm, or narrow and rapid-filled.

A desert where every step reshapes the terrain. Dunes settle, new slopes emerge, and the next step is dictated by what you've just trampled. That's roughly how "sandscapes" work — a mathematical model inspired by living systems.

A dance of hundreds of particles: some pairs spin slowly and gracefully, others twitch in a frantic rhythm. To understand the essence of the dance, you need to watch the slow movements: they are the main choreography of the system. In molecular science, such slow movements are called 'slow modes,' and they determine how a protein folds into a ball or how a drug binds to its target.

A candle flame in a draft dances: now it flares up, now it almost goes out. Quasars behave much the same way — supermassive black holes in the centers of distant galaxies, surrounded by a scorching disk of infalling matter. Only they flicker not over seconds, but over months and years.

Many chemical transformations resemble assembling furniture in complete darkness: you see the starting parts and the finished cabinet, but the assembly itself remains a mystery. The intermediate states, like awkward positions of the craftsman's hands, vanish too quickly to be discerned.

Every time you mull over a risky move, whether to invest in stocks or hold off, an inner dialogue runs through your head. Usually, scientists try to reconstruct that dialogue from the final choice alone, as if a detective were reconstructing a conversation between two people from a photo of the meeting spot.

Two musicians play a duet. In a perfect hall, they hear each other perfectly and gradually fall into resonance. But try playing in a room where someone keeps slamming the door. You'd think the rhythm would be thrown off. But no: sometimes that very random slam makes both of them sync up instantly.

The heart beats steadily, then slips into arrhythmia. The climate holds its familiar rhythm for decades, then shifts to a new state over a single summer. Such regime shifts are like a flick of a switch, but in real data we see only a noisy one-dimensional signal in which the moment of the break is hidden.

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.

Speech is like an orchestra where every note must sound at just the right moment. The brain turns a thought into a chain of sounds in a fraction of a second, and until now it was unclear how it manages this hierarchy — from individual sounds to syllables to whole words.

A laser that usually produces a steady beam starts to tremble near its threshold, like a string about to sound. This trembling is not a defect but quantum noise: atoms inside the laser emit light randomly, without any external cause.

A laser beam strikes a gold-and-silver film for just a few nanoseconds — in that time, light travels less than a meter. At the point of impact, the metal boils, splatters, and freezes into a random pattern of tiny particles. Each pattern is as unique as cracks in ice or tree branches.

A film of a shattered cup played backward looks absurd: shards don't spontaneously reassemble. This intuitive sense of irreversibility is tied to the increase of entropy — a measure of disorder in a system. The second law of thermodynamics states that the entropy of an isolated system never decreases, setting time's direction from past to future.

A coin toss seems random, but in reality, physics decides everything: the force of the throw, air resistance, the surface's elasticity. If you knew all the details, you could predict the outcome. True randomness lives deeper — in the quantum world.

A solar panel on a roof produces little current in the morning, a lot at noon, and stays silent at night. Wind doesn't blow randomly either: it's often stronger in the morning and evening than during the day. Energy providers need to know in advance how much energy will be available in an hour or three, so they can switch on backup plants or store electricity in batteries in time.

A material is like a person: if you stretch it for a long time, it remembers and behaves differently. Engineers have spent years fitting formulas to describe such memory. Now a neural network does it on its own. The new architecture, iCKAN, takes test data and converts it into equations.

A tossed coin is the simplest source of randomness. But if the coin is shaved, it starts cheating. In the quantum world, light plays the role of the coin: its properties are fundamentally unpredictable. The catch is that the device that “flips” the quantum coin might be faulty or even swapped by an adversary.

A blunt cylinder flying at six times the speed of sound isn't the most elegant object, but it's the star of a new simulation. The surface of such a vehicle is never perfectly smooth: during atmospheric entry, the heat shield ablates, leaving bumps and pits.

Every thought leaves a trace — a faint electrical blip on the scalp. A device with dozens of wires, resembling a swim cap, catches these traces and turns them into wavy lines. From those lines, you can guess what a person feels even if they stay silent. In a new study, three computer programs competed to see which could read such lines most accurately and determine whether the mood was bad, neutral, or good.

A random fiber laser is a thin glass thread through which light travels, but instead of reflecting off mirrors, it reflects off the tiniest irregularities in the fiber itself. Such a laser doesn't produce a steady beam: its light flickers and flares like a candle flame in a draft.

A group of people searches for a lost item in a large park. Some get tired and leave, others join along the way. The number of searchers is constantly changing. How does this affect the search time? Until now, science had no clear answer. In a new study, a general model of such a process has been built. In it, independent participants randomly enter and leave the search.

In disordered materials, such as organic semiconductors or inks made from two-dimensional materials, electrons don't move smoothly like water through a pipe. Instead, they hop from one spot to another, like a ball bouncing across a rough surface. This kind of motion is called hopping transport.

In 1992, Hurricane Andrew tore through Florida, leaving insurers with $$15.5 billion in losses. Many companies went bankrupt, and the market devised a way to shift disaster risk onto investors: catastrophe bonds. These securities pay generous coupons as long as a specified disaster does not occur, such as a hurricane of a certain strength or an earthquake in a particular region.

Each Atlantic hurricane season is like a game of dice, where the number of spots depends on three major climate forces: El Niño, ocean temperature in the 'hurricane nursery,' and the North Atlantic Oscillation.

In a large concert hall, sound bounces off the walls, and even if you stand in a corner, the echo can reveal the shape of the room. That is essentially the problem the authors tackle: how to recover a system's hidden rhythms from the 'echo' in data collected from only part of the system.

The Galaxy is a forest where birds don't start singing right away. First, trees must grow — stars, around which planets eventually appear. The Drake equation counts how many birds might be singing in this forest right now, but it implicitly assumes that their number doesn't change: as many as there are, that's how many there have always been.

An orchestra plays a complex piece, and each group of instruments enters on its own schedule. The violins might slow down while the woodwinds keep the same tempo, and the percussion might suddenly burst into a roll. The conductor hears everything at once, but pinpointing the exact moment when the flutes, not the cellos, changed rhythm is nearly impossible.

The ice cover of a river looks solid, but beneath it lie cracks that alter the flow. So it is with a time series: behind the smooth surface of numbers hide moments when the rules of the game change. These moments are called change points, and finding them is like tapping the ice to locate weak spots.