Thermodynamics with thermodynamic variable first-passage time. I. From stochastic trajectories to nonlinear transport equations

Imagine you are watching a crowded dance floor. In the old days of physics, scientists tried to describe the crowd's movement by assuming everyone was dancing in a perfect, predictable rhythm, like a well-rehearsed ballet. They believed that if you knew the music (the temperature) and the floor space (the pressure), you could predict exactly where every dancer would step next. This worked well for slow, calm dances. But what happens when the music speeds up, the lights flash, and the dancers start bumping into each other, panicking, or suddenly deciding to sprint for the exit? The old "ballet" rules break down. The dancers aren't just moving; they are waiting, hesitating, and making random, chaotic choices. This is the world of nonequilibrium thermodynamics—the study of systems that are changing fast, like a superheated liquid about to boil, a metal cooling down too quickly, or a virus trying to escape a cell.

For a long time, scientists had to guess how long these chaotic systems would last before they "broke" or changed state. They used a made-up number called a "relaxation time" to describe how quickly a system settles down, kind of like guessing how long it takes a shaken soda to stop fizzing. But this guesswork felt a bit like cheating. It didn't explain why the soda fizzes the way it does. This paper dives into that mystery. It asks: Can we stop guessing and start measuring the actual "lifetime" of these chaotic states? Can we treat the time it takes for a system to reach a breaking point as a real, measurable thing, just like temperature or pressure? The answer, the authors suggest, is a resounding yes, and it changes how we understand everything from boiling water to the flow of heat in tiny computer chips.

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The Great Escape: Measuring the "Lifetime" of Chaos

Meet V. V. Ryazanov, a physicist who decided to look at the universe not as a smooth, flowing river, but as a collection of frantic runners trying to cross a finish line. In this story, the "runners" are atoms and molecules, and the "finish line" is a point of no return—like a bubble growing so big in hot water that the whole pot explodes into steam.

In the past, scientists tried to describe these runners using a map of the dance floor (phase space). They knew the runners were jittery and random, but they treated the time it took to reach the finish line as a vague, invisible background noise. Ryazanov's paper says: "Stop ignoring the clock!" He proposes a new way of thinking where the First-Passage Time (FPT)—the exact moment a particle first touches that dangerous boundary—becomes a main character in the story. It's not just a side note; it's a full-blown thermodynamic variable, just as real as temperature or pressure.

The Absorbing Wall: A Metaphor for Boiling

Imagine a room full of people (the molecules) wandering around. On one side of the room is a wall made of glass. If a person bumps into it, they don't bounce back; they disappear (this is called an "absorbing boundary"). In a superheated liquid, the "people" are tiny bubbles. Most of the time, they pop and disappear. But sometimes, one bubble gets lucky, grows bigger, and hits the "critical size" (the glass wall). Once it hits, the whole system changes state: the liquid boils.

Ryazanov's big idea is to treat the time it takes for that lucky bubble to hit the wall as a macroscopic coordinate. Think of it like this: if you are driving a car, you usually care about your speed and your location. But what if you also cared about "how much time you have left before you run out of gas"? That "time left" becomes a crucial part of your driving strategy. In this paper, the "time left before the system explodes" (the FPT) is treated exactly like your speed or location. It's a real number you can plug into the equations of physics.

The Magic Formula: From Random Walks to Predictable Rules

Here is where it gets really cool. The paper connects three different ways of looking at the world:

1. Extended Irreversible Thermodynamics (EIT): The "big picture" view that tries to fix the old rules for fast processes.
2. Zubarev's Statistical Operator: A super-complex mathematical tool that tries to track every single particle's history.
3. First-Passage Time (FPT): The "stopwatch" view of how long it takes to hit a boundary.

Ryazanov shows that these three aren't just friends; they are actually the same thing wearing different hats. By using the "stopwatch" (FPT) as a key ingredient in the complex math, the paper derives a new version of a famous equation called the Maxwell-Cattaneo equation.

In the old version, this equation had a "relaxation time" (let's call it $\tau$) that was just a constant number, like a fixed setting on a thermostat. It was a bit of a guess. But in this new version, that constant $\tau$ is replaced by the average First-Passage Time ($\langle T_{fpt} \rangle$).

Why does this matter? Because the average time it takes to hit the wall isn't a constant. It changes depending on how hot the room is, how crowded it is, and how fast the runners are moving. By swapping the constant for this dynamic "time-to-crash," the equations become nonlinear. This means the system's behavior depends on its own current state in a much more realistic way. It's like saying, "The faster you run, the more likely you are to trip, and the more you trip, the slower you get," rather than just saying "You run at a fixed speed."

The "Memory" of the System

One of the most playful discoveries in the paper is about memory. In the old days, scientists thought systems forgot their past almost instantly (like a goldfish). But in these chaotic, fast-moving systems, the past matters a lot. If a particle got stuck in a corner for a long time, it remembers that delay.

The paper uses a branch of math called Renewal Theory (think of it as the math of "waiting for the next bus") to show that the "memory" of the system is directly linked to the distribution of these "waiting times" (the FPT). If the particles are waiting in a pattern that looks like a "heavy tail" (meaning some wait a really long time, like a bus that's 20 minutes late), the system's memory stretches out.

The authors prove that the "memory kernel" (the mathematical term for how much the past affects the present) is exactly determined by the statistics of these first-passage times. It's not a mystery; it's a direct calculation. If you know how long the particles usually wait to hit the wall, you know exactly how the system will remember its past. This explains why some materials (like polymers or glass) act weirdly slow and sticky—they have a "heavy tail" of waiting times.

The "Thermodynamic Force" of Time

The paper also introduces a new "force" called $\gamma$ (gamma). In normal physics, forces push things like gravity pushes an apple. Here, $\gamma$ is a force that pushes the system toward the "finish line" (the boundary). It's a measure of how intense the system's urge is to change state.

By treating the "time to hit the wall" as a variable and $\gamma$ as its partner, the authors create a new kind of Generalized Thermodynamic Potential. Think of this as a new energy map. Just as you can find the height of a hill by looking at a map, you can find the average "time to crash" by looking at this new map. Even better, by looking at the curvature of this map (the second derivative), you can calculate how much the "time to crash" will fluctuate.

This is huge for tiny systems (like nanosystems). In a tiny drop of water, the time it takes to boil isn't a single number; it's a wild guess. One drop might boil in 1 second, the next in 10. The paper shows that these fluctuations are massive—so massive that they can't be ignored. The "spread" of the waiting time is about 82% of the average time! This means the "noise" is almost as loud as the signal. The new theory handles this noise naturally, whereas old theories would just try to smooth it over and miss the point.

What This Means for the Real World

The paper doesn't claim to have solved every problem in the universe. It doesn't say, "We can now predict the exact second a volcano will erupt." Instead, it offers a rigorous framework. It shows that the messy, random waiting times of individual particles can be turned into clean, predictable rules for the whole system.

It rules out the idea that "relaxation time" is just a fixed, empirical number you have to measure in a lab and plug in. Instead, it proves that this time is a dynamic property of the system itself, derived from the microscopic chaos. It also rules out the idea that we need to ignore the "heavy tails" of waiting times in complex materials; the paper shows that these tails are the very source of the system's "memory" and its strange, slow behavior.

In short, Ryazanov's work is like giving a chaotic dance floor a new set of rules. Instead of trying to force the dancers into a ballet, the new rules say: "Let's count how long it takes for the first dancer to hit the wall, and let that time drive the music." It turns the randomness of the crowd into a predictable rhythm, allowing scientists to describe fast, explosive, and chaotic processes with a level of precision that was previously impossible. It bridges the gap between the jittery world of single atoms and the smooth world of flowing liquids, showing that the "stopwatch" is just as important as the "thermometer."