🔬 condensed matter

Thermodynamics with thermodynamic variable first-passage time. II. Dynamics of explosive boiling of superheated liquid, critical indices, fractional calculus

This paper substantiates the emergence of fractional Caputo derivatives in hydrodynamic equations through stochastic first-passage time thermodynamics and power-law distributions, applying this framework to explosive boiling to derive nonlocal entropy production and a critical dissipation index that characterizes kinetic catastrophes and the violation of Prigogine's minimum entropy production principle near spinodals.

Original authors: V. V. Ryazanov

Published 2026-07-28

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: V. V. Ryazanov

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are watching a pot of water on a stove. Usually, when water boils, it happens gently: tiny bubbles form, rise, and pop. But if you heat water very carefully in a perfectly smooth container, you can push it past its normal boiling point without it bubbling. This is called "superheating." The water is in a precarious state, like a pencil balanced perfectly on its tip. It wants to boil, but it's waiting for a tiny nudge—a random jolt from a molecule—to start the explosion.

For a long time, scientists described this waiting game using simple rules, assuming that every second the water waits is just like the one before it, with no memory of the past. But this paper suggests that reality is much more chaotic and "sticky." It proposes that the liquid remembers every moment it has spent waiting, and that this memory changes how the explosion happens. The authors use a concept called "First-Passage Time," which is just a fancy way of asking: "How long does it take for a random walker to hit a wall?" In this case, the "walker" is a tiny bubble trying to grow, and the "wall" is the point where it becomes too big to stop growing. By treating this waiting time as a living, breathing variable rather than a simple clock, the paper builds a new mathematical framework to predict exactly when and how violently a superheated liquid will explode.

The Story of the Liquid That Forgot How to Wait

This paper is a deep dive into the chaotic world of explosive boiling. It asks a simple but tricky question: When a superheated liquid finally decides to boil, does it happen smoothly, or does it happen like a sudden, catastrophic avalanche? The authors, using a toolkit called Thermodynamics with First-Passage Time (TFPT), argue that the answer is the latter. They suggest that the liquid doesn't just "wait" for a bubble to form; it accumulates a kind of "memory" of its struggle, and this memory makes the final explosion much more violent and unpredictable than old theories predicted.

The Old Way vs. The New Way

Imagine you are trying to push a heavy boulder up a hill. In the old way of thinking (classical thermodynamics), you assume the boulder rolls back down at a steady, predictable rate if you stop pushing. It's like a ball rolling on a smooth track. But the authors say that in a superheated liquid, the "track" is actually a sticky, fractal mess. The liquid has viscoelasticity—it acts like both a liquid and a rubber band. When a tiny bubble tries to grow, the surrounding liquid resists, remembers that resistance, and drags the bubble back.

The paper argues that we need to stop treating the "waiting time" for boiling as a simple number. Instead, it should be treated as a random variable with a "heavy tail." Think of it like this: In a normal game of chance, if you flip a coin, you expect heads or tails eventually. But in this "heavy-tailed" world, you might flip tails a thousand times in a row, and then suddenly, boom, you get heads. The liquid might sit quietly for a long time, and then, in a split second, the whole thing explodes.

The Magic of "Fractional" Math

To describe this sticky, memory-filled behavior, the authors introduce a new kind of math called fractional calculus. You might know that a derivative (like speed) tells you how fast something is changing right now. A fractional derivative is like a "ghost speed." It doesn't just look at the current moment; it looks at the entire history of the object's movement, weighing the past moments differently.

In this paper, the authors show that when you apply this math to boiling, the equations change. Instead of a simple equation saying "heat goes here," you get an equation that says "heat goes here, but it depends on how much the liquid struggled in the last hour, the last day, and every moment in between." This leads to non-locality, meaning the liquid at one spot "feels" the pressure of events happening in its entire past.

The Explosion Index: When Things Go Wrong

The most exciting part of the paper is the calculation of a critical index (let's call it z). This number tells us how violent the explosion will be.

In the old world: If the liquid loses its memory (becomes "Markovian"), the explosion is a mathematical singularity—a sudden, infinite spike that happens instantly. It's like a light switch flipping on.
In the new world: Because the liquid has memory (represented by a number called $\alpha$ , which is between 0 and 1), the explosion is still huge, but it's "smeared" out over time. The authors calculate that the entropy production (the messiness/energy release) doesn't just jump; it follows a power-law curve that shoots up to infinity as the moment of explosion approaches.

They found that this index $z$ is strictly determined by the integral history of the flow. The liquid "feels" the upcoming catastrophe by integrating the intensity of the flow over the entire future interval leading up to the moment of phase explosion. It's as if the liquid is screaming louder and louder the closer it gets to the edge, and the math captures that scream perfectly.

The Pressure Cooker Effect

The paper also explores what happens if you squeeze the liquid with external pressure. Imagine putting a heavy lid on the pot.

What happens: The pressure pushes the "absorbing boundary" (the point where the bubble explodes) further away. It makes the hill the bubble has to climb much steeper.
The result: The liquid's memory gets "frozen." The non-Markovian parameter $\alpha$ drops toward zero. The chaotic, heavy-tailed waiting time turns back into a more predictable, exponential wait.
The takeaway: High pressure smooths out the explosion. Instead of a violent, sudden kinetic catastrophe, the boiling becomes a slow, controlled process. The "singularity" (the infinite spike) is tamed.

Why This Matters

The authors suggest that this framework isn't just for boiling water. It could explain:

Polymer crystallization: How long chains of plastic molecules untangle and snap into place.
Nanofluidics: How ions move through tiny channels in our bodies or in computer chips.
Quantum systems: How long a particle stays in a specific state before jumping.

The paper concludes that by treating the "lifetime" of a state as a fundamental thermodynamic variable, we can finally describe systems that are stuck in a "metastable" state—waiting, waiting, and then suddenly changing. It challenges the old idea that systems always try to minimize their energy production (Prigogine's principle). Near the edge of an explosion, the paper shows that systems actually maximize their chaos in a very specific, mathematically predictable way.

In short, this paper tells us that when a superheated liquid is about to boil, it's not just waiting; it's remembering every second of its struggle, and that memory is what makes the final explosion so spectacularly violent. The math they've built doesn't just describe the explosion; it predicts exactly how the liquid's history shapes its future.

Technical Summary: Thermodynamics with Thermodynamic Variable First-Passage Time. II. Dynamics of Explosive Boiling of Superheated Liquid, Critical Indices, Fractional Calculus

Problem Statement
The paper addresses the limitations of classical thermodynamics and standard kinetic theories (such as the Frenkel-Zel'dovich nucleation theory) in describing high-intensity, fast-moving processes and low-dimensional systems, particularly near phase transition points and spinodals. Classical approaches, which rely on Markovian assumptions and exponential distributions for state lifetimes, fail to account for strong fluctuations, memory effects (non-Markovian properties), and spatial non-locality observed in metastable media. Specifically, the paper seeks to rigorously describe the dynamics of explosive boiling in superheated liquids, where the "lifetime" of the metastable state becomes a critical, fluctuating variable rather than a fixed parameter.

Methodology
The study employs a synthesized framework combining three theoretical pillars:

Stochastic First-Passage Time (FPT) Thermodynamics (TFPT): Treats the FPT ( $T_{fpt}$ ), or the "lifetime" of a local metastable state, as a fundamental stochastic variable analogous to energy in equilibrium thermodynamics.
Zubarev's Nonequilibrium Statistical Operator (NSO) Method: Used to derive macroscopic balance equations from microscopic Liouville dynamics, incorporating the FPT distribution explicitly into the quasi-equilibrium statistical operator.
Power-Law Distributions and Fractional Calculus: The approach utilizes heavy-tailed distributions (Mittag-Leffler and Lévy) to model the FPT near critical points. This mathematical structure naturally leads to the emergence of fractional Caputo derivatives in hydrodynamic equations, replacing constant phenomenological relaxation times with state functionals dependent on the system's history.

The methodology involves constructing a generalized thermodynamic potential $\Phi(\gamma)$ , where $\gamma$ acts as a thermodynamic force regulating fluctuations. By analyzing the scaling properties of this potential and applying renewal theory, the authors derive non-Markovian transport equations and calculate critical indices for dissipation and lifetime fluctuations.

Key Contributions and Results

Derivation of Fractional Hydrodynamic Equations: The paper substantiates the appearance of fractional Caputo derivatives in hydrodynamic equations. By substituting the Laplace transform of the Mittag-Leffler distribution into the memory kernel structure, the authors derive fractional differential equations of order $\alpha$ ( $0 < \alpha < 1$ ). These equations describe the transport of mass and heat as processes with fractal memory and intermittency, where the fractional operator acts as an integrator of the medium's viscous and thermal resistance history.
Nonlocal Entropy Production Formula: A rigorous integral formula for nonlocal entropy production ( $\sigma$ ) during boiling is derived. The result demonstrates that entropy generation at a specific instant $t$ is strictly determined by the integral history of the macroscopic matter flow $J(t)$ over the entire evolutionary interval up to the moment of phase explosion ( $t_{vzb}$ ). The formula is given by:
$\sigma(t) \propto \frac{J(t)}{\left[ \int_t^{t_{vzb}} J(t') dt' \right]^{1/(1-\alpha)}}$
This highlights the profound temporal nonlocality of dissipative processes in metastable media.
Critical Indices and Kinetic Catastrophe: The paper calculates the critical index $z$ for the intensity of dissipation near spinodals:
$z = \frac{2\alpha - 1 + \alpha}{1 - \alpha}$
This index characterizes the "kinetic catastrophe" of explosive boiling. As the system approaches the spinodal, the non-Markovian parameter $\alpha$ dictates the divergence of entropy production. In the Markov limit ( $\alpha \to 1$ ), the singularity becomes a delta-function type; for $0 < \alpha < 1$ , the divergence is a power-law burst, describing an avalanche-like release of entropy.
Violation of Prigogine's Principle: The study shows that near spinodals, the principle of minimum entropy production is violated. Instead of a smooth steady state, the system exhibits intermittent, singular bursts of entropy production driven by the power-law tails of the FPT distribution.
Effect of External Pressure: The paper analyzes how external pressure ( $P_{ext}$ ) modifies the dynamics. Increasing pressure shifts the absorbing boundary (critical radius) to larger values, exponentially increasing the barrier waiting time. Crucially, pressure reduces the non-Markovian parameter $\alpha$ (towards 0), which increases the denominator of the critical index $z$ . This effectively "smoothes" the singularity, transforming the explosive nature of homogeneous nucleation into a kinetically controlled, smooth phase transformation.
Comparison of Distributions: The authors argue that while the Gamma distribution serves as a mathematical intermediate, the Mittag-Leffler distribution is physically more adequate for describing the incubation period and protracted precritical states of boiling, as it correctly models the transition from exponential decay at short times to power-law tails at long times.

Significance and Claims
The paper claims to provide a rigorous microscopic justification for the use of fractional calculus in continuous media physics, demonstrating that fractional derivatives arise naturally from the non-Markovian structure of Zubarev's NSO method when applied to power-law FPT distributions, rather than being artificial phenomenological devices.

The synthesized approach is presented as a closed scheme that unifies stochastic trajectories with nonlinear transport equations. Its primary significance lies in:

Predictive Power: The ability to calculate critical indices for the lifetime of metastable phases and fluctuation divergences, allowing for the prediction of spinodal points and the intensity of thermal explosions.
Universality: The framework is claimed to be applicable beyond boiling, offering a universal language for describing steady-state macrotransport, noise, and viscoelastic memory in diverse systems such as nanofluidics, polymer crystallization, and charge transport in nanostructures.
Resolution of Non-Closure: The method addresses the non-closure problem of macroscopic balance equations by explicitly including the FPT variable, thereby linking microscopic fluctuations to macroscopic dissipation without heuristic approximations for memory kernels.

The paper concludes that while phase transitions serve as an extreme testing ground for the theory due to visible singularities, the introduction of FPT as a thermodynamic variable is a fundamental tool for describing any open dissipative system with finite spatiotemporal boundaries.