🔬 condensed matter

Cooling rate and glassy behavior in the Fermi--Pasta--Ulam system

This study numerically demonstrates that a Fermi--Pasta--Ulam system cooled at a rate $\xi$ falls out of equilibrium below a stochastic threshold, retaining a residual energy $E_0$ at zero temperature that scales as $(\xi N)^{2/3}$ with system size $N$ and cooling rate $\xi$ .

Original authors: Matteo Razza, Andrea Carati, Luigi Galgani

Published 2026-07-16

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CC BY 4.0

Original authors: Matteo Razza, Andrea Carati, Luigi Galgani

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

The Great Freeze: When Things Get Too Cold to Settle Down

Imagine you are making a batch of homemade ice cream. If you stir it slowly and let it cool down gently, the cream molecules have plenty of time to find their perfect, orderly spots, turning into a smooth, solid block of ice. But if you throw that same cream into a super-cold freezer and blast it with a fan, the molecules get scared and confused. They try to settle down, but they move so fast that they crash into each other before they can find a spot. The result isn't a perfect crystal; it's a "glassy" solid—a frozen mess that looks solid but is actually stuck in a chaotic, disordered state. This is the story of how things freeze, and it's a huge mystery in the world of physics.

Scientists have long wondered why some materials get stuck in this "glassy" state instead of becoming perfect crystals. The usual suspect is that as things get colder, the time it takes for the atoms to rearrange themselves and find order gets longer and longer—so long that the cooling process finishes before the atoms can ever settle. To study this without needing a lab full of expensive chemicals, physicists use a famous mathematical toy called the Fermi–Pasta–Ulam (FPU) system. Think of this system as a long line of beads connected by springs. It's the simplest possible model of a solid crystal. By watching how these beads jiggle and interact, scientists can test the rules of how energy moves and how things cool down. The big question is: if you cool this system down as fast as possible, does it ever truly reach a state of zero energy, or does it get stuck with some leftover energy that refuses to leave?

The Paper's Discovery: The "Ghost" Energy That Won't Let Go

In this study, the researchers decided to play a game of "chase the cold" with their line of beads. They set up a simulation where their FPU system (the beads on springs) was in contact with a "gas" (a bunch of other particles bouncing around). They then started cooling the gas down, effectively pulling the heat out of the system. They did this at different speeds (cooling rates) and with different numbers of beads (system sizes) to see what would happen.

What they found was quite surprising. As they cooled the system down, they expected the energy of the beads to drop right along with the temperature, eventually reaching zero when the temperature hit zero. However, they discovered a "stochastic threshold"—a kind of invisible speed limit for chaos. Below a certain temperature, the beads stopped behaving like a normal, calm system in equilibrium. Instead, they got stuck. Even when the temperature of the surrounding gas was lowered all the way to absolute zero, the FPU system didn't stop moving. It kept a tiny, stubborn amount of energy that refused to be removed. The authors call this leftover energy the "zero-point energy" ( $E_0$ ). It's as if the beads are shivering in the cold, not because they are warm, but because they are trapped in a disordered state and can't find the exit door to stop moving.

The most exciting part of their discovery is how this leftover energy behaves. The researchers ran thousands of computer simulations and found that this "ghost" energy isn't random; it follows a very specific pattern. The amount of leftover energy depends on two things: how fast they cooled the system down (the cooling rate, $\xi$ ) and how many beads were in the line (the number of particles, $N$ ).

Through their data, they suggest that the leftover energy scales according to a power law. Specifically, the energy $E_0$ seems to be proportional to the cooling rate and the number of particles multiplied together, raised to the power of two-thirds. In their own words, the relationship looks like this:
$E_0 \sim (\xi N)^{2/3}$

This means that if you have a huge system (a large $N$ ) and you cool it down quickly (a large $\xi$ ), you are guaranteed to be left with a significant amount of this trapped energy. Even if you slow down the cooling, if the system is big enough, that energy won't disappear. The authors note that for this leftover energy to vanish completely in a real, macroscopic world, the cooling rate would have to be impossibly slow—faster than $1/N$ .

The paper doesn't claim to have solved the mystery of why this happens or to have found a new law of nature that applies to every material in the universe. Instead, the authors suggest that this behavior is a strong hint that even in simple models, the transition to a glassy state prevents systems from reaching true thermal equilibrium. They point out that while the math is clear in their simulations, the reason why the exponents for the cooling rate and the system size are so similar (both close to $2/3$ ) is still a puzzle. They conclude that if this behavior holds true for real-world solids interacting with gases, it could mean that any real-world cooling process we try to perform will always leave behind a non-negligible amount of energy, fundamentally changing how we understand the thermodynamics of cold objects.

Technical Summary: Cooling Rate and Glassy Behavior in the Fermi–Pasta–Ulam System

Problem Statement
This work investigates the cooling dynamics of a Fermi–Pasta–Ulam (FPU) system, a prototypical model for a solid, when placed in thermal contact with an ideal gas whose temperature is gradually reduced. The study is motivated by the phenomenon observed in real liquids where rapid cooling prevents thermalization, leading to a supercooled liquid state and eventually a glassy solid. The authors hypothesize that this lack of thermalization in complex systems can be understood through the theory of perturbations of Hamiltonian systems in the thermodynamic limit. Specifically, they aim to determine if the FPU system, known for its diverging thermalization times at low specific energies, exhibits a "glassy" behavior where it falls out of equilibrium during cooling, retaining a residual energy even as the temperature approaches zero.

Methodology
The authors employ numerical simulations of a coupled Hamiltonian system consisting of:

The FPU System: A linear lattice of $N+2$ particles with fixed endpoints, interacting via a Lennard-Jones potential. The system is initialized in a non-equilibrium state and allowed to equilibrate.
The Cooling Bath: A one-dimensional ideal gas of $N+2$ non-interacting particles confined to a segment, interacting with the FPU system via a short-range interaction potential.

The cooling process is simulated through a sequence of discrete steps. In each step, the velocities of the gas particles are scaled down by a factor $\eta = \sqrt{T_{new}/T_{old}}$ to lower the target temperature, followed by a re-equilibration period. Two cooling protocols are tested: a linear reduction of temperature above a threshold and an exponential reduction below it.

To handle statistical fluctuations inherent in finite-size systems, the authors perform $M=128$ independent cooling trajectories. They utilize two averaging techniques: a "moving average" to visualize continuous trends and a "bin-averaging" procedure to generate statistically independent data points for quantitative analysis. The simulations vary the system size ( $N$ ) and the cooling rate ( $\xi$ ), defined as the ratio of temperature step to equilibration time ( $\Delta T / \Delta t$ ).

Key Results
The numerical simulations reveal two primary findings regarding the cooling process:

Departure from Equilibrium and Zero-Point Energy:
A distinct "stochastic threshold" is identified. Below a specific temperature (approximately $T < 0.035$ in the units used), the FPU system ceases to follow the equilibrium relation where specific energy $u$ equals temperature $T$ . Instead, the system's specific energy remains higher than the bath temperature. As $T \to 0$ , the system retains a finite residual energy, termed the "zero-point energy" ( $E_0$ ). This suggests the system enters a non-equilibrium state analogous to a glass, where latent heat (or energy) is not fully released to the thermostat.
Scaling Laws for Residual Energy:
The magnitude of the residual energy $E_0$ depends on both the system size ( $N$ ) and the cooling rate ( $\xi$ ).
- System Size Dependence: For a fixed cooling rate, $E_0$ increases with the number of particles, following a power law $E_0 \sim N^{0.61}$ .
- Cooling Rate Dependence: For a fixed system size, $E_0$ increases with the cooling rate, following a power law $E_0 \sim \xi^{0.66}$ .
- Combined Scaling: The authors propose that these dependencies combine into a single scaling law:
  $E_0 \sim (\xi N)^{2/3}$
  This implies that in the thermodynamic limit ( $N \to \infty$ ), the cooling rate $\xi$ must vanish faster than $1/N$ for the residual energy to disappear.

Significance and Claims
The paper concludes that even in a simple model like the FPU system, the details of the cooling process are non-trivial. The existence of a residual energy $E_0$ that persists at vanishing temperatures indicates that the system fails to thermalize completely under finite cooling rates, a behavior characteristic of glassy systems.

The authors modestly note that while the phenomenon is observed, the physical mechanism behind the specific exponent ( $2/3$ ) and the similarity between the dependence on $N$ and $\xi$ remains unexplained analytically. However, they suggest that if this behavior persists in more realistic models of solid-gas interactions, it could have implications for macroscopic experiments. Specifically, for macroscopic bodies where $N$ is large, the required cooling rate to achieve true equilibrium (vanishing $E_0$ ) might be so small that any practical cooling process would inevitably result in a non-negligible residual energy, thereby affecting the thermodynamic behavior of the cooled body.

The Great Freeze: When Things Get Too Cold to Settle Down

The Paper's Discovery: The "Ghost" Energy That Won't Let Go

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