🔬 condensed matter

Optimal preparation and reachable-state constraints in the Mpemba effect

This paper investigates the Mpemba effect in granular fluids by formulating state preparation as an optimal-control problem, revealing that while the optimal protocol is a one-bang strategy, the stochastic thermostat imposes reachable-state constraints that bound non-Gaussianities and limit the maximum attainable effect.

Original authors: C. Ríos-Monje, A. Patrón-Castro, C. A. Plata, B. Sánchez-Rey, A. Prados

Published 2026-07-15

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: C. Ríos-Monje, A. Patrón-Castro, C. A. Plata, B. Sánchez-Rey, A. Prados

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 have two identical pots of soup. One is boiling hot, and the other is just warm. Common sense tells you the warm soup should cool down faster, right? But sometimes, in a weird twist of physics, the boiling pot actually chills down to the fridge temperature before the warm one does. This counterintuitive trick is called the Mpemba effect.

For a long time, scientists looked at this effect like a magician pulling a rabbit out of a hat. They would just say, "Okay, let's assume the hot soup starts with a secret, weird ingredient that makes it cool faster," and then they'd calculate the result. They picked these "secret ingredients" (specifically, a number called excess kurtosis, which measures how "spiky" or non-standard the speed of the particles in the soup is) out of thin air, without asking how you could actually make the soup have that ingredient in the first place.

This paper says: "Wait a minute. We can't just wave a magic wand. We need to know the recipe."

The Race to the Finish Line

The authors decided to treat the preparation of the soup like a high-stakes driving race. Imagine you are driving a car (the soup) and you want to reach a specific destination (the final cold state) as fast as possible. But you aren't just driving; you are the engineer and the driver. You get to control the gas pedal (how much energy you pump into the system) and the brakes (how much you let it cool down) to set the car up for the perfect start.

The system they studied is a granular fluid—think of it as a giant box of billions of tiny, bouncy steel balls that lose a little bit of energy every time they crash into each other. To keep them moving, you shake the box (a "stochastic thermostat"). The goal was to figure out: What is the absolute best way to shake the box to create the most extreme "spiky" speed distribution, so that when we stop shaking, the balls cool down as fast as physics allows?

The "One-Bang" Discovery

The scientists used a powerful mathematical tool called Pontryagin's Maximum Principle (think of it as the ultimate GPS for finding the fastest route in a complex maze) to solve this.

They found something surprisingly simple. The best way to prepare the system isn't to gently ease the gas pedal up and down, or to drive in a fancy zig-zag pattern. The optimal strategy is a "one-bang" protocol.

In everyday terms, this means you either floor it (shake the box as hard as physically possible, $\chi_{max}$ ) or you cut the engine completely (stop shaking it, $\chi_{min}$ ). You don't do anything in between.

If the balls are "bouncy" enough (a property called the restitution coefficient $\alpha$ is low, meaning they lose a lot of energy when they hit), you should shake them as hard as you can.
If they are "springy" (high $\alpha$ , they bounce back well), you should stop shaking them entirely.

There is no middle ground. The math proves that any attempt to be "moderate" or "gradual" is a waste of time if you want to reach the extreme limits of this effect.

The Invisible Wall

Here is the most important part of the story, and the part that rules out a lot of previous ideas.

The authors discovered that there is an invisible wall in the physics of this system. No matter how hard you shake the box, or how clever your "one-bang" strategy is, you cannot create just any kind of weird speed distribution.

The "spikiness" of the particles (the excess kurtosis) is strictly bounded.

The Lower Limit: You can't make the distribution less spiky than a perfect, smooth bell curve (a Gaussian state, where the spikiness is 0).
The Upper Limit: You can't make it more spiky than the state the balls naturally fall into when they are just cooling down on their own without any shaking (called the Homogeneous Cooling State, or HCS).

The paper explicitly argues against the idea that you can just pick any initial condition to get the Mpemba effect. You are trapped inside a specific "playground" defined by the laws of the system. The strongest Mpemba effect you can ever achieve is limited by how far you can push the system to the edge of this playground.

What They Actually Did

The authors didn't just guess this. They did two things to be sure:

Math: They solved the equations using the "one-bang" rule and found the exact boundaries.
Simulation: They ran a massive computer simulation with $10^6$ (one million) particles. They tested different shaking strengths, from a gentle nudge ( $\chi_{min} = 0.001$ ) to a violent shake ( $\chi_{max} = 1000$ ).

The results from the computer matched the math perfectly. The simulations showed that as they made the shaking stronger or weaker, the "spikiness" of the particles got closer and closer to the theoretical limits (0 and the HCS value), but never broke through them.

The Takeaway

So, what does this mean for the Mpemba effect?
It means the effect isn't just a magic trick you can summon by picking random numbers. It is a race against the rules of the game. The "strength" of the Mpemba effect (how much faster the hot sample cools) is directly tied to how close you can get to the edge of the "reachable" states.

If you want the hot soup to beat the cold soup, you have to prepare it using the "floor it" or "cut the engine" strategy. But even then, you can't win by an infinite margin. The universe puts a speed limit on how weird the soup can get before it starts cooling down. The paper suggests that to get an even bigger effect in the future, we might need to find a different way to "shake the box" (a different driving mechanism), because the current method has a hard ceiling on how much it can do.

In short: The Mpemba effect is real, but it's not magic. It's a carefully engineered race where the starting line is strictly drawn by the laws of physics, and the best you can do is run as fast as the track allows.

Technical Summary: Optimal Preparation and Reachable-State Constraints in the Mpemba Effect

Problem Statement
The Mpemba effect, characterized by an initially hotter system relaxing faster than a colder one toward a common final state, is typically analyzed within kinetic frameworks by assuming non-stationary initial conditions selected a priori. However, this approach often neglects the physical constraints imposed by the specific aging protocol used to prepare these states. In the kinetic formulation, the effect relies on the memory of the system's thermal history, encoded in internal variables that deviate from equilibrium. While recent work has introduced protocols (e.g., Pontus–Mpemba) that account for preparation time, the fundamental question remains: which non-equilibrium states are physically accessible through admissible driving mechanisms, and how do these accessibility constraints limit the magnitude of the Mpemba effect? This paper addresses the problem of determining the optimal aging protocol to maximize the Mpemba effect within a kinetic framework, specifically for uniformly heated granular fluids, while explicitly accounting for the reachable set of states imposed by the stochastic thermostat.

Methodology
The authors model a $d$ -dimensional granular gas of hard spheres with inelastic collisions (restitution coefficient $\alpha$ ) driven by a stochastic thermostat (white noise). The system's mesoscopic state is described by the granular temperature $T$ and the excess kurtosis $a_2$ , which quantifies the non-Gaussianity of the velocity distribution function (VDF). The analysis is conducted within the first Sonine approximation, where the evolution of $T$ and $a_2$ is governed by a closed set of coupled differential equations.

The preparation stage is formulated as an optimal control problem:

State Variables: The granular temperature $T$ and excess kurtosis $a_2$ .
Control Variable: The intensity of the stochastic thermostat, $\chi(t)$ , bounded by $[\chi_{\min}, \chi_{\max}]$ .
Objective: To find the control protocol $\chi(t)$ that extremizes the excess kurtosis $a_2$ at a final time $t_f$ . Since the cooling rate at a fixed temperature is directly proportional to $a_2$ , extremizing $a_2$ corresponds to maximizing or minimizing the subsequent relaxation rate.
Solution Technique: The authors apply Pontryagin's Maximum Principle (PMP) to derive necessary conditions for optimality. They analyze the Hamiltonian system, costate equations, and transversality conditions to determine the structure of the optimal control.

Key Contributions and Results

Optimal Protocol Structure: The application of PMP reveals that the optimal preparation protocol is always a "one-bang" protocol. This means the control $\chi(t)$ remains constant at either its maximum ( $\chi_{\max}$ ) or minimum ( $\chi_{\min}$ ) value throughout the entire preparation interval $(0, t_f)$ . The specific choice depends on the sign of the excess kurtosis in the homogeneous cooling state (HCS), which is determined by the restitution coefficient $\alpha$ .
- For "large" inelasticity ( $\alpha < \alpha_c \equiv 1/\sqrt{2}$ ), the optimal control to minimize $a_2$ is $\chi_{\max}$ .
- For "small" inelasticity ( $\alpha > \alpha_c$ ), the optimal control to minimize $a_2$ is $\chi_{\min}$ .
- The logic is reversed for maximizing $a_2$ .
Reachable-State Constraints: A central finding is the existence of non-trivial reachable-state constraints. The set of accessible non-equilibrium states in the $(T, a_2)$ plane is bounded. Specifically, the accessible values of the excess kurtosis are confined between the Gaussian state ( $a_2 = 0$ ) and the value of the excess kurtosis in the homogeneous cooling state ( $a_2^{\text{HCS}}$ ).
- In the ideal limit where $\chi_{\min} = 0$ (free cooling) and $\chi_{\max} \to \infty$ (instantaneous strong heating), the extrema of the excess kurtosis are exactly $0$ and $a_2^{\text{HCS}}$ .
- This implies that the stochastic thermostat cannot generate arbitrarily large non-Gaussianities; the magnitude of the non-Gaussianity is strictly bounded by the HCS value.
Numerical Validation: The theoretical predictions were verified using Direct Simulation Monte Carlo (DSMC) simulations. The simulations confirmed that the extrema of the excess kurtosis converge to the theoretical bounds ($0$ and $a_2^{\text{HCS}}$ ) as the control bounds are relaxed. The agreement between the optimal control theory and numerical data is excellent across the entire range of restitution coefficients.
Implications for the Mpemba Effect: The study demonstrates that the strength of the kinetic Mpemba effect is inextricably linked to the accessibility of the underlying non-equilibrium states. Because the reachable set of $a_2$ is bounded, there is a fundamental limit to the relaxation-rate asymmetry that can be achieved between hot and cold samples. The "large" non-Gaussianities previously hypothesized in seminal works cannot be generated solely by a stochastic thermostat.

Significance and Claims
The paper claims to establish a rigorous connection between anomalous relaxation, optimal control, and state accessibility in non-equilibrium systems. By recasting the Mpemba effect as a problem of optimal state preparation under dynamical reachability constraints, the authors argue that the effect cannot be discussed independently of the preparation protocol's physical limitations.

The work highlights that the excess kurtosis acts as a critical internal variable encoding the system's memory, similar to its role in the Kovacs effect. The results suggest that the maximum attainable Mpemba effect is not an intrinsic property of the material alone but is fundamentally constrained by the geometry of the reachable non-equilibrium state space defined by the driving mechanism. The authors note that while the stochastic thermostat imposes these specific bounds, alternative driving mechanisms might yield different reachable sets, potentially allowing for stronger relaxation anomalies, though this remains a subject for future investigation. The paper concludes that engineering relaxation pathways requires navigating the specific geometry of the accessible state space.