Mpemba effect in a sheared granular gas with velocity-dependent restitution

Using kinetic theory, this study demonstrates that a dilute sheared granular gas with a velocity-dependent restitution coefficient exhibits both temperature and viscosity Mpemba effects, where systems with higher initial temperatures relax faster than cooler ones, with the velocity dependence introducing an intrinsic timescale that enables multiple relaxation curve crossings.

The Gist

The Big Idea: The "Hot Water" Mystery in a Box of Bouncing Balls

You might have heard of the Mpemba effect. It's a counterintuitive phenomenon where hot water can sometimes freeze faster than cold water. It sounds impossible, but it happens because the "hot" water has a different internal structure or history that helps it cool down faster once the freezer door closes.

This paper investigates whether this same weird trick happens in a granular gas. Imagine a box filled with thousands of tiny, hard steel balls bouncing around. Unlike real gas molecules, these balls lose energy every time they hit each other (they don't bounce perfectly). To keep them moving, scientists "shear" the box, which means they slide the top of the box to the right and the bottom to the left, constantly stirring the balls like a mixer.

The researchers asked: If you have two boxes of these bouncing balls, and one is "hotter" (moving faster) than the other, can the hotter one actually settle down into a calm, steady rhythm faster than the cooler one?

The Two Starting Points

To test this, they set up two different scenarios (protocols) that both end up in the exact same "final state" (a specific speed of stirring):

1. The "Stirred" Start (FS Protocol): Imagine a box of balls that has already been stirred for a long time. They are moving in a specific, organized, but chaotic pattern. Then, suddenly, the stirring speed changes.
2. The "Still" Start (FI Protocol): Imagine a box of balls that was just sitting still (or cooling down on its own) with no stirring. At the exact same moment, the stirring starts up at the same new speed as the first box. Crucially, the balls in this box start out with a higher temperature (they are moving faster) than the balls in the first box.

The Result: The Hot One Wins the Race

In a normal world, you would expect the cooler box to reach the final steady state faster. But, just like the hot water freezing trick, the hotter box (the "Still" Start) caught up and passed the cooler box.

• Why? The "Stirred" box had a lot of internal stress and "bad habits" from being stirred before. When the speed changed, it had to untangle those old patterns, which slowed it down.
• The "Still" box, even though it was hotter, started with a clean slate (no internal stress). It was able to absorb the new stirring motion more efficiently and settle into the rhythm faster, despite starting with more energy.

This is the Temperature Mpemba Effect: The system with more energy relaxed faster.

The Twist: The "Viscosity" Trick

The paper found something even stranger. It's not just the temperature (speed of the balls) that shows this effect; the viscosity (how "thick" or resistant the gas feels to the stirring) does it too.

Usually, when you change how fast you stir a fluid, its thickness changes smoothly. But here, the researchers saw the viscosity curves cross each other multiple times. The "hotter" system didn't just overtake the "cooler" one once; it zig-zagged past it, then maybe fell behind, then overtook it again, before finally settling down.

The Secret Ingredient: The "Bounciness" Switch

Why did this happen? The key was a special rule they applied to the balls: The bounciness changes depending on how hard they hit.

• Soft hits: The balls are very bouncy (like a superball).
• Hard hits: The balls are less bouncy (like a lump of clay).

This creates a "switch" in the physics. Because the balls behave differently at different speeds, it introduces a second clock or timescale into the system.

Think of it like a car with two different gears. If you only have one gear, the car accelerates smoothly. But if you have a car that suddenly shifts gears depending on how fast you're going, the acceleration becomes jerky and complex. This "gear shift" in the physics of the balls is what causes the relaxation curves to cross multiple times, creating multiple Mpemba effects.

The Bottom Line

The paper proves that in a gas of bouncing balls where the "bounciness" depends on speed:

1. A hotter system can relax to a steady state faster than a cooler one (Temperature Mpemba Effect).
2. The "thickness" of the gas can also show this effect (Viscosity Mpemba Effect).
3. Because of the speed-dependent bounciness, these systems can cross paths multiple times on their way to stability, a behavior not seen in simpler models.

This is a purely mathematical and physical discovery about how energy and stress interact in granular materials, showing that "hotter" doesn't always mean "slower to settle."

Technical Summary

Technical Summary: Mpemba Effect in a Sheared Granular Gas with Velocity-Dependent Restitution

Problem Statement
The Mpemba effect, originally observed in water freezing, describes the counterintuitive phenomenon where a system initially at a higher temperature reaches equilibrium faster than an identical system starting at a lower temperature. While recognized as a generic feature of nonequilibrium relaxation in driven classical and quantum systems, the specific role of rheological effects and velocity-dependent dissipation in generating this behavior within granular gases remains largely unexplored. Specifically, it is unclear how a velocity-dependent restitution coefficient—which introduces an intrinsic velocity scale and additional relaxation times—affects the relaxation pathways of a sheared granular gas.

Methodology
The authors investigate a dilute sheared granular gas composed of frictionless hard spheres with a velocity-dependent restitution coefficient. The study employs two primary approaches:

1. Kinetic Theory: Based on Grad's moment method, the authors derive a closed set of coupled evolution equations for the granular temperature ($T$), temperature anisotropy ($\Delta T$), and shear stress ($P_{xy}$). The model incorporates a stepwise velocity-dependent restitution coefficient, $e(v_n)$, which introduces a characteristic threshold velocity $v_c$. This allows for the analytical calculation of the collisional energy dissipation rate ($\zeta$) and the effective collision frequency ($\nu$).
2. Direct Simulation Monte Carlo (DSMC): To validate the theoretical predictions, the authors perform DSMC simulations of $N=10^3$ particles under uniform simple shear flow using SLLOD dynamics and Lees–Edwards boundary conditions.

The study compares two distinct relaxation protocols following a sudden change in shear rate to a target value $\dot{\gamma}_{tar}$:

• From Sheared (FS) Protocol: The system starts from a steady sheared state with an initial shear rate $\dot{\gamma}_{ini}$.
• From Isotropic (FI) Protocol: The system starts from an unsheared, isotropic state with zero shear stress.

Key Contributions and Results
The paper demonstrates the existence of both temperature and viscosity Mpemba effects in this system, characterized by the crossing of relaxation curves.

• Temperature Mpemba Effect: The authors show that an FI system, despite starting with a higher initial temperature ($T^{(FI)}_{ini} \gtrsim T^{(FS)}_{ini}$), can relax to the final steady-state temperature faster than an FS system. The mechanism arises from the initial balance between energy injection and dissipation: the FS system initially experiences suppressed relaxation due to negative shear stress, whereas the FI system, starting with zero shear stress, undergoes rapid initial dissipation.
• Viscosity Mpemba Effect: The study identifies a "viscosity Mpemba effect," where the shear viscosity relaxation curves of the two protocols cross. This is particularly pronounced in the FI protocol, where viscosity evolves from zero, creating a strong transient response that facilitates crossing.
• Multiple Mpemba Effects: A significant finding is the emergence of multiple crossings in the relaxation curves. Unlike systems with constant restitution coefficients which typically exhibit a single crossover, the velocity-dependent restitution coefficient introduces an additional intrinsic timescale. This leads to non-monotonic relaxation dynamics, allowing the trajectories to cross more than once. The phase diagrams reveal specific parameter regions (dependent on initial shear rates and temperatures) where the effect occurs twice.
• Rheological Connection: The emergence of multiple Mpemba effects is linked to parameter regimes where the steady-state rheology exhibits a non-monotonic (S-shaped) dependence on the shear rate, specifically when the restitution parameters satisfy $e_1 < e_2$.

Significance and Claims
The paper claims to provide a minimal, analytically tractable kinetic mechanism for multiple Mpemba effects in driven granular gases. The authors emphasize that these findings establish a direct link between anomalous relaxation and nonequilibrium rheology. Crucially, the mechanism is purely kinetic; it does not rely on dense packing, frictional contacts, or jamming effects. The study suggests that controlled Mpemba phenomena can be realized in granular flows by tuning microscopic dissipation mechanisms, specifically through the velocity dependence of the restitution coefficient. The work distinguishes itself from previous studies on steady-state rheology by focusing on transient relaxation dynamics, offering a framework to understand how internal degrees of freedom (stress and anisotropy) coupled with velocity-dependent dissipation govern relaxation pathways.