🔬 condensed matter

Energy non-equipartition in vibrofluidized particles

This study uses 3D DEM simulations to demonstrate that realistic tangential spring stiffness parameters in vibrofluidized granular systems lead to a breakdown of energy equipartition between translational and rotational modes, particularly for particles with high friction coefficients.

Original authors: Alok Tiwari, Manaswita Bose, V. Kumaran

Published 2026-05-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Alok Tiwari, Manaswita Bose, V. Kumaran

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 a box filled with thousands of tiny, bouncy balls. Now, imagine shaking the bottom of that box up and down very quickly. The balls jump around, collide with each other, and bounce off the walls. In physics, this is called a "vibrofluidized" system.

Usually, scientists expect that if you shake these balls long enough, the energy will spread out evenly. This is called the Equipartition Theorem. Think of it like a group of friends sharing a pizza: eventually, everyone gets an equal slice. In this case, the "pizza" is the energy of motion. Some of that energy makes the balls move forward, backward, up, and down (called translational energy), and some makes them spin around (called rotational energy).

The standard rule says: If the balls are identical and the system is stable, the energy spent on moving should equal the energy spent on spinning.

The Big Question
The authors of this paper asked: "What happens if the balls aren't perfectly smooth? What if they are rough, like sandpaper?" They wanted to see if the "roughness" (friction) and the "stiffness" of the balls would break the rule of equal energy sharing.

To find out, they didn't use a real box of balls. Instead, they built a giant, virtual 3D world inside a computer using a program called LAMMPS. They simulated millions of collisions between spherical particles.

The Two Key Ingredients
The researchers changed two main things in their simulation:

Friction (Roughness): They tested balls that were almost perfectly smooth (like ice) to balls that were extremely rough (like sandpaper).
Stiffness Ratio ( $\kappa$ ): This is a bit like the "springiness" of the ball. When two balls hit, they squish a tiny bit. The researchers looked at how stiff the ball is when it gets hit from the side (tangential) compared to when it gets hit head-on (normal). They tested two specific "recipes" for this stiffness:
- Recipe A: A specific ratio of 2/7.
- Recipe B: A specific ratio of 3/4.

What They Discovered

1. The Smooth Case (The "Ice" Scenario)
When the balls were very smooth and had low friction, the energy did stay separate. The balls moved around a lot but barely spun. It was like a group of people on a slippery ice rink: they could slide easily, but they couldn't get enough grip to spin. The energy for moving and the energy for spinning were very different.

2. The Rough Case with Recipe A (The "Sandpaper" Success)
When they used the 2/7 stiffness ratio and made the balls very rough (high friction), something magical happened. The balls finally shared the energy equally!

The Analogy: Imagine the balls are wearing heavy rubber boots. When they bump into each other, the friction is so strong that they "stick" together for a split second. This sticking allows the forward motion to easily turn into a spin.
The Result: The energy for moving and the energy for spinning became almost equal. The "pizza" was finally shared fairly.

3. The Rough Case with Recipe B (The "Broken" Rule)
This is where things got weird. When they used the 3/4 stiffness ratio and made the balls very rough, the energy did not share equally, even though the balls were rough.

The Analogy: Imagine the balls are wearing boots that are sticky but also have a weird spring mechanism. When they bump, they don't just stick; they bounce and slip in a way that wastes energy.
The Result: The balls kept moving forward with lots of energy, but they refused to spin up to the same level. The ratio of moving energy to spinning energy settled at about 5 to 1. The "pizza" was not shared; one group of friends got five slices, and the other got one.

Why Did This Happen?
The authors looked closely at the collisions to understand why Recipe B failed to share the energy.

In the successful case (Recipe A), the balls mostly entered a "sticking" mode where they held onto each other briefly. This was an efficient way to transfer energy from moving to spinning.
In the failed case (Recipe B), the balls mostly entered a "slipping" mode. They rubbed against each other but didn't stick. This created a lot of friction heat (energy loss) without effectively turning the forward motion into spin. The energy was lost to friction before it could be shared.

The Bottom Line
The paper concludes that how stiff a particle is (specifically the ratio of side-stiffness to front-stiffness) is just as important as how rough it is.

You cannot assume that rough particles will automatically share their energy equally. Depending on the specific "springiness" of the material, rough particles might either share energy perfectly (like the 2/7 case) or hoard it, leaving the spinning motion weak (like the 3/4 case).

In short: Roughness alone doesn't guarantee fairness. The material's internal "spring" matters just as much.

Technical Summary: Energy Non-equipartition in Vibrofluidized Particles

Problem Statement
The equipartition theorem, which posits that kinetic energy is equally distributed among all degrees of freedom in a fluid, frequently deviates in granular systems. While previous studies have documented non-equipartition in vibrofluidized beds involving smooth particles (leading to anisotropy in fluctuating kinetic energy) and rough particles, the specific influence of realistic particle interaction parameters remains a subject of investigation. Specifically, this work addresses how the ratio of the tangential to normal spring stiffness coefficient ( $\kappa = k_t/k_n$ ) affects the partitioning of energy between translational and rotational modes. The study seeks to determine whether realistic models of particle contact, particularly those involving high friction and specific stiffness ratios, lead to energy equipartition or sustained non-equipartition.

Methodology
The authors employed the Discrete Element Method (DEM) using the open-source software LAMMPS to simulate a three-dimensional, vertically vibrated granular system composed of spherical particles.

Interaction Model: Contact forces were modeled using a linear-spring dashpot (LSD) model. The normal spring stiffness constant ( $k_n = 10^8 mg/d$ ) was selected to be sufficiently large to ensure that collisions are predominantly binary ( $t_c/t_f \ll 1$ ).
Parameters: The study varied the friction coefficient ( $\mu$ ) from nearly smooth ( $\mu \approx 0$ ) to nearly perfectly rough ( $\mu = 10$ ). Two primary values for the stiffness ratio $\kappa$ were investigated: $2/7$ and $3/4$ . Additional simulations were conducted for $0.67 \le \kappa < 1$ .
Simulation Protocol: Simulations were run in two stages: an initial relaxation phase ( $7 \times 10^7 t_c$ ) to reach a steady state, followed by data collection with a smaller time step ( $\Delta t = t_c/100$ ).
Analysis: Instantaneous linear and angular velocities were analyzed to determine profiles of mean fluctuating translational ( $T$ ) and rotational ( $T_R$ ) kinetic energies. Ensemble averages were computed over more than $10^3$ configurations. The ratio $K = K_{ET}/K_{ER}$ (translational to rotational kinetic energy) was used as the primary metric for equipartition.

Key Results

Dependence on Stiffness Ratio ( $\kappa$ ): The behavior of the energy ratio $K$ is critically dependent on $\kappa$ .
- For $\kappa = 2/7$ : The ratio $K$ decreases monotonically as the friction coefficient $\mu$ increases. At high friction ( $\mu = 10$ ), $K$ approaches unity, indicating that energy equipartition between translational and rotational modes is achieved.
- For $\kappa = 3/4$ : The behavior is non-monotonic. After an initial reduction with increasing $\mu$ , $K$ increases and plateaus at approximately 5. This indicates a persistent lack of energy equipartition, even at very high friction coefficients.
- Simulations for $0.67 \le \kappa < 1$ confirmed that non-equipartition persists for particles with high friction coefficients in this range.
Collision Dynamics and Energy Exchange:
- Analysis of individual collisions revealed that for $\kappa = 2/7$ , the fraction of contacts in the "sticking" regime (where the rotational coefficient of restitution $\beta \approx 1$ ) increases with $\mu$ . At $\mu = 1$ , nearly 75% of contacts are in this energy-conserving regime, facilitating equipartition.
- For $\kappa = 3/4$ , the fraction of contacts in the stick-slip regime plateaus at roughly 55% for $\mu > 1$ , and $\beta < 1$ . Consequently, the energy exchange ratio $\Theta = E_{exch}/\Delta E$ remains less than 1 and decreases for very rough particles. The dissipation exceeds the energy exchange between modes, preventing equipartition.
Validation of Theory: The simulation results for the equilibrium granular temperature ( $T_o$ ) showed excellent agreement with theoretical predictions (based on leading-order moment expansions) for both perfectly smooth particles and rough particles in dilute regimes, provided the appropriate scaling parameters (involving $\beta$ ) were used.

Significance and Claims
The paper claims that the selection of the stiffness ratio $\kappa$ is a crucial factor in predicting the macroscopic flow behavior of realistic granular particles. The study demonstrates that energy equipartition is not a guaranteed outcome in vibrofluidized rough granular systems; rather, it is contingent upon the specific interplay between friction and the tangential-to-normal stiffness ratio.

Specifically, the authors conclude that:

Equipartition is observed for $\kappa = 2/7$ at high friction due to the dominance of energy-conserving sticking collisions.
Non-equipartition is observed for $0.67 \le \kappa < 1$ (including $\kappa = 3/4$ ) because the stick-slip collisions in these regimes are not energy-conserving, leading to a significant disparity between translational and rotational kinetic energies.

These findings suggest that realistic modeling of granular rheology requires careful consideration of the stiffness ratio, as it fundamentally alters the energy distribution mechanisms within the system. The work does not propose new experimental setups or applications but provides a systematic analysis of how microscopic interaction parameters dictate macroscopic energy distribution.

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