🔬 condensed matter

Granular thermostat implementation within the soft-sphere Discrete Element Method (DEM) framework, considerations and limitations

This paper addresses the limitations of conventional thermostats in dissipative granular systems by introducing two novel pairwise hybrid formulations that effectively control granular temperature while preserving essential dynamic correlations, thereby enabling the study of temperature-dependent rheological properties in Discrete Element Method simulations.

Original authors: Marco Previtali, Herbert Eric Huppert, Sergio Andres Galindo Torres

Published 2026-09-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Marco Previtali, Herbert Eric Huppert, Sergio Andres Galindo Torres

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

Sand, snow, and even coffee grounds share a strange duality. Under slow, steady pressure, they behave like a solid, locking together in a rigid network of friction. But shake them, pour them, or stir them fast enough, and they suddenly flow like a liquid. Scientists who study these materials, known as granular matter, rely on computer simulations to understand exactly how this switch happens. They track millions of tiny, individual particles as they collide, bounce, and grind against one another. A key challenge in these simulations is controlling the "temperature" of the system. In this context, temperature does not mean heat in the traditional sense of fire or boiling water; it is a measure of how wildly the particles are jiggling and fluctuating in their motion. If the particles stop jiggling, the simulation freezes into a solid state. If they jiggle too much, the material flies apart. To keep a simulation running in a specific, stable state, researchers use a digital tool called a thermostat. Just as a home thermostat turns a heater on or off to keep a room at a comfortable temperature, a digital thermostat adds or removes energy from the particles to keep their jiggling at a precise level.

For decades, scientists have used two main types of digital thermostats to manage these simulations. One type acts like a constant, gentle drag, slowing down fast particles and speeding up slow ones to maintain an average energy level. The other type acts like a feedback loop, constantly measuring the system's energy and applying a corrective force to keep it on target. However, when researchers applied these standard tools to dense, jostling piles of granular material, they found the methods were flawed. The drag-based approach was too aggressive; it smoothed out the natural, chaotic motion of the particles so much that it killed the very dynamics the scientists wanted to study. The feedback-based approach, while better at keeping the average energy steady, failed to break up the unnatural patterns that form when particles collide. In a real pile of sand, collisions are messy and random, but in these simulations, the particles began to cluster together and move in synchronized, unnatural waves, creating a state that was mathematically unstable and physically unrealistic.

To solve this, a team of researchers at Westlake University and the University of Cambridge developed a new, hybrid approach. They realized that the problem was trying to use a single tool to do two different jobs: controlling the overall energy level and breaking up the unnatural patterns. Their solution was to combine the strengths of the two old methods while removing their weaknesses. They created two new algorithms that work together. The first part of the system acts as a steady hand, ensuring the total amount of jiggling energy stays exactly where the researchers want it. The second part acts as a gentle, random nudge, designed specifically to disrupt the synchronized patterns and keep the particles moving in a chaotic, natural way. By separating these two tasks, the researchers could control the intensity of the jiggling without accidentally freezing the system or forcing the particles into fake, orderly lines.

The team tested these new methods by running thousands of simulations with different settings. They found that their hybrid tools could maintain a perfectly steady temperature even as the particles collided and lost energy, a feat the old methods struggled to achieve. More importantly, they discovered that the way they applied the random nudge changed the nature of the simulation in subtle but significant ways. One version of their new tool produced a system where the particles moved with a very specific, predictable randomness, while the other version created a slightly different statistical pattern. This proved that the "temperature" of a granular system is not just a single number. It is a complex state that depends on both how much energy the particles have and how that energy is distributed among them. Two systems could have the exact same average temperature but behave very differently depending on how the energy was injected.

To prove their method worked in a realistic scenario, the researchers applied it to a classic problem: a box of sand being sheared, or slid, between two walls. In this setup, they could hold the pressure constant while changing the temperature of the jiggling particles. They observed that as they increased the temperature, the friction between the particles decreased, making the material flow more easily. This matched what scientists had seen in previous experiments, confirming that their digital tool was accurately capturing the physics of the material. However, the results also showed that the relationship between temperature and friction was not as simple as a single formula could describe. The specific way the temperature was controlled mattered, suggesting that to fully predict how granular materials will behave, scientists need to account for more than just the average energy; they must also understand the detailed statistics of how the particles move and interact.

This work provides a new, reliable way to study the physics of sand, snow, and other granular materials. By giving researchers a tool that can hold the temperature steady without distorting the natural motion of the particles, it opens the door to more accurate models of landslides, industrial mixing, and geological flows. The study does not claim to have solved every mystery of granular matter, but it has removed a major obstacle. It shows that by carefully separating the control of energy from the control of motion, scientists can create a stable, realistic digital environment to test their theories. This allows them to ask sharper questions about how these materials flow and jam, moving closer to a complete understanding of the complex world of granular physics.

Technical Summary: Granular Thermostat Implementation within the Soft-Sphere Discrete Element Method (DEM)

Problem Statement
Dense granular flows are frequently modeled using the soft-sphere Discrete Element Method (DEM), yet large-scale applications often rely on continuum models requiring constitutive relations linking stress and deformation to state variables. Granular temperature ( $T$ ), defined as the variance of non-affine particle velocity fluctuations, is a critical variable for improving inertial rheological scaling (e.g., $\mu(I, \Theta)$ ). However, existing temperature-based relationships lack predictive power because they do not describe the production, transport, and dissipation of $T$ . Furthermore, experimental forcing methods (e.g., vibration) alter temperature alongside velocity statistics, correlations, and structural anisotropy, making it difficult to isolate the specific effects of temperature.

While thermostat algorithms are standard in Molecular Dynamics (MD) for controlling kinetic energy, their application to dissipative soft-sphere DEM systems is underexplored. Conventional thermostats face complementary limitations in granular systems:

Langevin Thermostats: To offset continuous collisional dissipation, they require strong coupling, which overdamps particle dynamics and suppresses diffusion.
Nosé-Hoover Thermostats: While they preserve dynamics better, they fail to disrupt the velocity correlations and segregation generated by repeated inelastic collisions. This leads to non-ergodic, clustered states and numerical instability in soft-sphere DEM.

Methodology
The authors investigate conventional thermostat algorithms and propose two new pairwise hybrid formulations designed to control granular temperature while allowing independent manipulation of velocity statistics. The study utilizes the open-source library Mechsys with GPU acceleration.

System Setup: Simulations employ a quasi-monodisperse particle size distribution in a periodic cubic domain with a solid fraction $\phi \approx 0.27$ . Interactions follow a linear elastic (Hookean) contact law with a viscous dashpot and Coulomb friction.
Hybrid Formulations:
1. Direct Addition (NA): Combines a deterministic Nosé-Hoover feedback loop (controlling mean temperature) with a standard Langevin term (stochastic forcing and damping).
2. Scaled Langevin (SL): Retains the Nosé-Hoover feedback but removes the explicit Langevin damping term. Instead, the stochastic forcing amplitude is scaled based on the instantaneous pair kinetic state ( $T_p$ ). The forcing vanishes if the pair temperature exceeds the target, mimicking the energy transfer behavior of mechanically vibrating boundaries.
Implementation: Both formulations use a pairwise approach (inspired by Dissipative Particle Dynamics) to ensure Galilean invariance and conserve pair center-of-mass momentum. An interaction range ( $r_t$ ) defines the thermostat pairs, and a smoothing function minimizes numerical noise at the range boundary.
Parametric Studies: The authors conducted three parametric studies varying:
1. Interaction range ( $R_t$ ) and coupling strengths ( $\xi_{La}, \xi_{NH}$ ).
2. Timestep size ( $\Delta t$ ), target temperature ( $\hat{T}$ ), and the dimensionless stochastic forcing strength ( $\pi_2 = \sqrt{\hat{T}}/(d\xi_{La})$ ).
3. Coefficient of restitution ( $e$ ) to test inelasticity effects.
Application: The framework was applied to a pressure-controlled simple shear test to explore the $(I, T)$ state space.

Key Results

Limitations of Conventional Thermostats: Pure Langevin thermostats failed to enforce the target temperature in dissipative systems without strong coupling, which suppressed diffusion and led to crystallization. Pure Nosé-Hoover thermostats maintained the target temperature but resulted in highly non-ergodic states with large temperature spikes, velocity segregation, and numerical instability.
Performance of Hybrid Formulations: Both NA and SL successfully enforced the prescribed scalar temperature while avoiding the numerical instabilities of pure Nosé-Hoover and the overdamping of pure Langevin.
Distinct Statistical Signatures: Despite controlling the same scalar temperature, the two hybrid formulations produced measurably different kinetic states:
- NA produced mesokurtic (Gaussian-like) velocity distributions and higher diffusivity.
- SL produced platykurtic distributions (flatter tails) and lower diffusivity.
- This demonstrates that fluctuation magnitude, decorrelation, and forcing statistics are separate aspects of the granular state.
Role of Stochastic Forcing: The dimensionless parameter $\pi_2$ governs the relative strength of stochastic decorrelation. Varying $\pi_2$ at a fixed temperature altered velocity statistics, diffusivity, and the radial distribution function peak $g(d)$ .
Shear Application: In pressure-controlled simple shear, increasing granular temperature at a fixed inertial number ( $I$ ) reduced the apparent friction, consistent with prior trends. However, the relationship did not strictly follow simple scaling laws, suggesting additional state variables are required.

Significance and Claims
The paper claims to provide a framework for generating controlled reference states in granular simulations. By decoupling the magnitude of velocity fluctuations (temperature) from the statistical character of the forcing (via $\pi_2$ and the choice of NA vs. SL), researchers can:

Identify which transport and rheological properties are governed primarily by scalar granular temperature versus those that depend on velocity statistics and correlations.
Compare differently forced systems (experimental or natural) on a state-based basis (matching $I$ and $T$ ) rather than a parameter-based basis.
Determine the specific variables required to close temperature-dependent constitutive models.

The authors emphasize that while the hybrid thermostats offer a method to isolate variables, the empirical relationships derived (e.g., dependence on $\pi_2$ and $e$ ) are specific to the investigated parameter range and require further validation with varying solid fractions and friction coefficients before serving as general constitutive closures. The work does not claim to reproduce specific experimental apparatus but rather to construct idealized states for comparative rheological analysis.

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