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Macroparticles with different weights relax to different temperatures in Particle-In-Cell simulations

This paper demonstrates that in Particle-In-Cell simulations, macroparticles with varying weights inevitably relax to an unphysical thermal equilibrium with unequal temperatures, but the authors derive a predictive equation for this evolution to identify strategies for delaying its onset.

Original authors: Remi Lehe, Arianna Formenti, Justin R. Angus, Jean-Luc Vay

Published 2026-08-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Remi Lehe, Arianna Formenti, Justin R. Angus, Jean-Luc Vay

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

In the vast, invisible world of plasma physics, scientists study a state of matter so hot and energetic that atoms break apart into a swirling soup of charged particles. To understand how this chaotic mixture behaves, researchers rely on powerful computer simulations. One of the most common tools for this job is a method called Particle-In-Cell. Imagine trying to track the movement of every single grain of sand on a beach; it would be impossible. Instead, a scientist might group the sand into handfuls and track those handfuls. In these simulations, the computer does something similar: it groups billions of real physical particles into single, virtual "macro-particles." Each of these virtual particles carries a specific weight, representing how many real particles it stands in for. This trick allows computers to simulate complex systems like stars or fusion reactors without crashing under the sheer number of calculations required.

For decades, researchers have operated under a comfortable assumption: as long as the total number of particles is correct, it shouldn't matter if some virtual particles represent more real ones than others. They believed that the average temperature of the plasma would settle into a natural, physical balance regardless of these virtual weights. However, a new study challenges this long-held view. By running detailed simulations of a simple plasma, researchers discovered that the computer's own method of handling these virtual particles creates a hidden flaw. When the virtual particles have different weights, the simulation does not settle into the expected physical balance. Instead, it drifts toward a strange, incorrect state where the temperature depends entirely on the arbitrary weight assigned to the particles, rather than the physics of the real world.

The researchers, working with a simulation code called WarpX, set up a virtual box filled with a uniform plasma containing two types of particles: electrons and ions. They started the simulation with the two types at different temperatures and watched what happened over time. In a perfectly physical world, the two types would exchange energy until they reached the same temperature, regardless of how many virtual particles represented them. But in the computer simulation, something different occurred. The virtual particles representing the lighter-weight groups absorbed energy and became hotter, while the heavier-weight groups lost energy and became cooler. The system did not stop at the correct physical temperature; it settled into a new, unphysical equilibrium where the temperature of each group was directly tied to its weight. The lighter the virtual particle, the hotter it got; the heavier the virtual particle, the cooler it became.

This unexpected behavior happens because the simulation algorithm, while incredibly sophisticated, has a blind spot. The computer sees only the virtual particles and their assigned weights; it has no memory of the fact that a light-weight particle is supposed to represent a massive number of real particles, or that a heavy-weight particle represents just a few. To the algorithm, every virtual particle is just a single entity with a specific charge and mass. Consequently, the computer naturally tries to share energy equally among these virtual entities, rather than among the real physical particles they represent. It is as if the computer is trying to balance the energy of the handfuls of sand rather than the individual grains, leading to a result that looks stable but is fundamentally wrong.

The study found that this error is not immediate; it takes time to develop. The speed at which the simulation drifts toward this incorrect state depends on how "noisy" the simulation is, which is controlled by the mathematical shapes used to smooth out the interactions between particles. The researchers discovered that by using more complex shapes to represent the particles, they could slow down this drift significantly. They also found that the time it takes for the error to appear changes depending on the size of the grid cells used in the simulation and the temperature of the plasma. In some cases, the simulation could run for a long time before the error became noticeable, but in others, the drift happened quickly enough to ruin the results of a short experiment.

The implications of this discovery are significant for anyone using these simulations to study real-world phenomena. If a scientist is studying a plasma where different types of particles have different weights—perhaps because they are modeling a rare impurity or a specific reaction—their results for temperature could be wrong. The paper suggests that researchers must be much more careful when setting up these simulations. They cannot simply assume that changing the number of virtual particles will not affect the outcome. Instead, they need to check if their results change when they adjust the weights, and they may need to use specific techniques, such as using more virtual particles or adjusting the grid size, to delay the onset of this error. The study does not claim that the method is broken, but rather that it has a hidden bias that must be understood and managed. By deriving a mathematical description of how the temperature changes over time, the authors provide a tool for scientists to predict exactly how long they can trust their simulation before this unphysical drift takes over.

Ultimately, this work serves as a reminder that even the most advanced computer models are built on approximations. The Particle-In-Cell method is a powerful way to see the invisible, but it requires a deep understanding of its own limitations. The researchers showed that the path to a correct answer is not always a straight line; sometimes, the simulation takes a detour into a world that looks real but follows different rules. By identifying this detour and mapping out how to avoid it, the study helps ensure that the virtual plasmas we create on our screens remain faithful to the real ones swirling in the universe.

Problem Statement
Particle-In-Cell (PIC) simulations utilize macroparticles to represent ensembles of physical particles, where the "weight" of a macroparticle denotes the number of physical particles it represents. While simulations often employ uniform weights, many practical scenarios necessitate varying weights, such as in simulations with spatially varying initial densities, cylindrical/spherical grids, trace impurities, low-probability Monte Carlo reactions, or adaptive splitting/merging algorithms. It has been generally assumed that while differing weights affect numerical noise, the average statistical properties of the plasma (e.g., temperature) remain independent of the weight ratio. This paper challenges that assumption, demonstrating that macroparticles with different weights relax to an unphysical thermal equilibrium where temperatures are unequal: lower-weight macroparticles reach a higher temperature than physically expected, while higher-weight macroparticles reach a lower one.

Methodology
The authors investigate this phenomenon through a combination of numerical experimentation and theoretical derivation:

Numerical Simulations: 3D PIC simulations were performed using the WarpX code with both electromagnetic (EM) and electrostatic (ES) algorithms. The simulations modeled a uniform plasma with electrons and ions initialized at different temperatures but equal physical density. The macroparticle weights ( $w_1, w_2$ ) were varied independently for the two species while maintaining constant physical density. The study utilized "energy-conserving" gathering schemes and third-order splines, excluding Monte Carlo collision modules to isolate the intrinsic collisionality of the PIC algorithm.
Theoretical Derivation: The authors derived a kinetic equation describing the evolution of the distribution function $f_\sigma$ for species $\sigma$ under the influence of effective PIC collisionality. This derivation extends the known Balescu-Guernsey-Lenard (BGL) collision operator to account for grid effects, macroparticle shape factors, and the specific weighting of macroparticles. By assuming Maxwellian distributions, they derived a closed-form equation for the time evolution of species temperatures ( $T_\sigma$ ).
Parameter Analysis: The theoretical predictions were validated against simulations across various parameters, including different spline orders (CIC, TSC, PQS), initial temperatures, and grid resolutions (specifically the ratio of Debye length to cell size, $\lambda_D/\Delta x$ ).

Key Contributions and Results

Unphysical Equilibrium State: The simulations revealed that the system evolves toward a steady state where $w_\sigma T_{\sigma, eq} = w_{\sigma'} T_{\sigma', eq}$ . Consequently, the equilibrium temperature is inversely proportional to the macroparticle weight. This contradicts the physical expectation of $T_{\sigma, eq} = T_{\sigma', eq}$ .
Mechanism: The unphysical energy transfer arises because the PIC algorithm inherently captures Coulomb collisions between charged macroparticles. However, the collision frequency is distorted by the grid and the macroparticle weight. Specifically, the collision frequency scales with the macroparticle weight ( $w_\sigma$ ), leading the algorithm to enforce energy equipartition among the macroparticles (mass $m_{macro} = w m_{phys}$ ) rather than the physical particles.
Predictive Equation: The authors derived an equation (Eq. 13) governing the time evolution of temperatures. This equation includes a modified Coulomb logarithm ( $\log \Lambda_{PIC}$ ) that accounts for the discrete grid, time step, and macroparticle shape.
Validation: The theoretical predictions showed excellent agreement with PIC simulations across a wide range of parameters. The relaxation timescale was found to depend on the effective collisionality, which is influenced by the spline order (higher orders reduce collisionality) and the resolution of the Debye length.
Approximate Expressions: For electron-ion plasmas with similar temperatures, the authors provided asymptotic expressions for the effective Coulomb logarithm. In the under-resolved regime ( $\lambda_D/\Delta x \ll 1$ ), $\log \Lambda_{PIC} \propto (\lambda_D/\Delta x)^4$ , while in the well-resolved regime ( $\lambda_D/\Delta x \gg 1$ ), it behaves logarithmically. This contrasts sharply with the physical Coulomb logarithm, which typically varies only between 2 and 20.

Significance and Implications
The paper establishes that the unphysical thermal equilibrium is an intrinsic feature of the PIC algorithm when macroparticle weights differ, regardless of whether the simulation is electrostatic or electromagnetic, or 1D, 2D, or 3D. The significance of this work lies in the following points:

Diagnostic Necessity: PIC simulations using varying weights require careful scrutiny. If the simulation duration exceeds the thermalization timescale, the measured temperatures will be incorrect. Even in shorter runs, the unphysical energy transfer may cause deviations from the true physical evolution.
Mitigation Strategies: The authors propose strategies to delay the onset of this unphysical equilibrium by reducing the effective collisionality of the algorithm. These include using more macroparticles (reducing weights), employing higher-order splines, or deliberately under-resolving the Debye length (if Debye-scale physics is not critical to the study).
Convergence Testing: The paper warns that standard convergence tests are insufficient if they only vary the weight ratio while keeping the ratio constant. To properly assess the impact of this effect, the weights of low- and high-weight species must be varied independently.
Future Directions: The authors note that this effect likely extends to cases where weights vary within a single species and that coupling PIC with Monte Carlo Collision (MCC) modules may create a competition between the physical equilibrium driven by MCC and the unphysical equilibrium driven by the PIC algorithm.

In conclusion, the authors provide a theoretical framework and empirical evidence to predict and manage unphysical thermalization in PIC simulations, offering a critical tool for ensuring the accuracy of plasma simulations involving non-uniform macroparticle weights.

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