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Equation of state and transport coefficients of warm dense aluminum from mixed deterministic-stochastic density functional theory

Using mixed deterministic-stochastic finite-temperature density functional theory, this study computes the equation of state and transport coefficients of warm dense aluminum up to 1000 eV, revealing significant deviations from existing model-based predictions that offer critical corrections for radiation-hydrodynamics simulations.

Original authors: Zi Li, Weijie Li, Cong Wang, Ping Zhang, Xianjue Peng

Published 2026-09-25

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CC BY 4.0

Original authors: Zi Li, Weijie Li, Cong Wang, Ping Zhang, Xianjue Peng

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 realm of high-energy-density physics, scientists study matter under conditions so extreme that atoms are stripped of their electrons and squeezed into states that do not exist naturally on Earth. This field is crucial for understanding how materials behave in powerful explosions, inside stars, or within the experimental chambers designed to replicate the power of the sun. To predict how these materials will react, researchers rely on mathematical descriptions called equations of state, which act like a rulebook for how pressure, temperature, and density interact. They also need to know how well these materials conduct electricity and heat, as these properties dictate how energy moves through the system. Aluminum is a standard reference point in this field; because its behavior is so well documented at normal conditions, it serves as a trusted benchmark for calibrating models used in everything from laboratory astrophysics to the design of fusion energy experiments. However, when aluminum is heated to thousands of degrees and compressed to high densities, the standard mathematical models used to predict its behavior begin to falter, leaving scientists unsure of exactly what the material is doing.

A team of researchers has now used a sophisticated computational method to map out the behavior of liquid aluminum at temperatures reaching up to 1,000 electron volts, a range where the material exists in a "warm dense" state that is difficult to study with traditional tools. Instead of relying on older, simplified models that approximate how electrons move, the team employed a hybrid approach that combines precise calculations for the most active electrons with a statistical sampling method for the vast number of less active ones. This technique allowed them to generate a detailed map of how aluminum's pressure and internal energy change as it is heated and compressed. Their results reveal that for temperatures below 200 electron volts, the older models can be off by more than 10 percent at high densities, with errors reaching over 30 percent at lower temperatures. This means that when scientists use these old models to predict how aluminum will behave under extreme compression, they are calculating a path that diverges significantly from reality, leading to incorrect predictions about how the material will respond in high-speed shock experiments.

The study also looked at how well this superheated aluminum conducts electricity and heat. The researchers found that at temperatures around 100 electron volts, the actual conductivity values differ from the predictions of standard models by anywhere from 26 percent to 63 percent. These discrepancies are not just minor technical errors; they are large enough to throw off the simulations used to design massive experimental facilities, such as the Z-pinch machines that use magnetic fields to compress fuel for fusion energy. To make their findings useful for engineers and other scientists, the team translated their complex simulation data into simple mathematical formulas that describe how conductivity changes with density and temperature. They also explored how strong magnetic fields, which are present in these fusion experiments, affect the flow of electricity and heat. They discovered that while the Drude model often used to describe these effects suggests a certain level of suppression, a more rigorous framework (the Epperlein–Haines model) indicates that the field suppresses the flow of energy significantly more than the Drude model predicts.

By providing these new, highly accurate data points, the researchers have offered a clearer picture of how aluminum behaves when pushed to its limits. Their work highlights that the gap between simplified theoretical models and the actual quantum mechanical behavior of matter is significant in the warm dense regime. This new information serves as a vital correction for the computer codes used to simulate radiation and hydrodynamics, ensuring that future experiments in fusion energy and high-energy physics are built on a foundation of accurate data rather than approximations that break down under pressure. The findings confirm that while aluminum remains a reliable reference standard, understanding its true nature at these extreme temperatures requires moving beyond the old rules and embracing more complex, direct calculations of how its electrons behave.

Technical Summary: Equation of State and Transport Coefficients of Warm Dense Aluminum from Mixed Deterministic–Stochastic Density Functional Theory

Problem Statement
Aluminum serves as a critical reference standard in high-energy-density (HED) research and functions as a liner material in megampere Z-pinch facilities, such as those used in Magnetized Liner Inertial Fusion (MagLIF). Accurate hydrodynamic simulations of these systems require reliable tables of the equation of state (EOS) and transport coefficients (electrical and thermal conductivity) across a wide range of warm-dense matter (WDM) conditions. While Quantum Molecular Dynamics (QMD) based on finite-temperature Kohn–Sham density functional theory (DFT) has provided accurate data for aluminum at lower temperatures, standard diagonalization-based approaches become computationally prohibitive at electron temperatures ( $T$ ) exceeding $\sim 100$ eV due to the steep growth in thermally occupied bands. Consequently, radiation-hydrodynamics codes currently rely on average-atom models (e.g., Thomas–Fermi–Dirac) or ionization-equilibrium tables for thermodynamics, and semi-empirical fits (e.g., Spitzer–Härm, Lee–More) for transport. The extent of the deviation between these models and ab initio results in the warm-dense regime, particularly at high densities and temperatures up to 1000 eV, remains poorly understood.

Methodology
To address the computational limitations of standard DFT at high temperatures, the authors employ mixed deterministic–stochastic finite-temperature DFT (mDFT). This method partitions the electronic states into two components:

Deterministic: A small subset of low-lying, strongly occupied Kohn–Sham orbitals is treated via exact diagonalization.
Stochastic: The high-lying, weakly occupied tail of the thermal band manifold is sampled using stochastic orbitals orthogonalized to the Kohn–Sham subspace.

The electron density is calculated by evaluating the trace of the Fermi–Dirac operator using a Chebyshev expansion, avoiding full diagonalization. This approach balances accuracy and computational cost, enabling calculations up to 1000 eV.

Transport coefficients are derived using the Kubo–Greenwood formalism adapted for the mDFT framework. Instead of relying on eigenvalues, the method evaluates the current response function via the trace of current operators acting on time-evolved Kohn–Sham and stochastic orbitals. The frequency-dependent conductivities are fitted to the Drude model to extract zero-field electrical ( $\sigma_0$ ) and thermal ( $\kappa_0$ ) conductivities.

Calculations were performed using the ABACUS code with norm-conserving pseudopotentials (13 valence electrons) and the Perdew–Burke–Ernzerhof (PBE) exchange–correlation functional. The study covers densities from 1 to 18 g/cm³ and temperatures from 10 to 1000 eV.

Key Contributions and Results

Equation of State (EOS) and Hugoniot:
The authors constructed a wide-range EOS for liquid aluminum by combining mDFT electronic thermal contributions with low-temperature QMD data and fitted ion-thermal contributions.
- Deviations from Models: At high densities ( $\rho \ge 6$ g/cm³) and low temperatures ( $T < 200$ eV), mDFT results diverge significantly from Thomas–Fermi–Dirac (TFD) and IONEQ models. At $\rho = 12$ g/cm³ and $T = 10$ eV, the TFD model overestimates pressure by 59%, while IONEQ underestimates it by 31%. The internal energy from TFD deviates by up to 45% at low temperatures, whereas IONEQ remains within 10% of mDFT across all conditions.
- Hugoniot Curve: These thermodynamic discrepancies lead to significant differences in the principal Hugoniot at high compression. While all methods agree for $\rho < 8$ g/cm³, the pressure gap widens rapidly at higher densities. At $\rho = 12$ g/cm³, TFD and IONEQ pressures differ from mDFT by 120% and 31%, respectively. Furthermore, the mDFT Hugoniot exhibits two local compression maxima (at $\rho \approx 13.5$ and $13.2$ g/cm³), whereas the TFD model shows only one, and the IONEQ model shifts the maximum to higher densities.
Transport Coefficients:
Zero-field electrical and thermal conductivities were calculated for $\rho = 2.7–10$ g/cm³ and $T = 20–500$ eV.
- Trends: Electrical conductivity ( $\sigma_0$ ) increases monotonically with temperature (by a factor of 20 from 20 to 500 eV at ambient density) but shows weak dependence on density. Thermal conductivity ( $\kappa_0$ ) increases more dramatically, by a factor of $\sim 300$ over the same temperature range.
- Model Comparison: At $T \sim 100$ eV, the mDFT results deviate from the Lee–More model predictions by 26% to 63%. Specifically, at $T=100$ eV and $\rho=2.7$ g/cm³, the electrical and thermal conductivities deviate by 42% and 63%, respectively.
- Fitting: To facilitate use in hydrodynamic codes, the authors provided bivariate polynomial fits for $\sigma_0$ and $\kappa_0$ as functions of $\log(\rho)$ and $\log(T)$ , valid within the calculated regime.
Magnetic Field Effects:
The study examined the suppression of perpendicular transport coefficients under magnetic fields using the Drude model and the Epperlein–Haines (EH) framework.
- At high temperatures ( $T=500$ eV) and strong magnetic fields ( $B=10^4$ T), the Drude model predicts a scaling factor of $\sim 0.34$ for electrical conductivity.
- However, the EH framework, which accounts for fully ionized plasma effects, predicts significantly stronger suppression, with scaling factors as low as 0.06 (19% of the Drude value). This indicates that classical Drude scaling may overestimate transport perpendicular to the magnetic field in these regimes.

Significance
The paper demonstrates that standard model-based approaches (TFD, IONEQ, Lee–More) can introduce substantial errors (exceeding 10–30% in pressure and up to 63% in transport coefficients) when simulating warm-dense aluminum at high densities and temperatures below 200 eV. By providing ab initio data up to 1000 eV, the authors supply critical input for radiation-hydrodynamics codes used in Z-pinch and inertial confinement fusion research. The constructed EOS tables and fitted transport coefficients offer a more accurate foundation for modeling the evolution of metallic liners through warm-dense states, particularly where model discrepancies affect the prediction of compression maxima and shock profiles.

Technical Summary: Equation of State and Transport Coefficients of Warm Dense Aluminum from Mixed Deterministic–Stochastic Density Functional Theory

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