Thermal Conductivity of the Lennard-Jones Fluid

Paper Information

Citation: Evans, D. J. (1986). Thermal conductivity of the Lennard-Jones fluid. Physical Review A, 34(2), 1449-1453. https://doi.org/10.1103/PhysRevA.34.1449

Publication: Physical Review A, 1986

What kind of paper is this?

This is primarily a Methodological Paper ($\Psi_{\text{Method}}$), with a significant secondary component of Discovery ($\Psi_{\text{Discovery}}$).

It focuses on validating a specific algorithm (the “Evans method”) for Non-Equilibrium Molecular Dynamics (NEMD) by comparing its results against experimental benchmarks. However, it also uncovers physical anomalies - specifically “long-time tails” in the heat flux autocorrelation function that deviate significantly from theoretical predictions - marking a discovery about the physics of the Lennard-Jones fluid itself.

What is the motivation?

The primary motivation is to overcome the limitations of simulating heat flow using physical boundaries (e.g., walls at different temperatures), which causes severe interpretive difficulties due to density and temperature gradients.

The “Evans method” uses a fictitious external field to induce heat flow in a periodic, homogeneous system. This paper serves to:

1. Validate this method across a wide range of state points (temperatures and densities) beyond the triple point.
2. Investigate the system’s behavior near the critical point, where transport properties are known to be anomalous.

What is the novelty here?

The core contribution is the rigorous stress-testing of the homogeneous heat flow algorithm (Evans method) combined with a Gaussian thermostat.

Specific novel insights include:

• Linearity Validation: Establishing that, away from phase boundaries, the effective thermal conductivity is a monotonic, linear function of the external field, justifying the extrapolation to zero field.
• Critical Anomaly Detection: Finding that near the critical point, conductivity becomes a non-monotonic function of the field, challenging standard simulation approaches in this regime.
• Tail Amplitude Discovery: Demonstrating that the “long-time tails” of the heat flux autocorrelation function have amplitudes roughly an order of magnitude larger than those predicted by mode-coupling theory.

What experiments were performed?

The author performed Non-Equilibrium Molecular Dynamics (NEMD) simulations using the Lennard-Jones potential.

• System: Mostly $N=108$ particles, with some checks using $N=256$ to test size dependence.
• Thermostat: A Gaussian thermostat was used to keep the kinetic energy (temperature) constant.
• State Points:
  • Critical Isotherm: $T=1.35$, varying density.
  • Supercritical Isotherm: $T=2.0$.
  • Freezing Line: Two points ($T=2.74, \rho=1.113$ and $T=2.0, \rho=1.04$).
• Validation: Results were compared against experimental data for Argon (using standard LJ parameters).
• Ablation:
  • Field Strength ($F$): Varied to check for linearity/non-linearity.
  • System Size ($N$): Comparison between 108 and 256 particles to rule out finite-size artifacts.

What were the outcomes and conclusions drawn?

• Agreement with Experiment: The Evans method yields thermal conductivities in broad agreement with experimental Argon data for most state points.
• Linearity: Away from the critical point, conductivity is linearly dependent on the field strength $F$, allowing for accurate zero-field extrapolation.
• Critical Region Failure: Near the critical point ($T=1.35, \rho=0.4$), the method struggles; the conductivity is non-monotonic with respect to $F$, and the zero-field extrapolation underestimates the experimental value by ~11%.
• Long-Time Tails: The decay of the heat flux autocorrelation function follows a $t^{-3/2}$ tail (consistent with mode-coupling theory), but the amplitude is ~6x larger than predicted.
• Phase Hysteresis: In high-density regions near the freezing line, the system exhibits hysteresis and bi-stability between solid and liquid phases depending on the field strength.

Reproducibility Details

Data

The simulation relies on the Lennard-Jones (LJ) potential to model Argon. No external training data is used; the “data” consists of the physical constants defining the system.

Algorithms

The core algorithm is the Evans Homogeneous Heat Flow method. To reproduce this, one must implement the specific Equations of Motion (EOM) derived from linear response theory.

Equations of Motion:

The trajectories are generated by: $$\dot{q}_i = \frac{p_i}{m}$$ $$\dot{p}_i = F_i^{\text{inter}} + (E_i - \bar{E})F(t) - \sum_{j} F_{ij} q_{ij} \cdot F(t) + \frac{1}{2N} \sum_{j,k} F_{jk} q_{jk} \cdot F(t) - \alpha p_i$$

Where:

• $F(t)$ is the fictitious external field driving heat flow.
• $E_i$ is the instantaneous energy of particle $i$.
• $\alpha$ is the Gaussian Thermostat multiplier (calculated at every step to strictly conserve kinetic energy/Temperature): $$\alpha = \frac{\sum_i [\dots]_{\text{force terms}} \cdot p_i}{\sum_i p_i \cdot p_i}$$

Conductivity Calculation:

• Zero-Frequency: $\lambda = \lim_{F \to 0} (J_Q / FT)$.
• Frequency-Dependent: $\lambda(\omega) = \frac{V}{3k_B T^2} \int_0^\infty dt \, e^{i\omega t} \langle J_Q(t) \cdot J_Q(0) \rangle$.

Models (Physical System)

The “model” here is the physical simulation setup.

• Particle Count: $N = 108$ (primary), $N = 256$ (validation).
• Boundary Conditions: Periodic Boundary Conditions (PBC).
• Thermostat: Gaussian Isokinetic (Temperature is a constant of motion, not just an average).

Evaluation

The primary metric is the Thermal Conductivity ($\lambda$).

Hardware

• Requirements: While 1986 hardware is obsolete, reproducing this requires a standard MD code capable of non-conservative forces (NEMD).
• Compute Cost: Low by modern standards. 108 particles for $\sim 10^5 - 10^6$ steps is trivial on modern CPUs.