Paper Summary

Citation: Rahman, A. (1964). Correlations in the Motion of Atoms in Liquid Argon. Physical Review, 136(2A), A405–A411. https://doi.org/10.1103/PhysRev.136.A405

Publication: Physical Review, 1964

What kind of paper is this?

This is a foundational “big idea” and method paper. It is widely considered the birth of molecular dynamics (MD) simulation as a tool for studying condensed matter. It’s a landmark “computer experiment” that demonstrated the feasibility and power of numerically integrating Newton’s equations of motion for a system of interacting particles to gain fundamental insights into the behavior of liquids. For scientific AI, this paper represents a proto-example of generating high-fidelity, first-principles data to test and falsify simpler analytical models.

What is the motivation?

The motivation was to create a computational microscope to directly observe atomic motion in a liquid. At the time, physicists relied on analytical theories and approximations (like the Vineyard convolution approximation) to interpret experimental data from neutron scattering, which probes the time-dependent correlations between atoms. Rahman’s goal was to use a computer to generate an “exact” numerical solution for a realistic model of liquid argon, thereby creating a ground-truth dataset to directly test the validity of these theoretical models and provide a clearer picture of liquid dynamics than was possible experimentally.

What is the novelty here?

The core novelty was the simulation of a liquid via direct integration of the classical N-body problem on a digital computer, a monumental task in 1964. This work introduced or popularized several key techniques that are now standard in scientific computing:

  1. Large-Scale Simulation: Solving the coupled equations of motion for a system of 864 particles interacting via a Lennard-Jones potential.
  2. Periodic Boundary Conditions: A clever method to simulate a small, computationally tractable number of particles in a way that mimics an infinite bulk fluid, eliminating surface effects.
  3. Direct Calculation of Time-Correlation Functions: This was the first time that fundamental statistical quantities like the velocity autocorrelation function and the Van Hove space-time correlation functions ($G_s(r, t)$ and $G_d(r, t)$) were computed directly from a particle-based simulation. These functions describe the evolution of the probability distribution of particle positions and are central to the theory of liquids.

What experiments were performed?

The “experiment” was the molecular dynamics simulation itself. The procedure involved:

  1. System Definition: 864 particles with the mass of argon atoms were placed in a cubic box with a density matching liquid argon at 94.4 K. The particles interacted via a truncated Lennard-Jones 6-12 potential.
  2. Numerical Integration: The classical equations of motion were solved using a finite-difference algorithm (a predictor-corrector method) with a time step of $10^{-14}$ seconds on a CDC 3600 computer.
  3. Data Collection and Analysis: The positions and velocities of all particles at successive time steps were recorded. This trajectory data was then post-processed to compute:
    • Static Properties: The pair-correlation function $g(r)$.
    • Dynamic Properties: The mean-square displacement $\langle r^2(t) \rangle$, the velocity autocorrelation function $\langle \mathbf{v}(0) \cdot \mathbf{v}(t) \rangle$, and the self and distinct parts of the Van Hove correlation function, $G_s(r, t)$ and $G_d(r, t)$.
  4. Validation: The simulation’s outputs, such as $g(r)$ and the self-diffusion constant derived from $\langle r^2(t) \rangle$, were compared with experimental results from X-ray scattering and diffusion measurements to confirm the physical realism of the model.

What were the outcomes and conclusions drawn?

The simulation successfully reproduced key experimental properties of liquid argon, validating MD as a powerful scientific tool. More importantly, it revealed new physics that challenged existing simple theories of liquids:

  • Complex Atomic Motion: The velocity autocorrelation function was not a simple exponential decay (as in Langevin theory for Brownian motion). Instead, it became negative after ~0.3 picoseconds, directly showing that an atom, on average, “rebounds” off the cage formed by its neighbors. Its Fourier spectrum showed a broad peak, indicating quasi-oscillatory behavior, unlike a gas but not as sharp as in a crystal.

  • Non-Gaussian Diffusion: The function describing the probability of a particle’s displacement, $G_s(r, t)$, was shown to be Gaussian only at very short and very long times. At intermediate times ($t \approx 3 \times 10^{-12}$ s), it showed significant non-Gaussian behavior, a key feature of diffusion in dense fluids.

  • Failure of a Key Approximation: The simulation demonstrated that the widely used Vineyard convolution approximation for $G_d(r, t)$ was flawed, as it predicted a much-too-rapid decay of the liquid’s structural order. Rahman proposed an improved “delayed-convolution” approximation based on the simulation data.

In essence, Rahman’s work showed that a direct numerical simulation could not only validate experimental data but also provide unprecedented microscopic insight, revealing the shortcomings of existing analytical theories and paving the way for decades of computational physics and chemistry.

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