[Paper Review] Blast in the one-dimensional cold gas: From Newton to Euler and Navier-Stokes

Subhadip Chakraborti, Santhosh Ganapa+2 more|arXiv (Cornell University)|Feb 16, 2021

Fluid Dynamics and Turbulent Flows4 citations

TL;DR

This study tests the hydrodynamic limit of a one-dimensional cold gas by simulating a blast-wave using hard-point particles with alternating masses, demonstrating near-perfect agreement with Euler and Navier-Stokes hydrodynamics. Deviations in a small core region are attributed to heat conduction, providing strong numerical evidence for the emergence of continuum hydrodynamics from Newtonian mechanics.

ABSTRACT

It is known that a gas composed of a large number of atoms following Newtonian dynamics can be described by the continuum laws of hydrodynamics. Proving this rigorously is one of the outstanding open problems in physics and mathematics. Surprisingly, precise numerical demonstrations of the equivalence of the hydrodynamic and microscopic descriptions are rare. We test this equivalence in the context of the classic problem of the evolution of a blast-wave, a problem that is expected to be at the limits where hydrodynamics would work. We study a one-dimensional gas for which the hydrodynamic Euler equations for the conserved fields of density, momentum and energy are known to have self-similar scaling solutions. Our microscopic model consists of hard point particles with alternate masses, which is a non-integrable system with strong mixing dynamics. Our extensive microscopic simulations find a remarkable agreement with hydrodynamics, with deviations in a small core region that are understood as arising due to heat conduction.

Motivation & Objective

• To test the emergence of continuum hydrodynamics from Newtonian particle dynamics in a non-integrable system.
• To investigate the validity of hydrodynamic descriptions—Euler and Navier-Stokes—under extreme conditions like blast-wave evolution.
• To quantify discrepancies between microscopic simulations and hydrodynamic predictions, particularly in regions of strong gradients.
• To assess the role of heat conduction in explaining deviations from ideal hydrodynamics in finite systems.

Proposed method

• Simulate a one-dimensional gas of hard-point particles with alternating masses to induce strong mixing and non-integrability.
• Implement Newtonian dynamics for all particles using event-driven molecular dynamics to track collisions and trajectories.
• Evolve the system from an initial blast-wave configuration with localized energy and density perturbation.
• Extract macroscopic fields (density, momentum, energy) from particle data and compare with self-similar solutions of the Euler and Navier-Stokes equations.
• Apply spatial and temporal coarse-graining to bridge microscopic particle data with continuum fields.
• Use a viscous-heat-conduction model to explain small-scale deviations in the core region of the blast wave.

Experimental results

Research questions

• RQ1 To what extent do the macroscopic fields of a 1D many-body system with Newtonian dynamics match the predictions of the Euler equations?
• RQ2 How do deviations between microscopic simulations and hydrodynamic models scale with system size and resolution?
• RQ3 What physical mechanisms account for discrepancies in the central core region of the blast wave?
• RQ4 Can heat conduction effects explain the observed deviations from ideal hydrodynamics in the simulation?
• RQ5 Does the system exhibit self-similar behavior consistent with the known hydrodynamic scaling solutions?

Key findings

• The microscopic simulations show excellent agreement with the self-similar solutions of the Euler equations across the entire system, validating the hydrodynamic limit in a non-integrable 1D gas.
• Small deviations from hydrodynamics appear only in a narrow core region, indicating breakdown of ideal hydrodynamics under extreme gradients.
• These deviations are quantitatively explained by the inclusion of heat conduction in the Navier-Stokes framework, confirming its role in dissipative corrections.
• The system exhibits strong mixing dynamics due to alternating particle masses, supporting the ergodicity required for hydrodynamic emergence.
• The agreement persists even in the absence of integrability, suggesting robustness of hydrodynamic behavior in low-dimensional systems.