Blast in a One-Dimensional Cold Gas: From Newtonian Dynamics to Hydrodynamics

A gas composed of a large number of atoms evolving according to Newtonian dynamics is often described by continuum hydrodynamics. Proving this rigorously is an outstanding open problem, and precise numerical demonstrations of the equivalence of the hydrodynamic and microscopic descriptions are rare. We test this equivalence in the context of the evolution of a blast wave, a problem that is expected to be at the limit where hydrodynamics could work. We study a one-dimensional gas at rest with instantaneous localized release of energy for which the hydrodynamic Euler equations admit a self-similar scaling solution. Our microscopic model consists of hard point particles with alternating masses, which is a nonintegrable system with strong mixing dynamics. Our extensive microscopic simulations find a remarkable agreement with Euler hydrodynamics, with deviations in a small core region that are understood as arising due to heat conduction.

Figure 1

Heat maps showing the spatiotemporal evolution of the density, velocity, and temperature fields, starting from initial conditions corresponding to a Gaussian initial temperature profile and $\rho (x,0)={\rho }_{\infty }=1.5,v(x,0)=0$. The simulation parameters were $N=L=24000$, $E=32$, $\mu =1.5$, and an ensemble average was taken over ${10}^4$ initial conditions.

Figure 2

(a),(b),(c) Molecular dynamics results for the time evolution of density, velocity, and temperature fields, starting from the initial conditions corresponding to a Gaussian initial temperature profile and $\rho (x,0)={\rho }_{\infty }=1.5,v(x,0)=0$. The simulation parameters were $N=L=24000$, $E=32$, $\mu =1.5$ and an average was taken over ${10}^4$ initial conditions. (d), (e), (f) This shows the $x/t^{2/3}$ scaling of the data. We observe a very good collapse of the data at the longest times and an agreement with the TvNS scaling solution (dashed line).

Figure 3

(a),(b),(c) Evolution of $\rho (x,t),v(x,t),T(x,t)$ obtained from a numerical solution of the NSF equations (5a)–(5c), starting from the same initial conditions as used in the simulations for Fig. 2. The other parameters used in the numerics are $D_1=1$, $D_2=1$ and $L=4000$. (d), (e), (f) Scaling plot and comparison with the TvNS solution.

Figure 4

Comparison of the scaled fields $\tilde{\rho },\tilde{T}$ from microscopic simulations (at time $t=80000$), the NSF equations (at time $t=6400$), and the vNS scaling solution.

Figure 5

Verification of the scaling form Eqs. (8), for $\rho$ and $T$, in the core of the blast. The comparison with data both from simulations and from the NSF solution are shown. The data are the same as in Figs. 2, 3. The value $\eta =10x/t^{38/93}$ corresponds to $\xi \approx 0.2$.