Breakdown of hydrodynamics in a one-dimensional cold gas

Abstract

The following model is studied analytically and numerically: point particles with masses $m,μ,m, \dots$ ($m\geqμ$) are distributed over the positive half-axis. Their dynamics is initiated by giving a positive velocity to the particle located at the origin; in its course the particles undergo elastic collisions. We show that, for certain values of $m/μ$, starting from the initial state where the particles are equidistant the system evolves in a hydrodynamic way: (i) the rightmost particle (blast front) moves as $t^δ$ with $δ< 1$; (ii) recoiled particles behind the front enter the negative half-axis; (iii) the splatter -- the particles with locations $x\leq 0$ -- moves in the ballistic way and eventually takes over the whole energy of the system. These results agree with those obtained in S. Chakraborti et al, SciPost Phys. 2022, 13, 074, for $m/μ=2$ and random initial particle positions. At the same time, we explicitly found the collection of positive numbers $\{\mathcal{M}_i, i \in \mathbf{N} \}$ such that, for $m/μ= \mathcal{M}_i$, $i\leq 700$, the following holds: (a) the splatter is absent; (b) the number of simultaneously moving particles is at most three; (c) the blast front moves in the ballistic way. However, if, similarly as in S. Chakraborti et al, the particle positions are sampled from a uniformly distributed ensemble, for $m/μ= \mathcal{M}_i$ the system evolves in a hydrodynamic way.

Figures (4)

• Figure 1: The total number of collisions $\mathcal{C}_{\rm fin}$ as a function of $m$. The scattered blue dots correspond to the resolution $\Delta m=0.01$; black circles correspond to integer values of $m$; red disks correspond to $m=\mathcal{M}_i$.
• Figure 2: The normalized energy $\varepsilon(m) =2\mathcal{E}_{\rm fin}/m$ as a function of $m$. Symbols are as in Fig. \ref{['fig1']}. Red discs correspond to $\varepsilon(\mathcal{M}_i)=1$.
• Figure 3: Dependence of $\zeta_i$ on $i\leq 700$, see (\ref{['eq_cond']}).
• Figure 4: Blast front $\mathcal{R}(t)$ for some values of masses $m$, cf \ref{['Obs']}. The fitted values of $\delta$ are given in Table \ref{['tab1']}. Dashed line corresponds to the ballistic evolution with $\delta=1$. Dotted line serves as an eye guide to show the hydrodynamics evolution with $\delta \simeq 0.628$.