Impurity dynamics in a zero-temperature gas

TL;DR

This work analyzes impurity dynamics in a zero-temperature hard-sphere gas subjected to an infinitely strong blast. Using hydrodynamic and kinetic-theory reasoning, it derives core-region scalings for the impurity: the typical impurity displacement obeys $R_{\rm imp}=\lambda\left(R/\lambda\right)^{h_d}$ with $h_d=(4+3d^2)/(8+3d^2)$, the typical speed scales as $V_{\rm imp}=\sqrt{E}\,(a/\lambda)^{(d-1)/2}(\lambda/R)^{\omega_d}$ with $\omega_d = d/2 - d^2/(8+3d^2)$, and the number of collisions scales as $C_{\rm imp}=(R/\lambda)^{(8+2d^2)/(8+3d^2)}$; in 2D these give $R_{\rm imp}\sim t^{2/5}$, $V_{\rm imp}\sim t^{-2/5}$, and $C_{\rm imp}\sim t^{2/5}$. The impurity remains inside a dissipative core whose size scales as $H=\lambda\left(R/\lambda\right)^{h_d}$, with core density $n_0$ and temperature $T_0$ set by kinetic theory arguments. The spatial distribution of the impurity is argued to be Gaussian with variance $R_{\rm imp}^2$, while a closed Lorentz-Boltzmann equation governs the full joint distribution $\Pi({\bf r},{\bf v},t)$, though this equation is analytically intractable. Numerical simulations in 2D corroborate the predicted scalings and Gaussian form, linking macroscopic blast behavior to mesoscopic impurity diffusion within the core.

Abstract

If energy is suddenly released in a localized region of space uniformly filled with identical stationary hard spheres, the outcome is a blast with an asymptotically spherical shock wave separating moving and stationary hard spheres. The radius $R(t)$ of the region filled with the moving spheres grows as $t^{2/(d+2)}$, where $d$ is the spatial dimension. The simplest way to inject energy is to kick a few `impurity' particles. Using hydrodynamics and kinetic theory, we argue that the typical displacement of an impurity scales as $R_{\rm imp} \sim λ(R/λ)^{(4+3d^2)/(8+3d^2)}$, where $λ$ is the mean-free path in the initial state. The number of collisions experienced by each impurity grows as $(R/λ)^{(8+2d^2)/(8+3d^2)}$, while its average speed decreases as $t^{-d(8-2d+3d^2)/[(2+d)(8+3d^2)]}$. In $2D$, the predictions for impurity displacement, collision numbers, and speed are $t^{2/5},~t^{2/5}$ and $t^{-2/5}$, respectively. These predictions are in reasonable agreement with the results of molecular dynamics simulations.

Figures (3)

• Figure 1: Blast in a box of size with $20000$ particles at density $n_{\infty}=0.15$ (packing fraction $\phi=0.118$). $(A), (B), (C)$ and $(D)$ depicts the moving ($\bullet$) and stationary particles ($\bullet$) at times $t=500, 1000, 1500$ and $2000$ respectively in a single realization. Also shown are the trajectories of four impurity particles. Eq. (\ref{['eq:Rimp_comp']}) is used to plot the blue circles in $(A)-(D)$.
• Figure 2: Time evolution of various impurity and fluid properties plotted on log-log scale in case of a $2$-dimensional hard-sphere gas: (A) impurity displacement $R_{\text{imp}}$; (B) impurity speed $V_{\text{imp}}$; (C) impurity collision count $C_{\text{imp}}$; (D) fluid speed around the impurity $u_{\text{imp}}$; (E) central density $n_0$; (F) central temperature $T_0$. All quantities are asymptotically expected to have power-law time dependence of the form $t^\alpha$ and the insets show the running temporal exponents plotted with time (see Sec. (\ref{['sec:test']}) for details). The dark dashed lines correspond to the theoretical predictions. All figures and their insets include error bars. However, the estimated statistical uncertainties are smaller than the symbol size and the line thickness used in the plots; as a result, the error bars are not visually discernible.
• Figure 3: Scaled radial probability distribution function $(p_{\rm imp})$ for impurity (obtained from simulation) plotted at different times. The dashed curve in the figure is plotted by fitting Eq. (\ref{['eq:pr']})) with $R_{\text{imp}}=b~t^{2/5}$ against the PDF obtained from simulation data at $t=4000$. The parameter $b$ obtained by fitting is $1.75$ which is close to $b=1.71$, obtained directly from computation of $R_{imp}/t^{2/5}$ at $t=4000$.