Chapman-Enskog expansion for chirally colliding disks

Abstract

We study a two-dimensional fluid of hard disks undergoing chiral, energy- and momentum-conserving collisions. We show that despite the microscopic breaking of time-reversal symmetry, the H-theorem is obeyed, guaranteeing a relaxation towards equilibrium in the absence of external forces. In the dilute limit, a Chapman-Enskog expansion yields analytical expressions for the shear and odd viscosity and the thermal conductivity. Theoretical predictions are confirmed by nonequilibrium molecular dynamics simulations.

Figures (6)

• Figure 1: Depiction of collision with chiral effective particle radii $r (1 \pm \varepsilon)$. $\mathbf{g}$ is the relative velocity, $\mathbf{n}$ is the vector connecting the centers of the colliding objects, $\mathbf{g}'$ is the outgoing relative velocity, and $\chi$ is the scattering angle.
• Figure 2: Schematic picture of two ratchet-shaped particles moving towards each other in two chirally distinct ways.
• Figure 3: Scattering cross section as a function of scattering angle $\chi$ corresponding to the numerical collision experiment of ratchet-shaped particles (see Fig. \ref{['fig:placeholder123123']}). The particles follow the biased Maxwell-Boltzmann statistic of Eq. (\ref{['eq:MB_distr']}) with $\Omega = 0.2\sqrt{2k_bT/I}$. The red line corresponds to the corresponding scattering cross section for the collision rule with chiral effective radii (see Fig. \ref{['fig:placeholder']}), where we fitted $r=1.79$ and $\varepsilon = 0.09$. For more details see App. \ref{['eq:numericalcollisionexperiment']}.
• Figure 4: Extrapolation of numerically obtained shear viscosity $\eta_{{\text{e}}}$ (top) and odd viscosity $\eta_{{\text{o}}}$ (bottom) to infinitesimal shear rate $\gamma$ for $\varepsilon =0.5$.
• Figure 5: Shear viscosity $\eta_{\mathrm{e}}$ (top) and odd viscosity $\eta_{\mathrm{o}}$ (bottom) as a function of ${k_\text{b}} T$ for $\varepsilon =0.5$. The numerical results were obtained using the extrapolation shown in Fig. \ref{['fig:result1']}.