Fate of Boltzmann’s Breathers: Stokes Hypothesis and Anomalous Thermalization

Boltzmann showed that in spite of momentum and energy redistribution through collisions, a rarefied gas confined in a isotropic harmonic trapping potential does not reach equilibrium; it evolves instead into a breathing mode where density, velocity, and temperature oscillate. This counterintuitive prediction is upheld by cold atoms experiments. Yet, are the breathers eternal solutions of the dynamics even in an idealized and isolated system? We show by a combination of hydrodynamic arguments and molecular dynamics simulations that an original dissipative mechanism is at work, where the minute and often neglected bulk viscosity eventually thermalizes the system, which thus reaches equilibrium.

Figure 1

Sketch of the system. The particles, shown with the disks, are confined in the (red) parabolic potential. Under generic initial conditions, the system evolves toward a breathing state, oscillating between a dense configuration with high temperature (left) and a more dilute configuration with a smaller temperature and a higher potential energy (right).

Figure 2

$⟨r^2⟩/{\rho }^2$ as a function of the dimensionless time, $\omega t$, for $\varphi =9\times {10}^{-3}$. The circles are the simulation results and the solid line the Boltzmann theoretical prediction (8). In the inset, the envelope of the oscillations (dashed line) is plotted on a much longer timescale (the dotted line at unity is plotted for reference).

Figure 3

${(\mathrm{\Omega }/\omega )}^2$ as a function of the dimensionless density, $\varphi$. The dots are the simulation results and the solid line is the theoretical prediction (13).

Figure 4

$\omega \tau$ as a function of the dimensionless density, $\varphi$, in logarithmic scale. The points are the simulation results, the solid line is the theoretical prediction for $\tau$ as defined above (13), and the dashed line is the linear fitting of the simulation results.