Extended Data Fig. 1: Thermodynamics of an oscillating granular gas.
a. Schematic of the system setup. We simulate a quasi-2D granular gas composed of frictional particles, which are forced to oscillate vertically at a constant frequency f but free to move horizontally. Interparticle collision between two oscillating particles could lead to their translational motions in the xy-plane. In the middle is a zoomed-in, top view of this many-body system. Horizontal translation of a particle is denoted by its tail, whereas its vertical oscillation is color-coded in the tail: gradient from a dark end to a bright front means the particle is moving towards the xy-plane, vice versa; purple denotes z < 0 whereas red denotes z > 0. Δt denotes the averaged collision duration. b. Maxwell distribution. The x-component of translational velocity displays a Gaussian distribution P(vx) at various oscillating frequency f. An effective temperature Teff is defined using the halfwidth of P(vx). Dependence of Teff on f is shown on the right. c. Boltzmann distribution. We put the system in a potential well \(U({{{{{\bf{r}}}}}})=-0.5{k}_{{{{{\mathrm{B}}}}}}{T}_{{{{{\mathrm{eff}}}}}}\ \left[1+\,{{{{\mathrm{cos}}}}}\,(\pi r/R)\right]\) for r < R, where r denotes the distance from the center of the system. The resultant spatial distribution of the particles turns out to follow the Boltzmann statistics \(n(-r)\propto \,{{{{\mathrm{exp}}}}}\,\left[-U(r)/{k}_{{{{{\mathrm{B}}}}}}{T}_{{{{{\mathrm{eff}}}}}}\right]\) (purple curve) as well. d. Green–Kubo relation. Shear viscosity of this many-body system can be either directly measured using linear response towards an applied shear or indirectly inferred from the Green–Kubo relation by calculating the integral of the stress–stress correlation function, known as the Green–Kubo relation. The predicted and measured shear viscosity η is compared at a wide range of frequencies f. The Kubo predictions with Teff and renormalized \({T}_{\,{{{{\mathrm{eff}}}}}\,}^{* }\) are marked as the dashed and solid lines, respectively. We have defined f0 = 1/Δt.
