Minimal Model of Stochastic Athermal Systems: Origin of Non-Gaussian Noise

For a wide class of stochastic athermal systems, we derive Langevin-like equations driven by non-Gaussian noise, starting from master equations and developing a new asymptotic expansion. We found an explicit condition whereby the non-Gaussian properties of the athermal noise become dominant for tracer particles associated with both thermal and athermal environments. Furthermore, we derive an inverse formula to infer microscopic properties of the athermal bath from the statistics of the tracer particle. We apply our formulation to a granular motor under viscous friction and analytically obtain the angular velocity distribution function. Our theory demonstrates that the non-Gaussian Langevin equation is the minimal model of athermal systems.

Figure 1

Schematic of a system driven by thermal ${\widehat{F}}_T(t;\widehat{v})$ and athermal ${\widehat{F}}_A(t;\widehat{v})$ forces. There is net energy current $J$ from the athermal to the thermal environment.

Figure 2

Typical trajectories depending on $\gamma$: (a) For $\gamma =0$, the athermal force is relevant to both fluctuation and dissipation (Diss.), where the non-Gaussianity is irrelevant. (b) For $\gamma \gg {\gamma }_A$, the athermal fluctuation is irrelevant to dissipation because of the dominance of the thermal friction. The athermal non-Gaussianity then becomes relevant.

Figure 3

(a) Schematic of a rotor driven by the thermal force ${\widehat{F}}_T(t;\widehat{\omega })$ from the viscous fluid and the athermal force ${\widehat{F}}_A(t;\widehat{\omega })$ caused by collisions of the granular particles. The heat current $J$ flows from the granular gas to the viscous fluid. We fix parameters as $M=I=h=e=\rho =v_0=1$, $T=l=0$, $m=0.01$, and $w=\sqrt{12}$ for numerical simulations. (b) Collisional rules are illustrated. (c) Numerical demonstration of the emergence of $\kappa \equiv ⟨{\widehat{\omega }}^4⟩/⟨{\widehat{\omega }}^2{⟩}^2-3$ corresponding to the increases of the viscosity $\gamma$.

Figure 4

(a) Steady PDF of the angular velocity $\widehat{\omega }$ obtained from the Monte Carlo simulation of the Boltzmann-Lorentz equation (9) for $\gamma =2$ (cross points), our theoretical line (11) (solid line), and the conventional Gaussian theory (5) (dashed line). (b) The granular VDF estimated from ${\mathcal{P}}_{\mathrm{SS}}(\mathrm{\Omega })$ using Eq. (12). Note that the accuracy of the point at $v\simeq 0.5$ is not good because of the singularity in Eq. (12) around $v=0$.