Large Fluctuations in Driven Dissipative Media

We analyze the fluctuations of the dissipated energy in a simple and general model where dissipation, diffusion, and driving are the key ingredients. The full dissipation distribution, which follows from hydrodynamic fluctuation theory, shows non-Gaussian tails and no negative branch, thus violating the fluctuation theorem as expected from the irreversibility of the dynamics. It exhibits simple scaling forms in the weak- and strong-dissipation limits, with large fluctuations favored in the former case but strongly suppressed in the latter. The typical path associated with a given dissipation fluctuation is also analyzed in detail. Our results, confirmed in extensive simulations, strongly support the validity of hydrodynamic fluctuation theory to describe fluctuating behavior in driven dissipative media.

Figure 1

Scaling of the dissipation LDF in the quasielastic limit ($\nu \ll 1$) for $N=50$, $T=1$, and varying $\nu$. The solid and dashed lines are the HFT prediction and Gaussian estimation, respectively. Small points around the peak were obtained in standard simulations. Inset: Convergence of time-averaged dissipation and sketch of the probability concentration associated with the large deviation principle.

Figure 2

Optimal energy profiles for varying $d/⟨d⟩$ measured for $\nu ={10}^{-3}$ (symbols) and $\nu ={10}^{-2}$ (dashed lines) and HFT predictions (solid lines and inset).

Figure 3

Top: Dissipation LDF measured for increasing values of $\nu$ [shifted vertically for convenience, $G(⟨d⟩)=0\forall \nu$]. Open symbols correspond to $N=50$ and $M={10}^3$ clones, while for filled symbols $N=100$ and $M={10}^4$. Solid lines are HFT predictions obtained by numerical integration of Eq. (5). Notice the (slow) convergence toward HFT as $N$ and $M$ increase. Inset: Measured average dissipation, its variance as a function of $\nu$, and Gaussian estimation. Bottom: Optimal energy profiles measured for $\nu =10$, $N=50$, and $M={10}^3$ (thick lines) and HFT theory.