Force-Force Correlator for Driven Disordered Systems at Finite Temperature

When driving a disordered elastic manifold through quenched disorder, the pinning forces exerted on the center of mass are fluctuating, with mean $f_c=-\overline{F_w}$ and variance $\Delta(w)=\overline{F_w F_0}^c$, where $w$ is the externally imposed control parameter for the preferred position of the center of mass. $\Delta(w)$ was obtained via the functional renormalization group in the limit of vanishing temperature $T\to 0$, and vanishing driving velocity $v\to 0$. There are two fixed points, and deformations thereof, which are well understood: The depinning fixed point ($T\to 0$ before $v\to 0$) rounded at $v>0$, and the zero-temperature equilibrium fixed point ($v\to 0$ before $T\to 0$) rounded at $T>0$. Here we consider the whole parameter space of driving velocity $v>0$ and temperature $T>0$, and quantify numerically the crossover between these two fixed points.