Generalized Edwards thermodynamics and marginal stability in a driven system with dry and viscous friction

We consider a spring-block model with both dry and viscous frictions, subjected to a periodic driving allowing mechanically stable configurations to be sampled. We show that under strong driving, the scaling of the correlation length with the energy density is incompatible with the prediction of the Edwards statistical approach, which assumes a uniform sampling of mechanically stable configurations. A crossover between the Edwards scaling and nonstandard high-energy scaling is observed at energy scales that depend on the viscous friction coefficient. Generalizing Edwards thermodynamics, we propose a statistical framework, based on a sampling of marginally stable states, that is able to describe the scaling of the correlation length in the highly viscous regime.

Figure 1

(a) Correlation length $\lambda$ as a function of the average energy density $\varepsilon$ of the sampled MSCs, for different values of the viscous friction coefficient $\gamma$, indicated by different symbols. Dashed lines emphasize the linear ($\lambda \sim \varepsilon$) and square-root ($\lambda \sim \sqrt{\varepsilon }$) behaviors reached for low and high energies, respectively. (b) First rescaling $\gamma \lambda =F_1\left(\gamma \varepsilon \right)$ around the departure from the linear regime. (c) Second rescaling $\gamma \lambda =F_2\left({\gamma }^2\varepsilon \right)$ around the onset of the square-root regime.

Figure 2

Mean-square displacement of spring elongation, $\langle {\left[\mathrm{\Delta }\xi \right]}^2\left(r\right)\rangle =\langle {[{\xi }_{i+r}-{\xi }_i]}^2\rangle$. (a) When only dry friction is present ($\gamma =0$), the mean-square displacement is diffusive, $\langle {\left[\mathrm{\Delta }\xi \right]}^2\left(r\right)\rangle \sim r$. (b) For strong enough viscous friction, the mean-square displacement is ballistic, $\langle {\left[\mathrm{\Delta }\xi \right]}^2\left(r\right)\rangle \sim r^2$ ($\gamma =0.3$). Data are collapsed by plotting $\langle {\left[\mathrm{\Delta }\xi \right]}^2\left(r\right)\rangle /\varepsilon$ as a function of the rescaled distance $r/\lambda \left(\varepsilon \right)$, with $\lambda \left(\varepsilon \right)$ the correlation length.

Figure 3

(a) A typical MSC sampled at high energy in the presence of viscous friction. The total force $f_i^{\mathrm{el}}={\xi }_{i+1}-{\xi }_i$ acting on each mass is plotted as a function of the mass index $i$ ($\mu =0.6$). (b) Correlation of the elastic force $f_i$ at the end of the driving phase, before relaxation takes place. (c) Correlation length $\lambda$ of spring elongations, a function of the energy $\varepsilon$, as obtained from the transfer operator method. A Gaussian approximation (with standard deviation $\sigma =0.5$) of the delta function has been used.