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.