Edwards Thermodynamics for a Driven Athermal System with Dry Friction

We obtain, using semianalytical transfer operator techniques, the Edwards thermodynamics of a one-dimensional model of blocks connected by harmonic springs and subjected to dry friction. The theory is able to reproduce the linear divergence of the correlation length as a function of energy density observed in direct numerical simulations of the model under tapping dynamics. We further characterize analytically this divergence using a Gaussian approximation for the distribution of mechanically stable configurations, and show that it is related to the existence of a peculiar infinite temperature critical point.

Figure 1

(a),(b),(c) Correlation functions $C(|i-j|)$ for different tapping protocols, while in each panel the different curves correspond to different energy values. (a) $F\in [20,128]$, $\rho =0.3$, $\sigma =0$; (b) $F\in [20,140]$, $\rho =0.8$, $\sigma =0$; (c) $F\in [20,128]$, $\rho =1$, $\sigma =F/4$. $\tau =60$ in all simulations. Inset $C(|i-j|)$ vs $|i-j|/\ell (e)$, showing a good collapse of the different curves. (d) Correlation length $\ell (e)$ as a function of the energy density $e$ of the MSC, for the different tapping protocols shown in (a) diamonds, (b) circles, and (c) squares, showing a linear increase $\ell (e)\propto e$ with a protocol-dependent slope.

Figure 2

Distributions of springs elongations $P(\xi )$ at different temperatures, on a semi-log scale. Points represent $P(\xi )$ in MSCs sampled via tapping, full lines represent $P(\xi )$ obtained from Edwards theory, computed at the temperature $T_{\mathrm{Ed}}$ yielding the same energy density $e$ as the tapping. (a) Uncorrelated regime [$\ell (e)<1$], where $e\sim T_{\mathrm{Ed}}$; $T_{\mathrm{Ed}}=0.0008$ (blue squares) and 0.0026 (orange circles). (b) Correlated regime [$\ell (e)>1$], where $e\sim \sqrt{T_{\mathrm{Ed}}}$; $T_{\mathrm{Ed}}=1.19$ (orange circles) and 3.84 (blue squares).

Figure 3

Energy density $e$ of MSC as a function of $T_{\mathrm{Ed}}$, from transfer operators (small red diamonds) or Gaussian approximation (full line), and as a function of dissipated energy $e_{\mathrm{diss}}$ for two tapping protocols: (i) $\rho =0.3$, $\sigma =0$ (orange circles); (ii) $\rho =1$, $\sigma >0$ (blue triangles). For $T_{\mathrm{Ed}}\ll {\mu }^2$ one finds an equilibriumlike regime, $e\sim T_{\mathrm{Ed}}$; for $T_{\mathrm{Ed}}\gg {\mu }^2$ the behavior is $e\sim \sqrt{T_{\mathrm{Ed}}}$. Top inset: Correlation length from exact calculation (transfer operators); the behavior $\ell (T_{\mathrm{Ed}})\sim \sqrt{T_{\mathrm{Ed}}}\sim e$ is clear when $T_{\mathrm{Ed}}\gg {\mu }^2$. Bottom inset: Same symbols, data sets of the main panel are collapsed by just rescaling the $x$ axis; up to a protocol-dependent prefactor we have $e_{\mathrm{diss}}\sim T_{\mathrm{Ed}}$.