We present results on a meso-scale model for amorphous matter in athermal, quasi-static (a-AQS), steady state shear flow. In particular, we perform a careful analysis of the scaling with the lateral system size, L, of: i) statistics of individual relaxation events in terms of stress relaxation, S, and individual event mean-squared displacement, M , and the subsequent load increments, ∆γ, required to initiate the next event; ii) static properties of the system encoded by x = σy − σ, the distance of local stress values from threshold; and iii) long-time correlations and the emergence of diffusive behavior. For the event statistics, we find that the distribution of S is similar to, but distinct from, the distribution of M . The exponents governing the scaling properties of P (S) completely determine the exponent α governing the finite size scaling of the load increment required to trigger the next event ∆γ ∼ L −α . P (M ) is analogous to but distinct from P (S). We find a strong correlation between S and M for any particular event, with S ∼ M q with q ≈ 0.65. This new exponent, q, completely determines the scaling exponents for P (M ) given those for P (S). For the distribution of local thresholds, we find P (x) is analytic at x = 0, and has a value P (x)| x=0 = p0 which scales with lateral system length as p0 ∼ L −a 1 . In our model, by construction, the minimum, xmin, of x in any particular configuration is precisely equal to ∆γ, and, also by construction, S = ∆γ . Extreme value statistics arguments lead to a scaling relation between the exponents governing P (x) and those governing P (S). Finally, we study the long-time correlations via single-particle tracer statistics. At short times, the displacement distributions are strongly non-Gaussian and consistent with exponentials as observed at short times in other driven and thermal glassy systems. At long times, a diffusive behavior emerges where the distributions become Gaussian. The value of the diffusion coefficient is completely determined by ∆γ and the scaling properties of P (M ) (in particular from M ) rather than directly from P (S) as one might have naively guessed. Our results: i) further define the a-AQS universality class with the identification of new scaling exponents unrelated to old ones, ii) help clarify the relation between avalanches of stress relaxation and longtime diffusive behavior, iii) help clarify the relation between local threshold distributions and event statistics and iv) should be important for any future work on the broad class of systems which fall into this universality class including amorphous alloys, glassy polymers, compressed granular matter, and soft glasses like foams, emulsions, and pastes. arXiv:1905.07388v1 [cond-mat.soft]

A mesoscopic model of amorphous plasticity is discussed in the context of depinning models. After embedding in a d + 1 dimensional space, where the accumulated plastic strain lives along the additional dimension, the gradual plastic deformation of amorphous media can be regarded as the motion of an elastic manifold in a disordered landscape. While the associated depinning transition leads to scaling properties, the quadrupolar Eshelby interactions at play in amorphous plasticity induce specific additional features like shear-banding and weak ergodicity break-down. The latters are shown to be controlled by the existence of soft modes of the elastic interaction, the consequence of which is discussed in the context of depinning.