Low-energy non-linear excitations in sphere packings

By Edan Lerner, G. During, Matthieu Wyart. 2013

118 citations recorded by the source corpus.

We study theoretically and numerically how hard frictionless particles in random packings can rearrange. We demonstrate the existence of two distinct unstable non-linear modes of rearrangement, both associated with the opening and the closing of contacts. The first mode, whose density is characterized by some exponent θ′, corresponds to motions of particles extending throughout the entire system. The second mode, whose density is characterized by an exponent θ ≠ θ′, corresponds to the local buckling of a few particles. Extended modes are shown to yield at a much higher rate than local modes when a stress is applied. We show that the distribution of contact forces follows P(f) ∼ fmin(θ′,θ), and that imposing the restriction that the packing cannot be densified further leads to the bounds and , where γ characterizes the singularity of the pair distribution function g(r) at contact. These results extend the theoretical analysis of [Wyart, Phys. Rev. Lett., 2012, 109, 125502] where the existence of local modes was not considered. We perform numerics that support that these bounds are saturated with γ ≈ 0.38, θ ≈ 0.17 and θ′ ≈ 0.44. We measure systematically the stability of all such modes in packings, and confirm their marginal stability. The principle of marginal stability thus allows us to make clearcut predictions on the ensemble of configurations visited in these out-of-equilibrium systems, and on the contact forces and pair distribution functions. It also reveals the excitations that need to be included in a description of plasticity or flow near jamming, and suggests a new path to study two-level systems and soft spots in simple amorphous solids of repulsive particles.

Subfields: Computational Mechanics, Materials Chemistry, Mechanics of Materials, Optimization & Training

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Citations in the corpus

The jamming transition as a paradigm to understand the loss landscape of deep neural networks (2018, background): “[57] Edan Lerner, Gustavo During, and Matthieu Wyart,”
Edwards Statistical Mechanics for Jammed Granular Matter (2016, methodology): “A marginal stability analysis provides γ = (2 + θe) −1 = (1 − θl)/2 (Müller and Wyart, 2015), which has also been demonstrated numerically (Lerner et al., 2013).”
Modeling semiflexible polymer networks (2014, background): “Finally, EMT’s were also developed to describe the dynamic shear modulus of a spring network embedded in a viscous medium (Lerner et al., 2013; Wyart, 2010; Yucht et al., 2013).”
Universality of jamming of nonspherical particles (2018, result): “This approach provides exact results for the critical exponents, which agree well with the numerical results (16), once localized excitation modes are carefully separated (17, 24).”
Erratum: On variational arguments for vibrational modes near jamming (2016, background): “This reasoning was confirmed numerically, as one finds that stretching a bond α by some amount ǫ leads to a displacement of all particles of order ǫ [27].”
Universality of the SAT-UNSAT (jamming) threshold in non-convex continuous constraint satisfaction problems (2017, methodology): “We now formulate in our abstract continuous CSP setting some well-known consequences of isostaticity [ 8,10 ] , i.e. the condition that the number of unsatisﬁed constraints is equal to the number of degrees of freedom, and c = 1.”
Fractal free energy landscapes in structural glasses (2014, background): “[40] Edan Lerner, Gustavo During, and Matthieu Wyart, “Low-energy non-linear excitations in sphere packings,” Soft Matter 9, 8252–8263 (2013).”