On the rigidity of amorphous solids

By Matthieu Wyart. 2005

269 citations recorded by the source corpus.

We poorly understand the properties of amorphous systems at small length scales, where a continuous elastic description breaks down. This is apparent when one considers their vibrational and transport properties, or the way forces propagate in these solids. Little is known about the microscopic cause of their rigidity. Recently it has been observed numerically that an assembly of elastic particles has a critical behavior near the jamming threshold where the pressure vanishes. At the transition such a system does not behave as a continuous medium at any length scales. When this system is compressed, scaling is observed for the elastic moduli, the coordination number, but also for the density of vibrational modes. In the present work, we derive theoretically these results, and show that they apply to various systems such as granular matter and silica, but also to colloidal glasses. In particular we show that: (i) these systems present a large excess of vibrational modes at low frequency in comparison with normal solids, called the "boson peak" in the glass literature. The corresponding modes are very different from plane waves, and their frequency is related to the system coordination; (ii) rigidity is a non-local property of the packing geometry, characterized by a length scale which can be large. For elastic particles this length diverges near the jamming transition; (iii) for repulsive systems the shear modulus can be much smaller than the bulk modulus. We compute the corresponding scaling laws near the jamming threshold. Finally, we discuss the implications of these results for the glass transition, the transport, and the geometry of the random close packing.

Subfields: Electronic, Optical and Magnetic Materials, Materials Chemistry, Mechanical Engineering, 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): “In the last decade several works have unveiled an analogy between the physical phenomenon of jamming [29, 30] and phase transitions taking place in certain classes of computational optimization and learning problems [31– 33], in particular the perceptron [33, 34] — the simplest neural network perfor”
A jamming transition from under- to over-parametrization affects generalization in deep learning (2018, background): “…very general, is of easier understanding when we consider particles that interact only within a ﬁnite range: upon increasing their density, such systems undergo a critical jamming transition [19, 20] when there is no longer space to accommodate all the particles without them touching one another.”
A jamming transition from under- to over-parametrization affects loss landscape and generalization (2018, background): “In the last decade physicists have unveiled an analogy between the physical phenomenon of jamming [17, 18], which characterizes the onset of rigidity in disordered packings of particles, and phase transitions taking place in certain classes of computational optimization and learning problems [19, 20”
Edwards Statistical Mechanics for Jammed Granular Matter (2016, background): “Therefore, one can conclude that the critical observables generally scale with the distance to isostaticity (Wyart, 2005).”
Mean-field theory of hard sphere glasses and jamming (2008, background): “It has also been proposed that the square-root singularity of g(r) is related to these modes (Wyart, 2005).”