On the rigidity of amorphous solids. Price fluctuations, Conventions and Microstructure of Financial Markets

Sur la rigidité des solides amorphes. Fluctuation des prix, conventions et microstructure des marchés financiers.

Matthieu Wyart (1)

1 SPEC - UMR3680 - Service de physique de l'état condensé

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 theo- retically 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 re- sults for the glass transition, the transport, and the geometry of the random close packing.

Sur la Rigidité des Solides Amorphes: On comprend mal les propriétés microscopiques des solides amorphes, comme le transport, la propagation des forces ou la nature de leur rigidité mécanique. Ces questions semblent liées à la présence d'un excès de modes vibratoires à basse fréquence, le ''pic boson". On explique la nature de ces modes dans les systèmes répulsifs à courte portée. On argumente que cette description s'applique aussi aux milieux granulaires, à la silice, et aux verres colloïdaux. Fluctuations des prix, Conventions et Microstructure des marches Financiers: Les fluctuations des cours de la bourse ont des propriétés étonnantes. La volatilité (l'amplitude de ces fluctuations) est environ un ordre de grandeur plus grand que les prédictions de la théorie des marches efficients, et est corrèlee sur des échelles de temps très longs. Les agents sur réagissent aux informations. On montre que ces propriétés apparaissent lorsque les agents agissent en fonction de leur expérience et du passé du marché. On étudie aussi la microstructure des marchés, qui régulent les échanges aux temps courts. On explique pourquoi le prix est diffusif bien que les ordres marchés (les chocs subis par les prix) soient très corrélés. On évalue la fourchette des prix par des arguments de symétrie.

Mots clés: Amorphous solids; Glass; Granular matter; Soft mode; Glass transition; Rigidity; Colloids; Boson peak; Volatility; Convention; Order-book; Spread

Mots clés: Pic boson; Granulaire; Solides amorphes; Verre; Rigidite; Colloides; Transition vitreuse; Carnet d'ordre; Volatilite; Fourchette; Mode mou

Domaines: Physique et Société [physics.soc-ph]

Matthieu Wyart. On the rigidity of amorphous solids. Price fluctuations, Conventions and Microstructure of Financial Markets. Physics and Society [physics.soc-ph]. Ecole Polytechnique X, 2005. English. ⟨NNT : ⟩. ⟨pastel-00001919⟩