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The force distribution of jammed disordered packings has always beenconsidered a central object in the physics of granular materials. However, manyof its features are poorly understood. In particular, analytic relations toother key macroscopic properties of jammed matter, such as the contact networkand its coordination number, are still lacking. Here we develop a mean-fieldtheory for this problem, based on the consideration of the contact network as arandom graph where the force transmission becomes a constraint optimizationproblem. We can thus use the cavity method developed in the last decades withinthe statistical physics of spin glasses and hard computer science problems.This method allows us to compute the force distribution $\text P(f)$ for randompackings of hard particles of any shape, with or without friction. We find anew signature of jamming in the small force behavior $\text P(f) \simf^{\theta}$, whose exponent has attracted recent active interest: we find afinite value for $\text P(f=0)$, along with $\theta=0$. Furthermore, we relatethe force distribution to a lower bound of the average coordination number $\,{\bar z}_{\rm c}^{\rm min}(\mu)$ of jammed packings of frictional spheres withcoefficient $\mu$. This bridges the gap between the two known isostatic limits$\, {\bar z}_{\rm c}(\mu=0)=2D$ (in dimension $D$) and $\, {\bar z}_{\rm c}(\mu\to \infty)=D+1$ by extending the naive Maxwell's counting argument tofrictional spheres. The theoretical framework describes different types ofsystems, such as non-spherical objects in arbitrary dimensions, providing acommon mean-field scenario to investigate force transmission, contact networksand coordination numbers of jammed disordered packings.

Cavity method for force transmission in jammed disordered packings of hard particles