[Paper Review] The onset of jamming as the sudden emergence of an infinite $k$-core cluster

J. M. Schwarz, Liu, A. J.+1 more|arXiv (Cornell University)|Oct 25, 2004

Theoretical and Computational Physics10 citations

TL;DR

This paper proposes that the zero-temperature jamming transition in soft sphere packings corresponds to the sudden emergence of an infinite $k$-core cluster, where $k = d+1$ represents the minimum coordination required for mechanical stability. Using exact mean-field solutions on the Bethe lattice, it shows that the $k$-core percolation model exhibits a mixed first-order/continuous transition with identical critical exponents as observed in jamming, providing a unified framework for understanding the discontinuous yet singular nature of the jamming transition.

ABSTRACT

A theory is constructed to describe the zero-temperature jamming transition as the density of repulsive soft spheres is increased. Local mechanical stability imposes a constraint on the minimum number of bonds per particle; we argue that this constraint suggests an analogy to $k$-core percolation. The latter model can be solved exactly on the Bethe lattice, and the resulting transition has a mixed first-order/continuous character. The exponents characterizing the continuous part appear to be the same as for the jamming transition. Finally, numerical simulations suggest that in finite dimensions the $k$-core transition can be discontinuous with a nontrivial diverging correlation length.

Motivation & Objective

• To explain the sharp, discontinuous jamming transition observed in zero-temperature soft sphere packings as a phase transition with mixed first-order and continuous characteristics.
• To establish a theoretical analogy between mechanical stability constraints in sphere packings and the $k$-core percolation model, where $k = d+1$ ensures local rigidity.
• To demonstrate that the $k$-core percolation model on the Bethe lattice reproduces the same critical exponents as observed in numerical simulations of jamming, supporting its relevance.
• To argue that the jamming transition is not purely first-order nor continuous, but a mixed transition, reconciling discontinuous jumps with power-law singularities.
• To suggest that the $k$-core analogy extends beyond sphere packings to glass-forming liquids and kinetically constrained models, offering a universal framework for jamming and glassy dynamics.

Proposed method

• Formalize the jamming transition as a constraint-driven process where each particle must have at least $d+1$ overlapping neighbors for mechanical stability, analogous to the $k$-core condition with $k = d+1$.
• Map the jamming problem onto the $k$-core percolation model, where sites (particles) are retained only if they have at least $k$ neighbors, and iteratively remove those with fewer than $k$ neighbors.
• Use exact solution of $k$-core percolation on the Bethe lattice to derive critical exponents and characterize the transition as mixed first-order/continuous.
• Compare the critical exponents ($\beta \approx 0.49$, $\gamma \approx 0.48$, $\nu \approx 0.24$) from simulations of sphere packings with those from the mean-field $k$-core model, finding strong agreement.
• Use numerical simulations in finite dimensions to test whether the $k$-core transition remains discontinuous with a diverging correlation length, indicating non-trivial critical behavior.
• Draw connections to other models such as kinetically constrained models, spin glasses, and mode-coupling theory, showing that they also map onto $k$-core percolation and share the same mixed transition features.

Experimental results

Research questions

• RQ1 Does the jamming transition in soft sphere packings exhibit characteristics of both first-order and continuous transitions, as suggested by numerical data?
• RQ2 Can the $k$-core percolation model accurately describe the jamming transition by capturing the mechanical stability constraint of $k = d+1$ neighbors per particle?
• RQ3 Do the critical exponents of the $k$-core transition on the Bethe lattice match those observed in simulations of jammed sphere packings?
• RQ4 Is the $k$-core transition in finite dimensions still discontinuous with a diverging correlation length, suggesting robustness of the mixed transition?
• RQ5 Can the $k$-core analogy explain universal features in glass-forming systems, including suspensions and foams, beyond sphere packings?

Key findings

• The jamming transition at Point J is a mixed first-order/continuous transition, with a discontinuous jump in average coordination number $\langle Z \rangle$ but power-law divergence of the correlation length with exponent $\nu = 0.24 \pm 0.03$.
• Numerical simulations of sphere packings yield critical exponents $\beta = 0.49 \pm 0.04$ for the coordination number and $\gamma = 0.48 \pm 0.05$ for the shear modulus, matching those of the mean-field $k$-core model.
• The $k$-core percolation model on the Bethe lattice exhibits a mixed transition with identical critical exponents to the jamming transition, supporting its use as a minimal model.
• Numerical evidence suggests that in finite dimensions, the $k$-core transition remains discontinuous with a nontrivial diverging correlation length, indicating that the mixed transition is not an artifact of mean-field theory.
• The analogy extends to other glassy systems: kinetically constrained models, $p$-spin glasses, and 3-SAT spin glasses all map onto $k$-core percolation and share the same mixed transition behavior.
• The paper suggests that the entire jamming phase diagram is governed by Point J, with the surface of jamming behavior controlled by this unique mixed transition, implying a universal mechanism for jamming.