The jamming transition in high dimension: an analytical study of the TAP equations and the effective thermodynamic potential

Abstract

We present a parallel derivation of the Thouless-Anderson-Palmer (TAP) equations and of an effective potential for the negative perceptron and soft sphere models in high dimension. Both models are continuous constrained satisfaction problems with a critical jamming transition characterized by the same exponents. Our analysis reveals that a power expansion of the potential up to the second order represents a successful framework to approach the jamming line from the SAT phase (the region of the phase diagram where at least one configuration verifies all the constraints), where the ground-state energy is zero. An interesting outcome is that close to jamming the effective thermodynamic potential has a logarithmic contribution, which turns out to be dominant in a proper scaling regime. Our approach is quite general and can be directly applied to other interesting models. Finally, we study the spectrum of small harmonic fluctuations in the SAT phase recovering the typical scaling $D(\omega )\sim {\omega }^2$ below the cutoff frequency but a different behavior characterized by a non-trivial exponent above it.

arXiv comment11 pages; a few typos corrected

References in corpus (23)

• Jammed Hard-Particle Packings: From Kepler to Bernal and Beyond
• The Jamming Transition in Granular Systems
• Fractal free energy landscapes in structural glasses
• Geometric origin of excess low-frequency vibrational modes in amorphous solids
• Universal Spectrum of Normal Modes in Low-Temperature Glasses: an Exact Solution
• Exact theory of dense amorphous hard spheres in high dimension. III. The full RSB solution
• Anharmonicity and quasi-localization of the excess low-frequency vibrations in jammed solids
• Exact theory of dense amorphous hard spheres in high dimension. II. The high density regime and the Gardner transition
• Replica Field Theory for Deterministic Models: Binary Sequences with Low Autocorrelation
• The simplest model of jamming
• Excess Vibrational Modes and the Boson Peak in Model Glasses
• Exact theory of dense amorphous hard spheres in high dimension. I. The free energy
• Self Induced Quenched Disorder: A Model for the Glass Transition
• The distribution of forces affects vibrational properties in hard sphere glasses
• On the rigidity of a hard sphere glass near random close packing
• Universal non-Debye scaling in the density of states of amorphous solids
• Dynamical transition of glasses: from exact to approximate
• The Complexity of Ising Spin Glasses
• Theory of the Jamming Transition at Finite Temperature
• The Gardner transition in finite dimensions
• Statics and dynamics of infinite-dimensional liquids and glasses: a parallel, compact derivation
• A mean-field hard-spheres model of glass
• Mean field theory of spin glasses

Cited by in corpus (21)

• Universality of the SAT-UNSAT (jamming) threshold in non-convex continuous constraint satisfaction problems
• Does a growing static length scale control the glass transition?
• Theory for Swap Acceleration near the Glass and Jamming Transitions
• Out-of-equilibrium dynamical mean-field equations for the perceptron model
• High-temperature Expansions and Message Passing Algorithms
• Constraint satisfaction mechanisms for marginal stability and criticality in large ecosystems
• Memory formation in jammed hard spheres
• Vibrational properties of hard and soft spheres are unified at jamming
• Instabilities of jammed packings of frictionless spheres under compression
• Impact of jamming criticality on low-temperature anomalies in structural glasses
• Quantum Jamming: Critical Properties of a Quantum Mechanical Perceptron
• Higher order corrections to the effective potential close to the jamming transition in the perceptron model
• Introduction to the dynamics of disordered systems: equilibrium and gradient descent
• Jamming Criticality of Near-Crystals
• Mean field theory of jamming of nonspherical particles
• How Generic are the Robust Theoretical Aspects of Jamming in Hard Sphere Models?
• Force balance controls the relaxation time of the gradient descent algorithm in the satisfiable phase
• Emergent Inter-particle Interactions in Thermal Amorphous Solids
• Hard-Sphere Jamming through the Lens of Linear Optimization
• Jamming, relaxation, and memory in a structureless glass former
• Comment on "Explicit Analytical Solution for Random Close Packing in $d=2$ and $d=3$"