Fractal free energy landscapes in structural glasses

By Patrick Charbonneau, Jorge Kurchan, Giorgio Parisi, Pierfrancesco Urbani, Francesco Zamponi. 2014

412 citations recorded by the source corpus.

Glasses are amorphous solids whose constituent particles are caged by their neighbours and thus cannot flow. This sluggishness is often ascribed to the free energy landscape containing multiple minima (basins) separated by high barriers. Here we show, using theory and numerical simulation, that the landscape is much rougher than is classically assumed. Deep in the glass, it undergoes a ‘roughness transition’ to fractal basins, which brings about isostaticity and marginal stability on approaching jamming. Critical exponents for the basin width, the weak force distribution and the spatial spread of quasi-contacts near jamming can be analytically determined. Their value is found to be compatible with numerical observations. This advance incorporates the jamming transition of granular materials into the framework of glass theory. Because temperature and pressure control what features of the landscape are experienced, glass mechanics and transport are expected to reflect the features of the topology we discuss here. Potential energy landscape models are often used to describe transitions in the glassy state. Here, the authors report that the landscape is much rougher than usually assumed, and demonstrate that it undergoes a transition to fractal basins before the jamming point is reached.

Subfields: Biomedical Engineering, Condensed Matter Physics, Materials Chemistry, AI for Physical Sciences

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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 glassy systems, such power-laws reveal properties that cannot be reached by studying the Hessian, in particular the fact that the dynamics occurs via broadly distributed avalanches [42–44], indicative of a hierarchical organization of the landscape [45].”
Universality of jamming of nonspherical particles (2018, neutral): “P(f )∼ { ∆p0(f ∆ −ν) (f ∼∆) f θ (f ∼ 1) , [15] with new critical exponents μ=κ/(4κ− 4) = 0.”
Statistical physics of inference: thresholds and algorithms (2015, background): “[48] Patrick Charbonneau, Jorge Kurchan, Giorgio Parisi, Pierfrancesco Urbani, and Francesco Zamponi.”
Unreasonable effectiveness of learning neural networks: From accessible states and robust ensembles to basic algorithmic schemes (2016, background): “14. Huang H, Kabashima Y (2014) Origin of the computational hardness for learning with”
Facets of glass physics (2015, background): “Specifically, the exact solution for the equilibrium phase diagram of hard spherical particles interacting in d spatial dimensions was rigorously obtained in the mathematical limit d→ ∞, where mean-field approximations become exact [7].”
Universality of the SAT-UNSAT (jamming) threshold in non-convex continuous constraint satisfaction problems (2017, methodology): “Finally, note that while we will be able to prove the existence of the scaling solution, and compute the values of a and κ, we will not be able to prove that ε is proportional to the distance from the jamming line: at present, this follows from the numerical solution of the equations [13].”
Edwards Statistical Mechanics for Jammed Granular Matter (2016, methodology): “We have reviewed the recent encouraging results of (Charbonneau et al., 2014b; Martiniani et al., 2017; Newman and Stein, 1999; Sharma et al., 2016) and have proposed a calculation for the 1-SF states in the SK model.”
Universal spectrum of normal modes in low-temperature glasses (2015, background): “, independent of the spatial dimension d for all d≥ 2 (26, 27), (ii) they can be analytically predicted through the exact solution in d→∞ (17, 28), and (iii) the perceptron model of neural networks, a prototypical CCSP, displays a jamming transition with the same exponents (25).”
The role of local structure in dynamical arrest (2014, background): “It has recently been shown that the energy landscape (for hard spheres) splits in a fractal way [84].”