Thesis title: Numerical study of the microscopic structure of jammed systems: from inferring their dynamics to finite size scaling
In this thesis I present most of the results obtained during my PhD, where I worked on different subjects regarding jamming in systems of frictionless spheres. In particular, I focused on microscopic properties of jammed packings, such as the distribution of contact forces and interparticle gaps, as well as the single particle dynamics that occur near the jamming point. Several of these results have already been presented in Refs. [1,2], but here I include a more detailed analysis of some of them. Besides, all the results of the second chapter are new, even if related to such works.
The thesis is structured as follows. Chapter 1 provides a quick introduction to the phenomenology of glassy systems, with an entire section dedicated to hard spheres (HS) as a minimalistic model of a glass former. I also include a brief survey of a recently developed mean-field theory, capable of providing an exact description of liquids, glasses and jammed systems in infinite dimensions. This theory severs as a framework for explaining many of the features of real glasses as “blurred” or imperfect analogies of the sharp transitions predicted in $d=\infty$. Moreover, its predictions about the distributions of contact forces and gaps seem to remain valid in low dimensions, i.e. $d=2, 3$. Given that here I only consider tridimensional system, its relevance is obvious. But not only, the physical picture based on the meta-basin structure of the free energy landscape will be a guiding principle for many of the topics in later chapters. The final section of this first chapter is devoted to a general discussion about the jamming transition. Naturally, special attention is given to the microscopic features of jammed packings. Thus, I explain in detail some of the most important results derived in previous works. In particular, I give a careful description of the network of contact forces, showing that it is entirely determined by the particles' position, and that it contains important information about the packings stability. Similarly, I reproduce the proof that the exponents of the distribution of forces and gaps are connected through stability inequalities. Because such bounds are saturated, critical jammed packings are only marginally stable. This last feature is rationalized in terms of the constant density of states for vanishingly small frequencies. I hope (the presumably many) readers will benefit from having all this material gathered in a single source.
Chapter 2 contains a detailed description of the Linear Programming algorithm we developed to generate jammed packings. This method has been previously introduced in Refs. [1,3] but here I present a complete explanation and proofs of several of its features. Moreover, I carried out a detailed characterization of it, although restricted to $3d$ systems. Additionally, in this chapter I review the Lubachevsky--Stillinger compression protocol, based on event-driven molecular dynamics simulations. This method is able to efficiently compress configurations of HS up to very high pressures, and at the same time allows to probe their glassy phase. Complementing this procedure with our crunching algorithm based on Linear Programming, results in a reliable and reasonably fast method to generate jammed packings of HS. Even more, I show that we can apply the same technique to jam configurations of the Mari--Kurchan (MK) model.
The dynamics of particles near the jamming point is explored in Chapter 3. Given that few works have addressed this issue before us, the first step was to characterise the statistical properties of the trajectories of individual particles in this dynamical regime. Then, the idea was to investigate if the information of the network of contacts can be used to make statistical inferences of the particles' motion close to their jamming point. We found that by considering only the contact vectors (i.e. ignoring the magnitude of the forces) we were able to construct a couple of structural variables that correlate well with the dynamical features of the particles. More precisely, the vectorial sum of contact vectors is a good descriptor of the preferential directions in single-particle trajectories, while the dot product between such vectors can be used as a predictor of particle's mobility. The correlations thus obtained are rather high, although mostly valid for short times. Importantly, our method proved to be superior than a normal modes approach, which fails to capture these dynamical features at the level of individual particles. I should also mention that most of the results discussed in this chapter have appeared in Ref. [1]
Finally, Chapter 4 deals with a thorough analysis of the critical distributions of contact forces and interparticle gaps in configurations of HS, soft spheres as well as in the MK model. In Ref. [2] polydisperse disks and near crystals with an FCC structure were also considered. The purpose is to verify the expected power-laws using an analysis based on the finite size effects of such critical distributions. As mentioned above, current numerical estimations of the power-law exponents suggest that jammed packings saturate the stability criteria, so carefully validating their value is clearly important. Moreover, theoretical predictions of these exponents imply that jamming defines an universality class, to which a broad range of constraint-satisfaction problems belong. However, the calculations involved, being based on mean-field theory, are only exact in the $d\to\infty$ limit. And yet, available numerical results indicate that such universal criticality holds in low dimensions as well. Proving that the same scalings occur in finite dimensions is thus of major theoretical interest, as it would represent the most precise prediction of the so called replica method in realistic materials models. Our approach uses the scaling collapse of the distributions caused by changing the system sizes and it amply confirms the predicted values in both models. However, it also reveals a striking difference in the size effects on the distribution of forces and gaps, namely, that such effects are negligible in the former but very pronounced in the latter. We rationalize this feature in terms of two correlation lengths that determine how fast the thermodynamic limit behaviour is reached.
I wish this thesis will be useful for new grad students entering the intriguing (and complex) world of jamming. With this in mind, I took advantage of the fact that the material of each chapter (after the first one) is independent and thus tried to make them as self-contained as possible. (Or, in any case, I refer to the relevant sections that provide the necessary context for the topics discussed.) For the same reasons, I decided to give independent conclusions in each chapter, rather than providing general ones at the end. However, I hope that it is sufficiently clear that the unifying thread of all the results presented here are the peculiar microscopic structural properties of jammed systems.