title: Mechanical disorder of sticky-sphere glasses. I. Effect of attractive interactions
authors: Karina González-López Mahajan Shivam Yuanjian Zheng Massimo Pica Ciamarra Edan Lerner
text: 
  
  
  
  
  Institute for Theoretical Physics, University of Amsterdam, Science Park 904, Amsterdam, Netherlands
  
  School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
  
  School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
  
  School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
  CNR-SPIN, Dipartimento di Scienze Fisiche, Università di Napoli Federico II, I-80126 Napoli, Italy
  
  e.lerner@uva.nl
  Institute for Theoretical Physics, University of Amsterdam, Science Park 904, Amsterdam, Netherlands
  
  
  
  
  
  Recent literature indicates that attractive interactions between particles of a dense liquid play a secondary role in determining its bulk mechanical properties. Here we show that, in contrast with their apparent unimportance to the bulk mechanics of dense liquids, attractive interactions can have a major effect on macro- and microscopic elastic properties of glassy solids. We study several broadly-applicable dimensionless measures of stability and mechanical disorder in simple computer glasses, in which the relative strength of attractive interactions — referred to as `glass stickiness' — can be readily tuned. We show that increasing glass stickiness can result in the decrease of various quantifiers of mechanical disorder, on both macro- and microscopic scales, with a pair of intriguing exceptions to this rule. Interestingly, in some cases strong attractions can lead to a reduction of the number density of soft, quasilocalized modes, by up to an order of magnitude, and to a substantial decrease in their core size, similar to the effects of thermal annealing on elasticity observed in recent works. Contrary to the behavior of canonical glass models, we provide compelling evidence indicating that the stabilization mechanism in our sticky-sphere glasses stems predominantly from the self-organized depletion of interactions featuring large, negative stiffnesses. Finally, we establish a fundamental link between macroscopic and microscopic quantifiers of mechanical disorder, which we motivate via scaling arguments. Future research directions are discussed. 
  
  
  Mechanical disorder of  sticky-sphere glasses. I. Effect of attractive interactions
      Edan Lerner
      August 27, 2020
  ===================================================================================
  
  
  
  
  
  § INTRODUCTION
  
  
  
  
  One of the key challenges in glass physics is establishing robust relations between structure and dynamics <cit.>. The various approaches to assessing glassy matter's structure can be roughly partitioned into two schools: approaches that focus on positional disorder <cit.>, and those that focus on mechanical disorder <cit.>. Mechanical disorder refers to the various forms of mechanical fluctuations that are seen to emerge in structural glasses, ranging from localized soft spots <cit.> to mesoscopic fluctuations in elastic moduli fields <cit.>. 
  
  
  The future utility of glasses in technological innovations and industrial applications depends both upon improving glass formation processes <cit.>, 
  and upon understanding the relations between glasses' composition — which, in turn, determines the nature of interactions between the constituent particles — and their mechanical properties <cit.>. While the process of glass formation and glass-forming ability have been the subject of enormous research efforts <cit.>, the effects of different types of microscopic interactions on the emergent mechanical disorder of glasses have not been fully explored yet.
  
  One field in which substantial success has been achieved in understanding relations between microscopic observables in disordered materials, and their macro- and microscopic elasticity, is that of unjamming <cit.>. The unjamming scenario precisely describes the changes in microscopic elasticity — in terms of e.g. characteristic frequency- and length-scales featured by vibrational modes <cit.> — and the changes in macroscopic elasticity — in terms of e.g. various elastic moduli ratios <cit.> — upon reducing the degree of connectedness of the network of interactions between the constituents of an amorphous solid. 
  
  While the phenomenology associated with the unjamming scenario has been largely understood, less focus has been dedicated by the materials-physics community to studying the effects of the form of interactions between the constituent particles of a glass on its elasticity, in systems largely removed from the unjamming point <cit.>. In Ref. <cit.> an effort was made to identify, trace and quantitatively compare between some of the trends observed in macroscopic elasticity of simple computer glasses, under changes of various parameters of interest. Here we adopt a similar strategy, except that we focus in particular on the role of attractive terms in pairwise interaction potentials on the elastic properties of the resulting glasses, in system for which the unjamming scenario is irrelevant <cit.>. To this aim, we employ two computer-glass models put forward in Refs. <cit.>, in which particles interact via a pairwise potential whose attractive term can be straightforwardly tuned. We study these models to shed light on the question: how do attractive interactions affect elastic properties, mechanical disorder and stability of glassy solids?
  
  Comparing the elastic properties of different types of glasses, with the aim of cleanly distilling the effects emanating from different features in the form of interparticle interactions, is generally not straightforward to accomplish. The difficulty to do so stems from the history-dependence that glasses' properties are notorious for, that might hinder comparison of elastic properties between different types of glasses on an equal footing.  In computer experiments, this difficulty can be circumvented; several works have shown that elastic properties of glasses instantaneously quenched from parent temperatures T_ — larger than some onset temperature T_ — become nearly independent of T_ <cit.>. The onset temperature T_ is found to be roughly twice the computer glass transition temperature T_g, defined here as the temperature at which the primary relaxation time is of order 10^4τ_0, where τ_0 is a microscopic timescale. Here we follow Refs. <cit.> and exploit this feature of glass-forming models, in order to achieve a meaningful, quantitative comparison between them. We create glassy samples by quenching high-temperature liquid states equilibrated at temperatures that are roughly a factor of 4 higher than their respective computer T_g, and therefore much larger than the onset temperature T_. The effects of varying T_ from high-temperature liquid states to supercooled viscous liquid states, on micro- and macroscopic elasticity of computer glasses featuring attractive interactions, is the focus of an accompanying paper <cit.>. 
  
  In addition to studying the effects of strong attractions between a glass constituents on its macroscopic elastic properties, here we focus much of our attention to the investigation of the effects of attractive interactions on what is referred to here as microscopic elasticity: the statistical, structural and energetic properties of soft, quasilocalized modes <cit.> (QLMs). These nonphononic soft excitations are observed in essentially all glasses quenched from a melt <cit.>, and are microscopic in nature: they feature a disordered core of linear size ξ_g <cit.> (defined and studied in detail in what follows), which is typically on the order of a few interparticle distances, and is decorated by algebraically-decaying Eshelby-like far fields <cit.>.
  
  The prevalence of QLMs or lack thereof have been shown to be extremely sensitive to the degree of supercooling of parent configurations from which glassy samples are instantaneously quenched <cit.>; in Ref. <cit.> it was shown that in glassy samples that underlie deeply supercooled equilibrium configurations, the number density N(T_) of soft QLMs can decrease by two orders of magnitude compared to that found in glasses quenched from high parent temperatures T_. In the same work it was also shown that the core size of QLMs decreases, and their typical frequency increases, with deeper supercooling of glasses' ancestral equilibrium states. 
  
  For these aforementioned reasons, the statistical and structural properties of QLMs can be associated with the notion of mechanical disorder, and are therefore studied carefully throughout this work under variations of the relative strength of attractive interactions between the constituent particles of our computer glasses. Our study echoes some of the findings of Ref. <cit.>, in which the same glass-forming model as (one of the two models) employed here was investigated, but a different set of observables were considered; in what follows we explain in detail the similarities and differences between our work and Ref. <cit.>. 
  
  Here we establish that the relative strength of attractive pairwise interactions — defined and referred to below as `glass stickiness' — can be a major contributor to glass stability. We show that increasing glass stickiness by employing steeper attractive terms in the interaction potential, or, alternatively, by decompressing a glass, can lead to a decrease in QLMs' core size, accompanied by the stiffening of QLMs' characteristic frequency scale. In some particularly interesting cases — discussed in detail in what follows —, we find that the number density of QLMs decreases by up to an order of magnitude compared to that found in more generic glass models <cit.>. These effects, as well as the effects of stickiness on macroscopic moduli, are reminiscent in essence, as well as in magnitude, to the trends seen in thermally-annealed <cit.> or otherwise-stabilized <cit.> glasses. 
  
  
  
  
  Another key conclusion of our study is that small changes in the shape of the attractive part of the employed pairwise potentials can lead to dramatic changes in the emergent micro- and macroscopic elastic properties of their resulting glasses. Consider, for example, the two pairwise potentials shown in Fig. <ref>(a), referred to as `glass 1' and `glass 2'. These two potentials do not differ significantly from each other in terms of the form of their respective attractive parts. Interestingly, despite this similarity, their resulting glasses' Poisson's ratio decreases by ≈ 18% between `glass 1' to `glass 2' (see Fig. <ref>b), and the density per frequency D(ω) of low-frequency, quasilocalized modes of these models differs substantially, by a factor of ≈ 6.5 (see Fig. <ref>(c)). In what follows we provide evidence indicating that these differences stem from the self-organized abundance or dearth of interactions featuring destabilizing negative stiffnesses.
  
  
  This work is structured as follows; in Sec. <ref> we reintroduce the two model systems employed in our investigation, and explain how the three sets of ensembles of computer glasses analyzed below were created. Sect. <ref> defines the notion of glass stickiness, and shows how it behaves in our computer glasses. In Secs. <ref> and <ref> we report the results of our investigation of the glass-stickiness-dependence of macro- and microscopic elasticity, respectively. In Sec. <ref> we hold several discussions in which the results of our computer-experiments are interpreted, explained, and discussed in the context of other recent work, together with various additional analyses. In Sec. <ref> we briefly summarize the main results of the manuscript, and Sec. <ref> provides an outlook of future research directions. Precise definitions of the observables considered in our work, some of their measurement methods, and additional supporting data pertaining to some of the discussions, are all deferred to the appendixes.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  § COMPUTER GLASS MODELS, GLASS ENSEMBLES AND UNITS
  
  
  
  
  
  In this study we employ two computer glass models of 50:50 binary mixtures of `small' and `large' particles of equal mass m in three dimensions (3D) at fixed volume V, interacting via radially-symmetric, pairwise interaction potentials φ_ij(r_ij), such that the total potential energy is written as U=∑_i<jφ_ij(r_ij). With these two models, we created three sets of glass ensembles, as detailed in Sec. <ref>. 
  
  
  
  
  
  
  
  
  
  
   §.§ Computer glass models
  
  
  
  
  
  
    §.§.§ q-sticky-spheres
  
  
  
  
  
  The first model employed is a generalized Lennard-Jones potential, also known as Mie potential <cit.>, in which the strength of the attractive and repulsive interactions can be tuned via an exponent q. The supercooled-liquid dynamics of this model was studied in Ref. <cit.>. The pairwise potential of this model reads
  
  
  
  
      φ_(r_ij) = ε[ (λ_ijr_ij)^2q - 2(λ_ijr_ij)^q + ∑_ℓ=0^3 c_(r_ijλ_ij)^2ℓ]
  
  for r_ij/λ_ij<x_c, and φ_(r_ij)=0 for r_ij/λ_ij≥x_c. Here ε is a microscopic energy scale, λ_ij is a length parameter (see below), and c_(q,x_c) are the q- and x_c-dependent coefficients (reported in Appendix. <ref>) that ensure the first and second derivatives of φ_ with respect to the pairwise distance r_ij are continuous at the dimensionless cutoff x_c. We chose the parameters q=6,8,12,20 and x_c=3.0,2.5,2.0,1.8, respectively. The length parameters are expressed in terms of the `small-small' interaction length λ_^, with λ_^=1.18λ_^ and λ_^=1.4λ_^. The potential as given by Eq. (<ref>) is plotted in Fig. <ref>a. We refer to this model as q-sticky-spheres.
  
  
  
  
  
  
    §.§.§ Piecewise-sticky-spheres
  
  
  
  
  The second model employed in this work is a Lennard-Jones-like potential, first introduced in Ref. <cit.>, in which the repulsive part of the pairwise potential is identical to the canonical Lennard Jones (LJ) potential, but the range — and therefore also the strength — of the attractive part can be readily modified, as shown in Fig. <ref>b. The supercooled liquid dynamics of this model was studied in Ref. <cit.>. The piecewise pairwise potential of this model reads
  
  
  
      φ_(r_ij) ={[                             4ε[ (λ_ij/r_ij)^12 - (λ_ij/r_ij) ^6],                                                   r_ij/λ_ij <  x_; ε[a(λ_ij/r_ij)^12 -b(λ_ij/r_ij)^6 + ∑_ℓ=0 ^3  c_(r_ij/λ_ij)^2ℓ] ,                                                 x_≤r_ij/λ_ij< x_c;                                                               0 ,                                                    x_c ≤r_ij/λ_ij ]. ,
  
  
  where ε is a microscopic energy scale, x_,x_c are the (dimensionless) locations of the minimum of the LJ potential and modified cutoff, respectively, and the λ_ij's are the same length parameters as detailed for the QSS model above. In what follows, we express the dimensionless cutoff x_c in terms of x_=2^1/6, for simplicity, by defining r_c≡x_c/x_; r_c serves as one of the key control parameters in our investigation, as explained below. The coefficients a,b,{c_} are chosen such that the attractive and repulsive parts of φ_, and its first two derivatives, are continuous at x_ and at x_c, see  Table <ref> in Appendix <ref> for the coefficients' numerical values. We refer to this model as piecewise-sticky-spheres.
  
  
  
  
  
  
   §.§ Computer glass ensembles and their preparation protocol
  
  
  We created 3 sets of ensembles of glasses, as follows:
  
      
    * Q-Sticky-Spheres (employing φ_; see Eq. (<ref>)), with q=6,8,12,20 and x_c=3.0,2.5,2.0,1.8, respectively; varying q in this model at fixed density leads to the variation of the dimensionless pressure p/p_0 where p_0 is a characteristic pressure scale defined precisely in Appendix. <ref>. For this reason, we tuned the density ρ≡mN/V such that p/p_0≈0.05 for all q, resulting in ρ=0.80, 0.74, 0.698, 0.677 for q=6,8,12,20, respectively. These ensembles are referred to as QSS. 
      
    * Piecewise-Sticky-Spheres (i.e., employing φ_; see Eq. (<ref>)), varying Density (referred to as DSS): piecewise-sticky-spheres glasses with fixed cutoff r_c=1.2 and varying density ρ=0.55,  0.60, 0.65, 0.70, 0.75, 0.80, 0.95, 1.20. 
      
    * Piecewise-Sticky-Spheres (i.e. employing φ_; see Eq. (<ref>)), varying Cutoff (referred to as CSS): piecewise-sticky-spheres glasses with fixed density ρ=0.60 and varying cutoff r_c=1.1, 1.15, 1.2, 1.3, 1.5. To avoid data cluttering, we present data only for r_c=1.15 in Figs. <ref>,<ref>,<ref>,<ref>,<ref>. 
  
  
  
  
  
  
  Ensembles of independent glassy samples were created as follows; we first equilibrated each system in the high temperature liquid state using standard molecular dynamics (MD) simulations. The equilibration temperature for each system was chosen to be at least 4 times higher than G_∞ a_0^3/30, where a_0≡(V/N)^1/3 is the typical interparticle distance, and G_∞ is the high-temperature limit of the sample-to-sample mean of the shear modulus of underlying inherent states (glasses) found by a quick quench. We empirically observe that G_∞ a_0^3/30 is a rough estimation of the computer glass transition temperature in many numerical models of supercooled liquids, which motivates our choice. The temperature was kept constant by employing the Berendsen thermostat <cit.>. After equilibration, the energy is minimized instantaneously using a standard conjugate gradient algorithm. We followed this protocol to generated n independent glasses for each ensemble. The total number of samples generated for each model, and the system sizes employed, are all detailed in Table <ref>. The dimensionless pressures of the QSS and CSS glass ensembles are shown in Fig. <ref> in Appendix <ref>. We finally note that the minimal coordination number over all of our glass ensembles, pertaining to the CSS ensemble with r_c=1.1, is ≈10.5, i.e. very far from the Maxwell threshold of 6 (in 3D). Consequently, none of the effects we observe in this work are related to the unjamming transition <cit.>. 
  
  
  
  
   §.§ Units
  
  In what follows, we report all lengths in terms of the characteristic interparticle distance a_0≡(V/N)^1/3, and all frequencies in terms of ω_0≡c_s/a_0, where the (ensemble dependent) speed of shear waves is given by c_s=√(G/ρ) with G denoting the (ensemble dependent) shear modulus and ρ≡mN/V denoting the mass density.
  
  
  
  
  § GLASS STICKINESS
  
  
  
  As stated in the introduction, we consider here computer glasses in which the strength of attractive interactions can be tuned. In order to assess the effective relative strength of attractive vs. repulsive forces in a meaningful way, we define a glass's `stickiness' as follows; for each particle i we identify the largest repulsive pairwise force f^_i=_j (-φ'_ij) and largest (most negative) attractive pairwise force f^_i=_j(φ'_ij). The stickiness s of a glass is then defined as
  
      s ≡_i f^_i/_i f^_i ,
  
  where mean_i denotes a mean over particles i. Clearly s=0 for purely-repulsive systems, and s>0 once attractive interactions are present. 
  
  Importantly, we note that stickiness is an emergent property of a glass, affected by its self-organized structure, which is controlled, in turn, by features of the pairwise interactions, and by mechanical-equilibrium and spatial-confinement constraints to which the glass is subjected. In Fig. <ref> we show our measurements of glass stickiness for our three sets of glass ensembles. It is clear that glass stickiness can be manipulated either by varying features of the employed pairwise potentials (cf. QSS and CSS models), or by compressing or decompressing a system with a fixed pairwise potential, as shown for the DSS system in Fig. <ref>b. This latter route to controlling glass stickiness might be relevant experimentally <cit.>.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  § EFFECT OF GLASS STICKINESS ON MACROSCOPIC ELASTICITY
  
  
  
  
  
   §.§ Elastic moduli and Poisson's ratio
  
  
  We start our investigation of macroelasticity by studying the sample-to-sample averages of athermal shear to bulk moduli ratio G/K, and Poisson's ratio ν (see Appendix <ref> for precise definitions), of the QSS, DSS, and CSS ensembles described in the previous section, as a function of their respective control parameters. The results are shown in Fig. <ref>; we observe that G/K increases significantly in the DSS and CSS ensembles by decreasing ρ and r_c, respectively, with a total variation of ≈180% in the former case. In addition, the high density plateau G/K≈0.15 observed for the DSS glasses is in excellent agreement with the values measured for computer glasses featuring purely repulsive, inverse-power-law pairwise potentials <cit.>. This indicates that at high densities the relative strength of attractive interactions is too small to play an important role in determining the elastic properties of the DSS glasses, as expected.
  
  
  
  In contrast, we observe in Figs. <ref>a and <ref>d that G/K, and therefore also ν, are largely insensitive to varying glass stickiness in the QSS model, indicating that glass stickiness alone is not predictive of elastic properties. In particular, we show in Fig. <ref> that the QSS pairwise interaction with q=20 (referred to as `glass 1' in that figure) has a very comparable attractive term to that of the CSS interaction with r_c=1.2 (referred to as `glass 2' in Fig. <ref>), but nevertheless features a much smaller G/K ratio, and therefore also a larger Poisson's ratio ν. The observed difference between the large-glass-stickiness elastic properties of the QSS ensemble, and those of the DSS and CSS ensembles, will be discussed further in what follows. 
  
  
  
  
  
   §.§ Elastic moduli's relative fluctuations
  
  
  
  In addition to ensemble-averaged values of the shear and bulk moduli, and of their ratio, we consider next two dimensionless, N-independent measures of their sample-to-sample fluctuations, defined respectively as 
  
      χ__G =√(N⟨ (G - ⟨ G ⟩)^2 ⟩)/⟨ G ⟩    χ__K=√(N⟨ (K - ⟨ K ⟩)^2 ⟩)/⟨ K ⟩ ,
   
  where ⟨∙⟩ denotes an ensemble average. These observables can be considered as quantifiers of mechanical disorder <cit.>; similar quantifiers based on spatial (coarse-grained) statistics rather than sample-to-sample statistics were put forward by Schirmacher <cit.> in the context of the vibrational density of states and transport properties of glasses, and also discussed in Ref. <cit.>. Recently, χ__G was shown in Ref. <cit.> to predict long-wavelength wave attenuation rates in computer glasses. Appendix <ref> provides details about how χ__G and χ__K were estimated for our glass ensembles. 
  
  Our results are shown in Fig. <ref>; we find stark differences between the way the two quantifiers χ__G and χ__K depend on each model's key control parameter. Remarkably, χ__G is roughly independent of q in the QSS system, in line with the underwhelming variation of G/K with q as seen in Fig. <ref>. At the same time χ__G significantly decreases upon increasing glass stickiness in the DSS and CSS ensembles, suggesting the increased mechanical stability of those glasses <cit.>. 
  
  
  
  In contrast, χ__K appears to depend strongly on q in the QSS glasses – it grows by roughly a factor of 6 with increasing q. Similarly, increasing glass stickiness in the DSS and CSS ensembles leads to an increase in the sample-to-sample fluctuations of K as captured by χ__K, opposite to the decrease we observe in χ__G upon increasing glass stickiness. We note, importantly, that the value of χ__K remains substantially smaller than χ__G under all considered scenarios. These trends and their origins are further discussed in Sec. <ref>.
  
  
  
  
  
  
  
  
  
  
  
  § EFFECT OF GLASS STICKINESS ON MICROSCOPIC ELASTICITY
  
  
  
  As discussed in the Introduction, the macroscopic mechanical stability of disordered solids is often related to the statistical and energetic properties of low-frequency, nonphononic, quasilocalized vibrational modes that emerge from the microscopic disorder and glassy structural frustration <cit.>. In this Section we thus focus on the microscopic elasticity of the glasses generated by the three protocols described in Sec. <ref>. Precise definitions of the observables studied can be found in Appendix <ref>.
  
  
  
  
  
  
  
  
  
  
   §.§ Density per frequency of quasilocalized modes
  
  
  
  Under a certain set of circumstances <cit.>, the vibrational density of states (vDOS) of structural glasses has been shown to grow from zero frequency as
  
      D(ω)=A_gω^4 ,
  
  independent of spatial dimension <cit.>, glass history <cit.>, or interaction details <cit.>. The vibrational modes that populate this asymptotic scaling regime were shown to be quasilocalized; they feature a disordered core of size ξ_g, decorated by algebraically-decaying fields ∼r^1- <cit.>. The prefactor A_g, featuring units of frequency^-5, has been argued <cit.> to encompass information regarding both the characteristic frequency ω_g of quasilocalized modes (QLMs), and their number density N (per particle), discussed further below. 
  
  Here we measure the prefactor A_g by fitting the scaling law Eq. (<ref>) to the low-frequency tail of the vDOS; see Figs. <ref>(a)-<ref>(c) for some examples. Figures <ref>(d)-<ref>(f) of Fig. <ref> report the extracted prefactors A_g for our different glass ensembles, as a function of their respective control parameters. In the QSS system, we observe that A_g remains almost constant, independent of the exponent q, consistent with and similar to the near independence of G/K on q. In contrast with this behavior, A_g varies by an order of magnitude in the DSS glasses, and by almost two orders of magnitude in the CSS glasses, upon increasing glass stickiness. We find the lowest prefactor to be featured by the CSS system with r_c=1.1, for which A_g≈4×10^-3. 
  
  
  
  
  Similar analyses were carried out in Refs. <cit.>, where the same CSS model was studied. In Ref. <cit.> it was argued that the prefactor B (in the notations of that work) of a (power-law) distribution of `plastic modes” (presumably the same objects referred to here as QLMs) stiffnesses was inferred by subtracting the Debye contribution <cit.> from the eigenvalue distribution of the Hessian matrix. The result of this inference was that B varies over a factor of ≈ 4.5, by varying r_c from 1.3 to 2.4, in good agreement with our measurements, (cf. Fig. <ref>), and notice that we probed smaller r_c's. Different from our analysis here, in the analysis of Ref. <cit.> the exponent of the plastic modes' stiffness distribution — known now to be universal <cit.> (however see Refs. <cit.>) — was treated as a fitting parameter (whose values are not reported), and the explicit finite-size-effects seen in the Debye contribution <cit.> were not accounted for. Nevertheless, we stress that the trend we observe for A_g with varying glass stickiness has been pointed out first in Ref. <cit.>.
  
  We next turn to examining the effect of glass stickiness on the core size of QLMs.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
   §.§ Core-size of quasilocalized modes
  
  
  Following Ref. <cit.>, the length ξ_g that represents the (linear) core size of QLMs can be obtained by considering the characteristic spatial-decay profiles c(r) of the linear response of a glass to local force dipoles, defined and explained in detail in Appendix <ref>. Figures <ref>(a)-(c) show the decay functions c(r), while Figs. <ref>(d)-(f) show the products r^6c(r), with the aim of visualizing the length ξ_g beyond which the continuum scaling c(r)∼r^-6 starts to hold. As demonstrated in Appendix <ref>, we define the length ξ_g as the location of the maximum of the products r^6c(r), marked by the vertical dashed line in Figs. <ref>(d)-(f), for the largest and smallest value of the relevant control parameter.
  
  
  
  Our estimations of the length ξ_g are shown in Fig. <ref>(g)-(i); we observe that ξ_g depends only weakly on q in the QSS system, while it appears to depend strongly on the respective control parameters of the DSS and CSS systems, varying by up to a factor of 2. This variation is large compared to the one that stems from very deep supercooling of equilibrium parent configurations: in Ref. <cit.> a decrease of approximately 40% was observed between ξ_g measured for glasses quenched from high parents temperatures, and that measured in glasses quenched from very deeply supercooled states.
  
  We next consider the effect of glass stickiness on the characteristic frequency and number density of QLMs.
  
  
  
  
  
  
  
  
  
  
   §.§ Quasilocalized modes' characteristic frequency and number density
  
  
  In parallel to the decrease of QLMs' core size upon thermal annealing or deep supercooling, the characteristic frequency of QLMs has been shown to increase, by up to a factor of ≈2 <cit.>. It is therefore of interest to examine how the strength of attractive interactions of our model glasses affects their embedded QLMs' characteristic frequencies.
  
  With the length ξ_g in hand (see Fig. <ref>), we are in position to estimate the characteristic frequency ω_g of QLMs in our different glass ensembles via the relation 
  
      ω_g=2πc_s/ξ_g ,
  
  confirmed recently in Ref. <cit.>, but proposed in a similar form earlier <cit.>. Our results are reported in Fig. <ref>(a)-(c); we find that QLMs in the QSS model tend to not stiffen in any appreciable manner upon changing the exponent q. In contrast, the DSS and CSS systems show a substantial increase in the characteristic frequency ω_g, by a factor of approximately 2, by increasing glass stickiness.
  
  Finally, thermal annealing processes have also been shown to deplete QLMs <cit.>, thus it is interesting to ask how the degree of glass stickiness affects the total number of QLMs in our glasses. Following Refs. <cit.>, we estimate the number density of QLMs for our different glass ensembles via
  
      N=A_gω_g^5 .
  
  
  Our results are displayed in Fig. <ref>(d)-(f). As expected from the weak q-dependence of both A_g and ξ_g (and thus of ω_g) in the QSS system, QLMs show no depletion in that system. The contrasting result for the DSS and CSS systems is rather interesting; Figs. <ref>(e)-(f) show that the number of QLMs in these system are not depleted by increasing glass stickiness, with the exception of r_c=1.1, which features an anomalously dilute population of QLMs: N∼ O(10^-2). This means that most of the change we find in A_g in these models (see Fig. <ref>) stems from the stiffening of QLMs with increasing glass stickiness — reflected by the variation of ω_g as shown in Fig. <ref>a-c — rather than from their depletion, as discussed at length in Refs. <cit.>. In the most extreme case, namely the CSS glasses with r_c=1.1, the change in the prefactor A_g cannot be fully accounted for by the stiffening of ω_g, indicating that QLMs are then depleted.
  
  One more interesting observation should be mentioned; up to some noticeable noise in N (stemming from uncertainties, primarily in the extraction of  A_g, but also of ξ_g, cf. Eqs. (<ref>) and (<ref>)), the value of N≈0.3 appears to be quasi-universal <cit.>, for all of the ensembles studied here, with the aforementioned exception of the CSS system with r_c=1.1, the most stable amongst all systems studied. Understanding the precise mechanism that leads not only to the observed substantial stiffening, but also to the depletion of QLMs, is left for future studies. 
  
  We note finally that N is not expected to remain bounded from above near the unjamming transition; in Refs. <cit.> it was shown that disordered packings of harmonic spheres confined at pressure p feature A_g∼p^-4. Associating ω_g with the characteristic frequency scale ω^⋆ of anomalous modes <cit.>, one expects ω_g∼ω^⋆∼√(p), then Eq. (<ref>) implies that N∼p^-3/2, i.e. it diverges near unjamming. Further work is needed to establish the relevance of Eq. (<ref>) to unjamming.  
  
  
  
  
  
  
  
  
  
  
  
  
  
  § INTERPRETATIONS, INSIGHTS AND PROPOSITIONS
  
  
  
  Using two simple computer glass models in which the strength of attractive interactions can be tuned, we demonstrated that glasses' macro- and microscopic elastic properties, mechanical disorder and stability, can depend dramatically on the strength and on the functional form of pairwise attractive interactions. In what follows we hold extensive discussions about several points of interest. 
  
  
  
  
   §.§ How do features of attractive interaction potentials affect glass stability?
  
  
  
  
  Why does increasing glass stickiness lead to the mechanical stabilization of glasses in some cases (cf. DSS and CSS glasses), but not in others (cf. QSS glasses)? Intuitively, it is not surprising that attractive forces themselves stabilize mechanical structures, be them ordered <cit.> or disordered <cit.>. In the absence of attractive forces or a sufficient confining pressure, solidity itself would be lost, as happens near the unjamming point <cit.>. To better understand how the observed attraction-induced stabilization comes about, we next examine how the energies of quasilocalized, nonphononic low-frequency modes are comprised out of different contributions of the linear stability (Hessian) operator of our computer glasses. 
  
  In models employing radially-symmetric pairwise potentials, the Hessian of the potential energy can be split into four terms <cit.>:
  
      M    =    ∑_φ_ij”>0φ_ij”∂ r_ij/∂⊗∂ r_ij/∂ + ∑_φ_ij”<0φ_ij”∂ r_ij/∂⊗∂ r_ij/∂
               + ∑_φ_ij'>0φ'_ij∂^2r_ij/∂∂ + ∑_φ_ij'<0φ'_ij∂^2r_ij/∂∂
         ≡     M”_+ +  M”_- + M'_+ + M'_-  ,
  
  where M”_+,M'_+ are positive definite, and M”_-,M'_- are negative definite. With the above decomposition of M, the energy ω^2=·M·>0 associated with any given, translation-free mode  can be written as
  
      ω^2 = k_+ + k_- + f_+ + f_-  ,
  
  where 
  
      k_+    ≡   ·M”_+· ,     k_-  ≡·M”_-· , 
      
      f_+    ≡   ·M'_+· ,     f_-  ≡·M'_-· ,
  
  and we note that k_+,k_-,f_+,f_- all have units of energy/length^2, assuming  is normalized and thus dimensionless.
  
  In widely-studied purely-repulsive models such as inverse-power-law or Hertzian spheres (see e.g., Ref <cit.>), f_+=k_-=0 identically, by construction. It is known that delicate cancellations between the remaining k_+>0 and f_-<0 terms in Eq. (<ref>)  to give rise to the low frequencies featured by quasilocalized modes (QLMs) <cit.>, and by other classes of nonphononic modes <cit.> in systems residing near the unjamming transition <cit.>. In the latter case, this delicate cancellation is referred to as `marginal stability' <cit.>, and allows one to deduce scaling relations between the mean coordination and elastic moduli <cit.>. 
  
  
  
  Introducing attractive interaction terms on top of repulsive ones gives rise to nonzero stabilizing f_+ and destabilizing k_- contributions to the energy ω^2 <cit.>, cf. Eq. (<ref>). It is the interplay between these different contributions that determines glass stability, i.e. its featured abundance of soft QLMs as reflected by the dimensionless prefactor A_gω_0^5. To assess the relative contributions of the different terms to the energy of a mode, we consider the rescaled energies k_+/ω_0^2, k_-/ω_0^2, f_+/ω_0^2 and f_-/ω_0^2, and recall that ω_0^2≡ c_s^2/a_0^2, with c_s denoting the speed of shear waves, and a_0 denoting an interparticle distance. We focus in particular on the QSS system with q=20 (see `glass 1' in Fig. <ref>) and the DSS system with ρ=0.55 (see `glass 2' in Fig. <ref>), whose pairwise potentials, Poisson's ratios, and probability distributions of low-frequency vibrational modes were reported in Fig. <ref>. The attractive parts of the respective pairwise potentials of these models have roughly similar forms, however the emergent elastic properties of the resulting glasses of these models are found to be quite different, see further details in Sec. <ref>.
  
  
  
  To the aim of revealing the origin of the aforementioned differences between the QSS, q=20 and DSS, ρ=0.55 glasses, we examine the distributions of the rescaled energy contributions as defined above, extracted for ensembles of soft, quasilocalized modes, calculated as explained in Appendix <ref> (one soft QLM per glass). The results are shown in Fig. <ref>. We make three key observations regarding these distributions.
  
  
  
  
  
  
  The first observation is that, despite that the frequencies of the QLMs we calculated (as described in Appendix <ref>) in both the QSS and DSS systems span over half a decade over the frequency axis, the distributions of the dominant contributions to the modes' energies feature relative widths of order unity. Crucially, we find that these relative widths are N-independent, conditioned that the frequencies of the modes calculated reside well within the ∼ω^4 scaling regime of the nonphononic vDOS. This indicates that while the softening mechanism of QLMs can be particular to each glass model, it remains predominantly independent of the modes' frequencies within a given glass model, at low frequencies.
  
  The second observation is that, in our sticky-spheres models, the dominant softening mechanism of QLMs comes from the exploitation of large negative pairwise stiffnesses, namely from the k_- contribution to the total energy ω^2. This stands in essential contrast with the softening mechanism of QLMs in purely repulsive systems, for which |f_-|∼k_+ <cit.>. In addition, and perhaps counter-intuitively, the stabilizing effect of attractive forces is relatively very small, approximately an order of magnitude smaller than the contribution of positive, stabilizing stiffnesses. 
  
  
  
  
  
  
  Finally, the third observation is that the approximate symmetry around the origin between the dominant pair p(k_+/ω_0^2) and p(k_-/ω_0^2), as seen in the QSS system, is violated in the DSS system: both p(k_+/ω_0^2) and p(k_-/ω_0^2) feature means (marked by dashed vertical lines in Fig. <ref>) larger in amplitude and in bias towards positive energy contributions, compared to the same distributions in the QSS system. We note, importantly, that as N→∞, the sum of the means of the 4 dimensionless contributions to modes' energies — which represents the mean minimal QLM frequency in a system of N particles — is expected to vanish as ∼N^-2/5 <cit.>, so long as the glasses analyzed feature gapless ∼ω^4 nonphononic spectra. However, the key observation here is that, in the more stable glasses (DSS with ρ=0.55), the destabilizing effect of the k_- term is relatively weaker, suggesting that the population of interactions featuring large, negative stiffnesses in the DSS glasses is depleted, compared to its size in the QSS glasses. We speculate therefore that the relative mechanical instability of the QSS glasses compared to the DSS glasses, as shown in Fig. <ref>, stems from the presence of a larger population of negative-stiffness interactions in the former.
  
   
  
  
  
  
  
  
  The depletion process suggested above is validated in Fig. <ref>, which shows that indeed as the pairwise potential features larger, more negative stiffnesses, the population of interactions in the resulting glasses that feature those negative stiffnesses is depleted. Their depletion — as indicated by the reduction of the radial distribution function g(r) — is stronger in the DSS system (Fig. <ref> (b)), where `small'-`small' interactions with the largest negative stiffnesses are, as a result, entirely absent. In contrast, we see a significantly weaker depletion of the `small'-`small' and `large-small' negative-stiffness interactions in the QSS system (compare Fig. <ref> (a) and Fig. <ref>(b)), and almost no negative-stiffness-induced depletion in the population of `large-large' interactions of the QSS system. To clarify this point further, in Fig. <ref> we parametrically-plot the pair correlation function g(r) against the (negative) dimensionless stiffness φ”(r)/ω_0^2 for the QSS and DSS glasses, in the range of distances r for which φ”(r)<0. This representation further sharpens the depletion picture discussed above. 
  
  
  
  
  
  
  
  To summarize, according to the physical picture proposed here, negative stiffnesses featured by attractive pairwise potentials can serve as the main softening mechanism of low-frequency nonphononic modes, as we showed for the QSS, q=20 glasses. However, when those negative stiffnesses are made very large (compared to characteristic positive interaction stiffness scales), as seen in the DSS, ρ=0.55 glasses, the population of interactions that possess those large negative stiffnesses becomes depleted, resulting in the stabilization of the glass, and the reduction of its shear modulus fluctuations, via the stiffening of its typical QLMs' frequencies.
  
  
  
  
  
   §.§ Fluctuations of elastic moduli
  
  
  
  Another interesting trend we have identified is the opposite variations with glass stickiness of relative fluctuations of shear and bulk elastic moduli, the latter are captured by χ__G and χ__K, respectively, and reported in Fig. <ref>. In particular, we find that χ__G decreases, while χ__K increases, upon increasing glass stickiness – with the interesting exception of χ__G in the QSS glasses, that appears to be invariant to changing glass stickiness. 
  
  
  
  What is the origin of these opposite trends? We proposed above that the stiffening and depletion of QLMs should lead to the reduction in moduli fluctuations, due to their diminished effect on moduli's respective nonaffine terms <cit.>. The latter are shown below to manifest the influence of soft modes on elastic moduli and their fluctuations. This relation between soft modes and elastic moduli can explain, on a qualitative level, the trends seen in χ__G (see Fig. <ref>b,c), which decreases substantially upon increasing glass stickiness in the DSS and CSS systems.
  
  
  
  One more physical factor that may control the effect of QLMs' properties on elastic moduli fluctuations of glasses can be identified by writing the nonaffine term of any elastic moduli E as <cit.>
  
      E_≡∂^2U/∂ϵ∂·M^-1·∂^2U/∂∂ϵ/V = 1/V∑_ℓ(ψ^(ℓ)·∂^2U/∂∂ϵ)^2/ω_ℓ^2 ,
  
  where ϵ represents a strain parameter (cf. the shear and dilational strain parameters γ and η of Eqs. (<ref>) and (<ref>) in Appendix <ref>, respectively), and ψ^(ℓ) is the eigenmode of M pertaining to the eigenvalue ω_ℓ^2. This expression clarifies that not only do the statistics of soft modes affect the typical values and fluctuations of elastic moduli, but so does the strength of their coupling to different deformation geometries. 
  
  
  
  Recently, a set of tools to extract the core properties of QLMs, including their deformation-coupling, was developed in Ref. <cit.>. Here we employ those tools to probe the way QLMs in our different computer glasses couple to external deformations; see a detailed explanation about the QLMs, their deformation-coupling definitions, and calculations in Appendix <ref>. The analysis determines the ratio of a QLM's dilatational strain coupling to shear strain coupling, referred to here as simply `dilation/shear'. This ratio is calculated for a single QLM extracted for each member (i.e. one QLM per glassy sample) of our different sets of glass ensembles.
  
  
  
  The results of these calculations are shown in Fig. <ref>; we find that glass stickiness has a pronounced effect on the mean dilation vs. shear coupling of QLMs (middle line on bars), which increases dramatically with increasing glass stickiness in the DSS and CSS ensembles. This is an interesting observation on its own right – that the form of the interaction potential of a glass has a strong and systematic effect on the structural and mechanical properties of that glass's QLMs. To the best of our knowledge, this observation has not been made in previous literature; an extensive investigation of these effects and their importance is left for future studies. 
  
  In the QSS system the mean dilation vs. shear coupling remains vanishing small for all q's (see leftmost bars in Fig. <ref>); however, the fluctuations of these coupling ratios increase dramatically with increasing q. As can be seen from Eq. (<ref>), the square of the couplings (ψ^(ℓ)·∂^2U/∂∂ϵ)^2 enter the expression for the nonaffine modulus, implying that fluctuations in the dilation to shear coupling ratio would be echoed by fluctuations of the associated elastic moduli, regardless of the former's fluctuations sign. This mechanism is suggested to qualitatively explain the behavior of χ__K as seen in Fig. <ref>d. Curiously, χ__K and the fluctuations in the shear to dilation coupling ratios discussed here are the only observables considered in our work that show a substantial dependence on the exponent q in the QSS glasses. 
  
  We note finally that similar ideas regarding QLMs' deformation couplings were put forward in the context of the elastic properties of metallic glasses in Ref. <cit.>. In that work a distinction is made between `shear softening' and `pressure softening' of `local soft regions', which echoes some of the differences we showed here between QLMs' properties in non-sticky and sticky glasses, respectively. We speculate that our framework of QLMs, their abundance, and their deformation-coupling properties, are concrete micromechanical realizations of the `shear softening' and `pressure softening' concepts put forward in Ref. <cit.>. 
  
  
  
  
  
  
   §.§ Glasses' STZ size vs. Poisson's ratio
  
  
  
  
  Plastic flow in structural glasses is known to proceed via dissipative, localized rearrangements of a few tens or hundreds of particles <cit.>. The precursors of those rearrangements — called Shear Transformation Zones (STZs) —, have been recently shown using computer simulations to correspond to a subset of soft QLMs <cit.>. This allows us to meaningfully compare between QLMs' core size measured in our computer glasses, to the sizes of shear transformation zones (STZs) measured in the experiments on bulk metallic glasses reported in Ref. <cit.>. To this aim we estimated the STZ size as c_ ξ_g^3, 
  with c_ a constant of order unity.
  
  
  
  The results are shown in Fig. <ref>, where we plot the estimated STZ size against the Poisson's ratio of the glasses that host those STZs. The agreement is satisfying, both in magnitude as well as in trend; in the laboratory metallic glasses, and in our computer glasses, the size of STZs grows upon increasing the hosting glass's Poisson's ratio. At high Poisson's ratios the scatter in STZ sizes increases, indicating that the correspondence between STZ size and Poisson's ratio is not one-to-one.
  
  
  
  
  
  
  
  
   §.§ Is the Poisson's ratio an indicator of a glass's susceptibility to plastic flow?
  
  
  We next comment on the possible relation between a glass's Poisson's ratio ν and its degree of ductility or brittleness, as suggested in previous work, e.g., Refs <cit.>. We first note that in Refs. <cit.> it was shown that CSS glasses with small r_c feature brittle failure, while larger r_c glasses feature ductile failure. Since we find that the prefactor A_g depends strongly on r_c — while N is mostly constant ( cf. Fig. <ref>) — we conclude that it is A_g that facilitates or inhibits plasticity via the abundance or sparsity of STZs, respectively. 
  
  
  
  For these reasons, we plot in Fig. <ref> the dimensionless prefactors A_gω_0^5 of our glass ensembles, against those glasses' mean Poisson's ratios ν. In addition, we measure and plot in Fig. <ref> the same observables for polydisperse, inverse-power-law soft-sphere computer glasses quenched from a broad range of parent temperatures (`POLY' in the figure legend, see model details and parameters in Ref. <cit.>), for Hertzian binary-mixture soft-sphere computer glasses (HRZ in the figure legend, see model details in Ref. <cit.>), for the Stillinger-Weber tetrahedral-network glass-former <cit.> (SW in the figure legend), for the Orthophenyl molecular-glass former (OTP in the figure legend, for details on the model see Ref. <cit.>), and for a polymeric glass (PG in the figure legend, see details also in Ref. <cit.>). We find that while a single model's Poisson's ratio ν seems to form a monotonically increasing function with that model's prefactor A_g, there is no deep, overarching relation between these observables, as indicated by the huge spread of A_g seen at almost all ν values, once  different model systems are considered. Similar statements were previously made in Refs. <cit.>.
  
  
  
  
  
  
   §.§ Micro- vs. macroscopic quantifiers of disorder
  
  
  The satisfying agreement we find between the experimental and simulational STZ sizes, and their correlated variation with the Poisson's ratio ν, is an interesting connection between micro- and macroelastic  observables. It suggests a possible connection between macroscopic and microscopic quantifiers of disorder, which we test next. 
  
  
  
  
  In Fig. <ref> we plot the dimensionless prefactor A_gω_0^5 of the vDOS (reported in Fig. <ref> ) against the dimensionless (and N-independent, see Eq. (<ref>)) measure χ__G of sample-to-sample fluctuations of the shear modulus (reported in Fig. <ref>). We also include data measured for the POLY, HRZ and SW glasses, see details about these models in the previous Subsection. Despite some measurable scatter, the good correlation is apparent and impressive, constituting a key result of our work, and another interesting link between microscopic mechanical fluctuations and macroscopic ones. 
  
  
  
  
  
  
  
  We end this discussion section with one final observation, shown in Fig. <ref>, where we plot the dimensionless length ξ_g/a_0 vs. χ__G, for the QSS, DSS and CSS systems, on logarithmic scales. The dashed line corresponds to a ξ_g/a_0∼χ__G^2/3 scaling, explained next.
  
  
  
  Recall that QLMs characteristic frequency ω_g is related to the crossover length ξ_g between micromechanical-fluctuations-dominated to continuum-elastic-like responses via ω_g/ω_0∼a_0/ξ_g, as established in Ref. <cit.> and discussed in Sec. <ref> above. A similar crossover between heat transport via propagating phonons, to heat transport via `diffusons' <cit.> occurs at the Ioffe-Regel frequency ω_, defined implicitly via the frequency-dependent transverse wave attenuation rate Γ(ω) as Γ(ω_)∼ω_. According to Heterogeneous Elasticity Theory (HET) <cit.>
  
      Γ(ω)∼χ__G^2ω^+1 ,
  
  at low wave frequencies ω, as verified recently in two-dimension computer glasses in <cit.>, and recall that  denotes the dimension of space. Assuming that Eq. (<ref>) persists up to ω_, we expect that ω_∼χ__G^-2/. 
  
  Given the above discussion, a reasonable proposition would be to to associate ω_g with ω_ (see also Refs. <cit.>), in which case we expect
  
      ξ_g/a_0 ∼χ__G^2/ ,
  
  as we indeed show in Fig. <ref> for our computer glasses, forming yet another interesting and potentially-useful connection between micro- and macroscopic quantifiers of mechanical disorder.
  
  
  
  § SUMMARY OF MAIN RESULTS
  
  
  
  In this work we have studied how increasing the strength of attractive interactions (referred to here as `glass stickiness') between the particles of simple computer glasses affect those glasses' micro- and macroscopic elastic properties, and their featured degree of mechanical disorder. Our main findings are the following:
  
  
    * The degree of mechanical disorder featured by glasses that were quickly quenched from high-T_p liquid-configurations can vary substantially due to changes in their interparticle potential. In particular, we find that the relative fluctuations χ__G of the shear modulus can change by nearly a factor of 4 (see Fig. <ref>), and that the linear size ξ_g of quasilocalized excitations can change by up to a factor of 2 (see Fig. <ref>) by varying glass stickiness alone. The relative magnitude of these effects is similar to that seen to be induced by thermal annealing <cit.>. 
  
  
    * The geometry of soft quasilocalized modes (QLMs) — which have been recently shown <cit.> to exist in any structural glass quenched from a melt — can be sensitive to details of the interaction potential. In particular, we show in Sec. <ref> that the dilatational component of the QLMs' eigenstrains can grow from essentially zero (in non-sticky glasses) to a few tens of percents of their dominant shear eigenstrain (in sticky glasses). 
  
  
    * The physical mechanism responsible for the emergence of soft QLMs is non-universal (see Fig. <ref> and associated discussion). In particular, the dominant contribution to decreasing QLMs' energies in sticky glasses is shown to be associated with negative curvatures – that are generically featured by attractive pairwise interactions. At the same time, we highlight in Sec. <ref> the role of large negative curvatures of the pairwise potential in suppressing the number fraction and increasing the characteristic frequencies of QLMs.
  
  
  
  
    * We find a quasi-universal relation (see Fig. <ref>) between two dimensionless quantifiers of mechanical disorder, namely the (dimensionless) prefactor A_g of the nonphononic vDOS, and the quantifier χ__G of elastic moduli fluctuations. This relation is shown to hold across 8 different glass models that span very large ranges of these two quantifiers.
  
  
  
  
  
  
  
  
  § OUTLOOK
  
  
  
  Our results in this work constitute a starting point for several further investigations. We first reiterate that in an accompanying paper <cit.> we show that glass stickiness affects the way thermal annealing — in the form of deep supercooling of parent equilibrium states — induces changes in elastic properties of simple computer glasses. 
  
  Next, we propose that the real-space counterpart of the analysis presented in Fig. <ref>, that shows how low-frequency QLMs' energies are composed of stabilizing and destabilizing terms, and the latters' relative contributions, should be systematically carried out. The aim would be to reach an understanding regarding the key destabilizing mechanisms and their variation as a function of the interparticle potential properties, and as a function of the emergent populations of interactions featuring positive and negative stiffnesses and forces. 
  
  The mechanical properties of the CSS model have been investigated in Refs. <cit.>, where it was shown that increasing glass stickiness leads to brittle-like failure under uniaxial tension. It would be interesting to investigate how the deformation-coupling properties of the STZs (which are a subset of the QLMs) in the CSS model, as shown in Fig. <ref> and discussed in Sec. <ref>, affect different failure modes such as uniaxial compression or hydrostatic tension. Furthermore, it is important to establish to what extent the analogy between thermal-annealing-induced stability and glass-stickiness-induced stability generally persists in dynamic mechanical tests of sticky-sphere glasses. 
  
  Finally, one of our key results, shown in Fig. <ref>, strongly suggests a close connection between the (sample-to-sample) fluctuations of macroscopic shear moduli, as captured by χ__G, and the crossover length ξ_g, which also represents the core-size of STZs <cit.>. This connection should be more firmly established, with the aim of building a unifying framework that will allow to effectively quantify the degree of mechanical disorder that a given glass possesses, and compare it — on the same footing — with other classes of disordered materials.
  
  
  
  
  
  
  
  
  
  
  
  We warmly thank Srikanth Sastry, Geert Kapteijns, David Richard, Eran Bouchbinder, and J. Chattoraj for fruitful discussions. We are indebted to David Richard for providing us Stillinger-Weber computer glasses and elasticity data. E. L. acknowledges support from the Netherlands Organisation for Scientific Research (NWO) (Vidi grant no. 680-47-554/3259). K. G. L gratefully acknowledges the computer resources provided by the Laboratorio Nacional de Supercómputo del Sureste de México, CONACYT member of the national laboratories network. M. P. C. acknowledges support from the Singapore Ministry of Education throughthe Academic Research Fund (MOE2017-T2-1-066(S)). Parts of this work were carried out on the Dutch national e-infrastructure with the support of SURF Cooperative.
  
  
  
  
  
  
  
  § DEFINITIONS OF OBSERVABLES
  
  
  In this section we list and provide precise definitions of the physical observables we focused on in this study, and some of the methods of their measurement. We divide the observables to macroscopic and microscopic ones, in the next Subsections.
  
  
  
   §.§ Macroscopic elasticity
  
  
  
  
  
    §.§.§ Elastic moduli
  
  
  We start with athermal (T=0) elastic moduli <cit.>; the shear modulus G is defined as
  
      G ≡1/Vd^2U/dγ^2= ∂^2U/∂γ^2-∂^2U/∂γ∂·M^-1·∂ ^2U/∂∂γ/V ,
  
  where d/dγ denotes the total derivative under the constraints of mechanical equilibrium <cit.>,  denotes particles' coordinates, M≡∂^2U/∂∂ is the Hessian matrix of the potential U, and γ is a shear-strain parameter that parameterizes the imposed affine simple shear (in the x-y plane) transformation of coordinates →H(γ)· with
  
      H(γ) =  ( [ 1 γ 0; 0 1 0; 0 0 1 ]) .
  
  
  
  The bulk modulus K is defined as
  
      K ≡ -1/dp/dη = ∂^2U/∂η^2 -∂ U/∂η- ∂^2U/∂η∂·M^-1·∂^2U/∂∂η/V^2 ,
  
  where 
  
      p≡-1/V∂ U/∂η
  
  is the pressure,  is the dimension of space, and η is an expansive-strain parameter that parameterizes the imposed affine expansive transformation of coordinates →H(η)· as
  
      H(η) =  ( [ e^η   0   0;   0 e^η   0;   0   0 e^η ]) .
  
  With the definitions of the shear and bulk moduli in hand, the Poisson's ratio ν of a 3D solid is given by
  
      ν≡3K-2G/6K+2G = 3-2G/K/6+2G/K .
  
  
  
  
    §.§.§ Characteristic pressure scale
  
  
  
  For models featuring radially-symmetric pairwise potentials — as employed throughout this work — the pressure can be decomposed into positive and negative contributions, as
  
      p = 1/V∑_f_ij>0 f_ijr_ij - 1/V∑_f_ij<0(-f_ij)r_ij≡ p_+ - p_- .
  
  It is natural to use this decomposition to define a characteristic scale p_0≡p_+ + p_- with respect to which the pressure can be assessed. In Fig. <ref> we report p/p_0 for two of the three glass ensembles investigated, in addition of the ratios p/K for comparison. We assert that since characteristic forces and characteristic stiffnesses do not vary together in the different q glasses, the correct way to compare these systems on the same footing is by (approximately) fixing the scale p_0, as we did for the QSS glasses. 
  
  
  
  
  
  
  
    §.§.§ Sample-to-sample elastic moduli fluctuations
  
  
  
  We also consider two dimensionless, N-independent measures of the sample-to-sample fluctuations of shear and bulk elastic moduli, defined respectively as 
  
      χ__G = √(N⟨ (G - ⟨ G ⟩)^2 ⟩)/⟨ G ⟩    χ__K = √(N⟨ (K - ⟨ K ⟩)^2 ⟩)/⟨ K ⟩ ,
  
  where ⟨∙⟩ denotes an ensemble average (these definitions are also spelled out in Eq. (<ref>) of Sect. <ref>). Estimating the sample-to-sample variance of G (appearing in the definition of χ__G above) is difficult, since its probability distribution is known to feature strong finite size effects <cit.>, similar to those discussed at length in Refs. <cit.>. In particular, it is common to observe outliers with large and negative G in small glasses, that can corrupt the estimated variance in our finite data set. 
  
  For these reasons we opt for estimating χ__G using the following approach; we start by calculating χ__G directly using the entire raw data set, following the definition as spelled out in Eq. <ref>. Then:
  
      
    * For each data point G_i we calculate χ__G^i, which is the same as χ__G but computed while excluding the i'th data point G_i from the calculation. Once χ__G^i is obtained, the i'th data point G_i is returned to the data set.
      
    * We define a percent difference δ_i≡100×(χ__G -χ__G^i)/χ__G for each data point i.
      
    * The data point with the largest percent difference δ_i is permanently removed from the total data set, and χ__G is re-calculated.
      
    * Goto 1.
  
  The scheme described above is repeated, until the maximal percent difference falls below a fixed percentage δ, which we set at 1%, as explained and motivated in Fig. <ref> and its caption.
  
  To test the scheme proposed above for estimating χ__G, we compare the result of our estimations with other approaches: a full Gaussian fit, a `3/4 Gaussian' fit, and a skewed Gaussian <cit.>, all using the built-in model from the LMFIT python library <cit.>. In addition, we test the effect of the threshold percentage δ employed. The results of these tests are displayed in Fig. <ref> and explained it its caption. Our proposed scheme for handling the finite-size-induced noise in χ__G, which is similar in spirit to the Jack-Knife method, was also employed for estimating χ__K.
  
  
  
  
   §.§ Microscopic elasticity
  
  
  
  
  
    §.§.§ Vibrational density of states
  
  
  The vibrational density of states (vDOS) is defined as
  
      D(ω) = 1/N<∑_ℓδ(ω - ω_ℓ)> ,
  
  where ω_ℓ is the vibrational frequency associated with the vibrational mode ^(ℓ) that solves the eigenvalue equation
  
      M·^(ℓ) = ω_ℓ^2 ^(ℓ) ,
  
  assuming all masses are identical and equal to unity. It is known that D(ω)=A_gω^4 as ω→0 in structural glasses quenched from a melt <cit.>, and see also Fig. <ref> of the main text. In the same figure, the prefactor A_g, which has units of a frequency^-5, is measured and reported for our glass ensembles. 
  
  
  
  
    §.§.§ Mesoelastic length and frequency scales
  
  
  
  In order to extract the lengthscale that characterizes soft, quasilocalized modes, we follow the procedure introduced in <cit.>; first, we impose a force dipole to a pair of interacting particles, of the form
  
      _ij = ∂ r_ij/∂ .
  
  The normalized response to this force is 
  
      _ij = M^-1·_ij/√(_ij·M^-2·_ij) .
  
  We define a correlation function c(r) as
  
      c(r) ≡< _r_ij,kℓ≈ r(_ij·_kℓ)^2 > ,
  
  where the median is taken over all pairs k,ℓ whose distance to the i,j pair is r, and the average is taken over independent samples. Examples of the functions c(r) are shown in Fig. <ref>a-c of the main text.
  
  
  
  Continuum elasticity tells us that, at large r, c(r)∼r^-2 <cit.>; we extract a mesoscopic length ξ_g by finding the maximum of the product r^6c(r), as shown in Fig. <ref>. With a mesoscopic length in hand, we follow <cit.> and define a characteristic frequency scale of quasilocalized modes as ω_g≡2π c_s/ξ_g. Finally, using the prefactor A_g of the vDOS, and the frequency scale ω_g, the number density of quasilocalized modes is obtained via N≡A_gω_g^5, as discussed in <cit.>. ω_g and N measured in our model glasses are shown in Fig. <ref> in the main text. 
  
  
  
    §.§.§ Deformation coupling of quasilocalized modes
  
  
  
  Studying QLMs' properties in computer glasses via a harmonic analysis can be challenging, due to the narrow set of conditions in which such an analysis is able to reveal those properties <cit.>. In particular, in situations in which A_g is small (e.g. in the CSS systems with small r_c, see Fig. <ref>), revealing QLMs by harmonic analyses is difficult, and requires unreasonably large ensemble sizes. For this reason, the deformation patterns associated with QLMs are most conveniently studied using the nonlinear quasilocalized excitations framework put forward in <cit.>. In this framework, nonlinear QLMs  solve the algebraic equation
  
      M· = M:/U””:: U””:·  ,
  
  where U””≡∂^4U/∂∂∂∂ is the rank-4 tensor of derivatives of the potential energy with respect to coordinates, and ::,:· denote quadruple and triple contractions (over N× components), respectively.
  This framework currently allows to compute the one of the softest quasilocalized modes given a computer glass, regardless of how its frequency relates to phonon frequencies, and of the degree of hybridization of harmonic, quasilocalized modes with phonons. We followed the protocol described in <cit.> to calculate a single soft nonlinear QLM in each glassy sample. 
  
  Recently, a set of tools to extract the core properties of QLMs was developed <cit.>. For a given QLM , its coupling to dilatational and shear deformations can be extracted as follows <cit.>; we define the tensor F_ as
  
      F_ = ∂^2U/∂ϵ∂· ,
  
  where ϵ is the strain tensor that quantifies the geometry and amplitude of imposed deformations. F_ is decomposed into a deviatoric (traceless) and isotropic terms as F_=F_^+F_^, where F_^=I(F_)/ (I is the identity tensor) and F_^=F_-F_^. Without loss of generality, if the eigenvalue of F_^ with the largest absolute magnitude is negative, we switch →- (note that  is defined up to a sign <cit.>). Denoting then the largest eigenvalue of F_^ by λ_, the ratio of dilation to shear coupling is given by <cit.>
  
      /≡(F_)//λ_ .
  
  This ratio is reported in Sect. <ref> for ensembles of nonlinear modes calculated in our glasses (one mode per glass). 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  § INTERACTION POTENTIALS' COEFFICIENTS
  
  
  
  We provide a mathematical expression to obtain the coefficients c_ as a function of the exponent q in the φ_ potential (see Eq. (<ref>). The expression reads as:
  
  
      [ c_(q); c_(q); c_(q) ]=[    1/4 x_c^-2q(2+q)(4 q x_c^q-2(q+1)); -1/2 q x_c^-2(q+1)((4+q)x_c^q-2(q+2));  1/4 q x_c^-2(2+q)((2+q)x_c^q-2(q+1)) ]
  
  
  Additionally, table <ref>, provides the coefficients that ensure the first and second derivatives of the sticky spheres interaction potential, φ_ to be smooth at the dimensionless interaction cutoff x_c=r_c× x_. 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  122
  fxundefined [1]
   ifx#1
  
  fnum [1]
   #1firstoftwo
   secondoftwo
   
  fx [1]
   #1firstoftwo
   secondoftwo
   
  
  
  
  
  
  noop [0]secondoftwo
  
  ref[1]@startlink#1@href
  href[1]#1@endlink
  anitize@url [0]`
  12`$12`&12`#12`1̂2`_12`%12
  startlink[1]
  endlink[0]
  
  rl [1]href #1
  
  
  
  
  
  
  
  
  
  
  
  
  @bib@innerbibempty
  
  [Falk and Langer(1998)]falk_langer_stz
    author author M. L. Falk and author J. S. Langer, 10.1103/PhysRevE.57.7192 journal
    journal Phys. Rev. E volume 57,  pages 7192 (year 1998)NoStop
  [Götze(2008)]gotze2008complex
    author author W. Götze, @noop  title Complex dynamics of
    glass-forming liquids: A mode-coupling theory, series
    International series of monographs on physics, Vol. volume
    143 (publisher OUP Oxford, year
    2008)NoStop
  [Sollich et al.(1997)Sollich, Lequeux, Hébraud, and  Cates]sgr_prl_1997
    author author P. Sollich, author F. m. c. Lequeux, author P. Hébraud,
     and author M. E. Cates,  10.1103/PhysRevLett.78.2020 journal journal Phys. Rev. Lett. volume 78, pages 2020 (year 1997)NoStop
  [Berthier and Jack(2007)]rob_jack_PRE_2007
    author author L. Berthier and author R. L. Jack, 10.1103/PhysRevE.76.041509 journal
    journal Phys. Rev. E volume 76,  pages 041509 (year 2007)NoStop
  [Widmer-Cooper et al.(2004)Widmer-Cooper, Harrowell, and Fynewever]harrowell_isoconfiguration
    author author A. Widmer-Cooper, author P. Harrowell,  and author H. Fynewever, 10.1103/PhysRevLett.93.135701 journal journal Phys. Rev. Lett. volume 93, pages 135701 (year
    2004)NoStop
  [Widmer-Cooper et al.(2008)Widmer-Cooper, Perry, Harrowell, and  Reichman]widmer2008irreversible
    author author A. Widmer-Cooper, author H. Perry, author P. Harrowell,   and author D. R. Reichman,  10.1038/nphys1025 journal journal Nature Phys. volume 4, pages
    711 (year 2008)NoStop
  [Manning and Liu(2011)]manning2011
    author author M. L. Manning and author A. J. Liu, 10.1103/PhysRevLett.107.108302 journal journal Phys. Rev. Lett. volume 107, pages 108302 (year
    2011)NoStop
  [Bouchbinder and Langer(2009)]Eran_Jim_STZ_PRE3_2009
    author author E. Bouchbinder and author J. S. Langer, 10.1103/PhysRevE.80.031133 journal journal Phys. Rev. E volume
    80, pages 031133 (year 2009)NoStop
  [Karmakar et al.(2014)Karmakar, Dasgupta, and Sastry]smarajit_review
    author author S. Karmakar, author C. Dasgupta,
     and author S. Sastry, 10.1146/annurev-conmatphys-031113-133848 journal
    journal Annu. Rev. Condens. Matter Phys. volume 5, pages 255 (year 2014)NoStop
  [Royall and Williams(2015)]paddy_huge_review_2015
    author author C. P. Royall and author S. R. Williams, https://doi.org/10.1016/j.physrep.2014.11.004
    journal journal Phys. Rep. volume 560, pages 1  (year 2015)NoStop
  [Lerner and Bouchbinder(2018a)]cge_paper
    author author E. Lerner and author E. Bouchbinder, 10.1063/1.5024776 journal
    journal J. Chem. Phys. volume 148,  pages 214502 (year 2018a)NoStop
  [Tong and Tanaka(2018)]Tanaka_PRX_2018
    author author H. Tong and author H. Tanaka,  10.1103/PhysRevX.8.011041 journal journal Phys. Rev. X volume 8, pages
    011041 (year 2018)NoStop
  [Janssen(2018)]Liesbeth_mct_review_2018
    author author L. M. C.  Janssen, 10.3389/fphy.2018.00097
    journal journal Front. Phys. volume 6, pages 97 (year 2018)NoStop
  [Nicolas et al.(2018)Nicolas, Ferrero, Martens, and  Barrat]Barrat_elastoplastic_review_2018
    author author A. Nicolas, author E. E. Ferrero, author K. Martens,   and author J.-L. Barrat,  10.1103/RevModPhys.90.045006 journal
    journal Rev. Mod. Phys. volume 90,  pages 045006 (year 2018)NoStop
  [Richard et al.(2020a)Richard, Ozawa,
    Patinet, Stanifer, Shang,
    Ridout, Xu, Zhang,
    Morse, Barrat, Berthier,
    Falk, Guan, Liu,
    Martens, Sastry, Vandembroucq, Lerner, and Manning]david_huge_collaboration
    author author D. Richard, author M. Ozawa,
    author S. Patinet, author E. Stanifer, author
    B. Shang, author S. A. Ridout, author B. Xu, author G. Zhang, author P. K. Morse,
    author J.-L. Barrat, author L. Berthier, author
    M. L. Falk, author
    P. Guan, author A. J. Liu, author K. Martens, author S. Sastry, author D. Vandembroucq, author E. Lerner,  and author M. L.  Manning, 10.1103/PhysRevMaterials.4.113609
    journal journal Phys. Rev. Materials volume 4, pages 113609 (year
    2020a)NoStop
  [Kurchan and Levine(2010)]Kurchan_Levine_2010
    author author J. Kurchan and author D. Levine, 10.1088/1751-8113/44/3/035001 journal journal J. Phys. A Math. Theor. volume 44, pages 035001 (year
    2010)NoStop
  [Tong and Tanaka(2020)]Tanaka_Tong_prl_2020
    author author H. Tong and author H. Tanaka,  10.1103/PhysRevLett.124.225501 journal
    journal Phys. Rev. Lett. volume 124,  pages 225501 (year 2020)NoStop
  [Turnbull and Cohen(1961)]free_volume_1961
    author author D. Turnbull and author M. H. Cohen, 10.1063/1.1731549 journal
    journal The Journal of Chemical Physics volume 34, pages 120 (year 1961)NoStop
  [Laird and Schober(1991)]Schober_Laird_numerics_PRL
    author author B. B. Laird and author H. R. Schober, 10.1103/PhysRevLett.66.636 journal journal Phys. Rev. Lett. volume 66, pages 636 (year 1991)NoStop
  [Wijtmans and Manning(2017)]manning_defects
    author author S. Wijtmans and author M. L. Manning, 10.1039/C7SM00792B journal
    journal Soft Matter volume 13,  pages 5649 (year 2017)NoStop
  [Lerner et al.(2016)Lerner,
    Düring, and Bouchbinder]modes_prl_2016
    author author E. Lerner, author G. Düring,   and author E. Bouchbinder,  10.1103/PhysRevLett.117.035501 journal
    journal Phys. Rev. Lett. volume 117,  pages 035501 (year 2016)NoStop
  [Schirmacher et al.(2007)Schirmacher, Ruocco, and Scopigno]Schirmacher_prl_2007
    author author W. Schirmacher, author G. Ruocco,
     and author T. Scopigno,  10.1103/PhysRevLett.98.025501 journal
    journal Phys. Rev. Lett. volume 98,  pages 025501 (year 2007)NoStop
  [Mizuno et al.(2013)Mizuno,
    Mossa, and Barrat]mizuno_mossa_barrat_pre_2013
    author author H. Mizuno, author S. Mossa,   and author J.-L. Barrat,  10.1103/PhysRevE.87.042306 journal journal Phys. Rev. E volume 87, pages 042306 (year 2013)NoStop
  [Goodrich et al.(2014)Goodrich, Dagois-Bohy, Tighe, van Hecke, Liu, and Nagel]everyone
    author author C. P. Goodrich, author S. Dagois-Bohy, author B. P. Tighe, author M. van Hecke,
    author A. J. Liu,  and  author S. R. Nagel, 10.1103/PhysRevE.90.022138 journal journal Phys. Rev. E volume 90, pages 022138 (year 2014)NoStop
  [Mizuno et al.(2020)Mizuno,
    Ruocco, and Mossa]mossa_prb_2020
    author author H. Mizuno, author G. Ruocco,   and author S. Mossa, 10.1103/PhysRevB.101.174206 journal journal Phys. Rev. B volume 101, pages 174206 (year 2020)NoStop
  [Löffler(2003)]LOFFLER2003529
    author author J. F. Löffler, https://doi.org/10.1016/S0966-9795(03)00046-3 journal
    journal Intermetallics volume 11,  pages 529  (year 2003)NoStop
  [Park and Kim(2005)]Park2005
    author author E. S. Park and author D. H. Kim,  10.1007/BF03027480 journal journal Met. Mater. Int. volume 11, pages 19 (year 2005)NoStop
  [Schroers(2010)]schroers_review
    author author J. Schroers, 10.1002/adma.200902776 journal
    journal Adv. Mater. volume 22,  pages 1566 (year 2010)NoStop
  [Schroers(2013)]Schroers_bmgs
    author author J. Schroers, 10.1063/PT.3.1885 journal
    journal Phys. Today volume 66,  pages 32 (year 2013)NoStop
  [Li et al.(2016)Li,
    Chen, and Liu]Li2016
    author author N. Li, author W. Chen,  and  author L. Liu, 10.1007/s11837-016-1844-y journal journal JOM  volume 68, pages 1246 (year
    2016)NoStop
  [Duan et al.(2008)Duan,
    Blauwe], Lind, Schramm, and  Johnson]composition1
    author author G. Duan, author K. D. Blauwe],
    author M. L. Lind, author J. P. Schramm,  and author W. L. Johnson, https://doi.org/10.1016/j.scriptamat.2007.10.001 journal
    journal Scr. Mater. volume 58,  pages 159  (year 2008)NoStop
  [Fan et al.(2006)Fan,
    Freels, Choo, Liaw,
    Li, Rhim, Johnson,
    Yu, and Wang]composition2
    author author G. J. Fan, author M. Freels,
    author H. Choo, author
    P. K. Liaw, author
    J. J. Z. Li, author
    W.-K. Rhim, author
    W. L. Johnson, author
    P. Yu,  and author
    W. H. Wang, 10.1063/1.2408634 journal journal Appl. Phys.
    Lett. volume 89, pages 241917
    (year 2006)NoStop
  [Le Houérou et al.(2003)Le Houérou, Sangleboeuf, Dériano,
    Rouxel, and Duisit]composition3
    author author V. Le Houérou, author J.-C. Sangleboeuf, author S. Dériano, author T. Rouxel,   and author G. Duisit, https://doi.org/10.1016/S0022-3093(02)01937-3 journal journal J. Non-Cryst. Solids volume 316, pages 54  (year
    2003)NoStop
  [Wang(2012)]wang_review_2012
    author author W. H. Wang, https://doi.org/10.1016/j.pmatsci.2011.07.001
    journal journal Prog. Mater. Sci. volume 57, pages 487  (year
    2012)NoStop
  [Launey et al.(2009)Launey,
    Hofmann, Johnson, and Ritchie]Launey4986
    author author M. E. Launey, author D. C. Hofmann,
    author W. L. Johnson,  and  author R. O. Ritchie, 10.1073/pnas.0900740106 journal journal
    Proc. Natl. Acad. Sci. U.S.A. volume 106,  pages 4986 (year 2009)NoStop
  [Xu et al.(2010)Xu,
    Wirth, Schroers, and Johnson]johnson_gfa_apl_2010
    author author D. Xu, author B. D. Wirth,
    author J. Schroers,  and  author W. L. Johnson, 10.1063/1.3462315 journal journal
    Appl. Phys. Lett. volume 97, pages
    024102 (year 2010)NoStop
  [Zhang et al.(2015)Zhang,
    Dice, Liu, Schroers,
    Shattuck, and O’Hern]Zhang_gfa_2015
    author author K. Zhang, author B. Dice,
    author Y. Liu, author
    J. Schroers, author
    M. D. Shattuck,  and author
    C. S. O’Hern, 10.1063/1.4927560 journal journal J. Chem.
    Phys. volume 143, pages 054501
    (year 2015)NoStop
  [O'Hern et al.(2003)O'Hern,
    Silbert, Liu, and Nagel]ohern2003
    author author C. S. O'Hern, author L. E. Silbert,
    author A. J. Liu,  and  author S. R. Nagel, 10.1103/PhysRevE.68.011306 journal journal Phys. Rev. E volume 68, pages 011306 (year 2003)NoStop
  [Liu and Nagel(2010)]liu_review
    author author A. J. Liu and author S. R. Nagel, 10.1146/annurev-conmatphys-070909-104045
    journal journal Annu. Rev. Condens. Matter
    Phys. volume 1, pages 347 (year 2010)NoStop
  [van Hecke(2010)]van_hecke_review
    author author M. van
    Hecke, http://stacks.iop.org/0953-8984/22/i=3/a=033101 journal journal J. Phys.: Condens. Matter volume 22, pages 033101 (year
    2010)NoStop
  [Wyart et al.(2005)Wyart,
    Silbert, Nagel, and Witten]matthieu_PRE_2005
    author author M. Wyart, author L. E. Silbert,
    author S. R. Nagel,  and  author T. A. Witten, 10.1103/PhysRevE.72.051306 journal journal Phys. Rev. E volume 72, pages 051306 (year 2005)NoStop
  [Silbert et al.(2005)Silbert, Liu, and Nagel]Silbert_prl_2005
    author author L. E. Silbert, author A. J. Liu,   and author S. R. Nagel, 10.1103/PhysRevLett.95.098301 journal journal Phys. Rev. Lett. volume 95, pages 098301 (year 2005)NoStop
  [Wyart(2010)]mw_EM_epl
    author author M. Wyart, http://stacks.iop.org/0295-5075/89/i=6/a=64001 journal journal Europhys. Lett. volume 89, pages 64001 (year
    2010)NoStop
  [DeGiuli et al.(2014)DeGiuli, Laversanne-Finot, During,
    Lerner, and Wyart]eric_boson_peak_emt
    author author E. DeGiuli, author A. Laversanne-Finot, author G. During, author E. Lerner,   and author M. Wyart, 10.1039/C4SM00561A journal journal
    Soft Matter volume 10, pages 5628
    (year 2014)NoStop
  [Yan et al.(2016)Yan,
    DeGiuli, and Wyart]new_variational_argument_epl_2016
    author author L. Yan, author E. DeGiuli,  and  author M. Wyart, http://stacks.iop.org/0295-5075/114/i=2/a=26003 journal
    journal Europhys. Lett. volume 114,  pages 26003 (year 2016)NoStop
  [Ellenbroek et al.(2009)Ellenbroek, Zeravcic, van Saarloos, and  van Hecke]Ellenbroek_2009
    author author W. G. Ellenbroek, author Z. Zeravcic, author W. van
    Saarloos,  and author M. van
    Hecke, 10.1209/0295-5075/87/34004 journal
    journal Europhys. Lett. volume 87,  pages 34004 (year 2009)NoStop
  [Kooij and Lerner(2019)]stefanz_pre_2019
    author author S. Kooij and author E. Lerner,  10.1103/PhysRevE.100.042609 journal journal Phys. Rev. E volume 100, pages 042609 (year 2019)NoStop
  [Braier et al.(1990)Braier,
    Berry, and Wales]Braier1990
    author author P. A. Braier, author R. S. Berry,   and author D. J. Wales, 10.1063/1.459263 journal journal J.
    Chem. Phys. volume 93, pages 8745
    (year 1990)NoStop
  [Sciortino(2002)]Sciortino2002
    author author F. Sciortino, 10.1038/nmat752 journal
    journal Nat. Mater. volume 1,  pages 145 (year 2002)NoStop
  [Fusco et al.(2010)Fusco,
    Albaret, and Tanguy]Tanguy_pre_2010_vary_lambda_sw
    author author C. Fusco, author T. Albaret,   and author A. Tanguy, 10.1103/PhysRevE.82.066116 journal journal Phys. Rev. E volume 82, pages 066116 (year 2010)NoStop
  [Karmakar et al.(2011)Karmakar, Lerner, Procaccia, and  Zylberg]potential_itamar_pre_2011
    author author S. Karmakar, author E. Lerner,
    author I. Procaccia,  and  author J. Zylberg, 10.1103/PhysRevE.83.046106 journal journal Phys. Rev. E volume 83, pages 046106 (year 2011)NoStop
  [Dauchot et al.(2011)Dauchot, Karmakar, Procaccia, and  Zylberg]itamar_brittle_to_ductile_pre_2011
    author author O. Dauchot, author S. Karmakar,
    author I. Procaccia,  and  author J. Zylberg, 10.1103/PhysRevE.84.046105 journal journal Phys. Rev. E volume 84, pages 046105 (year 2011)NoStop
  [Beltukov et al.(2016)Beltukov, Fusco, Parshin, and  Tanguy]beltukov_sw_pre_2016
    author author Y. M. Beltukov, author C. Fusco,
    author D. A. Parshin,  and  author A. Tanguy, 10.1103/PhysRevE.93.023006 journal journal Phys. Rev. E volume 93, pages 023006 (year 2016)NoStop
  [Ciamarra and Sollich(2013)]massimo_soft_matter_2013
    author author M. P. Ciamarra and author P. Sollich, 10.1039/C3SM51505B journal
    journal Soft Matter volume 9,  pages 9557 (year 2013)NoStop
  [Shi et al.(2014)Shi,
    Luo, Yuan, and Huang]shi_intrinsic_ductility
    author author Y. Shi, author J. Luo, author F. Yuan,  and author
    L. Huang, 10.1063/1.4862959 journal journal J. Appl.
    Phys. volume 115, pages 043528
    (year 2014)NoStop
  [Sun et al.(2016)Sun,
    Hu, Wang, Zhu, Wen, Wang, Liu, and Yang]experimental_inannealability
    author author B. Sun, author Y. Hu, author D. Wang, author
    Z. Zhu, author P. Wen, author W. Wang, author C. Liu,  and  author Y. Yang, https://doi.org/10.1016/j.actamat.2016.09.014 journal
    journal Acta Mater. volume 121,  pages 266  (year 2016)NoStop
  [Babu et al.(2016)Babu,
    Mondal, Sengupta, and Karmakar]smarajit_ductile_brittle_soft_matter_2016
    author author J. S. Babu, author C. Mondal,
    author S. Sengupta,  and  author S. Karmakar, 10.1039/C5SM02200B journal journal
    Soft Matter volume 12, pages 1210
    (year 2016)NoStop
  [Molnár et al.(2017)Molnár, Ganster, and Tanguy]Tanguy_pre_2017_silicate_glass
    author author G. Molnár, author P. Ganster,   and author A. Tanguy, 10.1103/PhysRevE.95.043001 journal journal Phys. Rev. E volume 95, pages 043001 (year 2017)NoStop
  [Lerner(2019)]boring_paper
    author author E. Lerner, https://doi.org/10.1016/j.jnoncrysol.2019.119570 journal
    journal J. Non-Cryst. Solids volume
    522, pages 119570 (year 2019)NoStop
  [foo(a)]footnote
    @noop  journal journal The irrelevance of
    unjamming is evident by counting the number of interactions per particle,
    found in our models to be much larger than the Maxwell threshold of twice the
    spatial dimension <cit.>  NoStop
  [Mie(1903)]mie_potential
    author author G. Mie, 10.1002/andp.19033160802 journal
    journal Ann. Phys. (Berl.) volume
    316, pages 657 (year 1903)NoStop
  [Sastry et al.(1998)Sastry,
    Debenedetti, and Stillinger]Sastry1998
    author author S. Sastry, author P. G. Debenedetti,  and author F. H. Stillinger, 10.1038/31189 journal
    journal Nature volume 393, pages 554 (year 1998)NoStop
  [Sastry(2002)]SASTRY2002267
    author author S. Sastry, https://doi.org/10.1016/S0378-4371(02)01222-0
    journal journal Physica A Stat. Mech. Appl.  volume 315, pages 267  (year
    2002), note slow Dynamical Processes in NatureNoStop
  [Ashwin et al.(2004)Ashwin,
    Brumer, Reichman, and Sastry]Ashwin2004
    author author S. S. Ashwin, author Y. Brumer,
    author D. R. Reichman,  and  author S. Sastry, 10.1021/jp047137m journal journal J.
    Phys. Chem. B volume 108, pages 19703
    (year 2004)NoStop
  [Wang et al.(2019)Wang,
    Ninarello, Guan, Berthier,
    Szamel, and Flenner]LB_modes_2019
    author author L. Wang, author A. Ninarello,
    author P. Guan, author
    L. Berthier, author
    G. Szamel,  and author
    E. Flenner, 10.1038/s41467-018-07978-1 journal journal Nat.
    Commun. volume 10, pages 26 (year 2019)NoStop
  [Richard et al.(2020b)Richard, González-López, Kapteijns, Pater,
    Vaknin, Bouchbinder, and Lerner]modes_prl_2020
    author author D. Richard, author K. González-López, author G. Kapteijns, author R. Pater,
    author T. Vaknin, author E. Bouchbinder,  and author E. Lerner, 10.1103/PhysRevLett.125.085502 journal journal
    Phys. Rev. Lett. volume 125, pages
    085502 (year 2020b)NoStop
  [González-López et al.(2021)González-López, Shivam,
    Zheng, Ciamarra, and Lerner]inannealability
    author author K. González-López, author M. Shivam, author Y. Zheng,
    author M. P. Ciamarra,  and  author E. Lerner, 10.1103/PhysRevE.103.022606 journal journal Phys. Rev. E volume 103, pages 022606 (year 2021)NoStop
  [Kapteijns et al.(2018)Kapteijns, Bouchbinder, and Lerner]modes_prl_2018
    author author G. Kapteijns, author E. Bouchbinder,  and author E. Lerner, 10.1103/PhysRevLett.121.055501 journal journal Phys. Rev. Lett. volume 121, pages 055501 (year
    2018)NoStop
  [Rainone et al.(2020a)Rainone, Bouchbinder, and Lerner]pinching_pnas
    author author C. Rainone, author E. Bouchbinder,  and author E. Lerner, 10.1073/pnas.1919958117 journal
    journal Proc. Natl. Acad. Sci. U.S.A. volume 117, pages 5228 (year
    2020a)NoStop
  [Schober and Oligschleger(1996)]SchoberOligschleger1996
    author author H. R. Schober and author C. Oligschleger, 10.1103/PhysRevB.53.11469 journal journal Phys. Rev. B volume
    53, pages 11469 (year 1996)NoStop
  [Rainone et al.(2020b)Rainone, Bouchbinder, and Lerner]corrado_dipole_statistics_2020
    author author C. Rainone, author E. Bouchbinder,  and author E. Lerner, 10.1063/5.0005655 journal
    journal J. Chem. Phys. volume 152,  pages 194503 (year 2020b)NoStop
  [Kapteijns et al.(2019)Kapteijns, Ji, Brito, Wyart, and Lerner]fsp
    author author G. Kapteijns, author W. Ji,
    author C. Brito, author M. Wyart,  and author
    E. Lerner, 10.1103/PhysRevE.99.012106 journal journal Phys.
    Rev. E volume 99, pages 012106
    (year 2019)NoStop
  [Sengupta et al.(2011)Sengupta, Vasconcelos, Affouard, and  Sastry]sri_potential
    author author S. Sengupta, author F. Vasconcelos, author F. Affouard,  and author S. Sastry, 10.1063/1.3660201 journal
    journal J. Chem. Phys. volume 135,  pages 194503 (year 2011)NoStop
  [Chattoraj and Ciamarra(2020)]Massimo_supercooled_PRL
    author author J. Chattoraj and author M. P. Ciamarra, 10.1103/PhysRevLett.124.028001 journal journal Phys. Rev. Lett. volume 124, pages 028001 (year
    2020)NoStop
  [Berendsen et al.(1984)Berendsen, Postma, van Gunsteren,
    DiNola, and Haak]Berendsen
    author author H. J. C.  Berendsen, author J. P. M.  Postma, author W. F.  van Gunsteren, author A. DiNola,  and author J. R.  Haak, http://dx.doi.org/10.1063/1.448118
    journal journal J. Chem. Phys. volume 81, pages 3684 (year
    1984)NoStop
  [Xiao et al.(2015)Xiao,
    Tofteskov, Christensen, Dyre, and Niss]XIAO2015190
    author author W. Xiao, author J. Tofteskov,
    author T. V. Christensen,
    author J. C. Dyre,  and  author K. Niss, https://doi.org/10.1016/j.jnoncrysol.2014.08.041 journal
    journal J. Non-Cryst. Solids volume
    407, pages 190  (year 2015)NoStop
  [Schirmacher(2006)]Schirmacher_2006
    author author W. Schirmacher, 10.1209/epl/i2005-10471-9 journal journal Europhys. Lett. volume
    73, pages 892 (year 2006)NoStop
  [Marruzzo et al.(2013)Marruzzo, Schirmacher, Fratalocchi, and  Ruocco]Marruzzo2013
    author author A. Marruzzo, author W. Schirmacher, author A. Fratalocchi,  and author G. Ruocco, https://doi.org/10.1038/srep01407 journal
    journal Sci. Rep. volume 3, pages 1407 EP  (year 2013), note
    articleNoStop
  [Riggleman et al.(2010)Riggleman, Douglas, and de Pablo]riggleman_soft_matter_2010
    author author R. A. Riggleman, author J. F. Douglas,  and author J. J. de Pablo, 10.1039/B915592A journal
    journal Soft Matter volume 6,  pages 292 (year 2010)NoStop
  [Mizuno et al.(2014)Mizuno,
    Mossa, and Barrat]barrat_pnas_2014
    author author H. Mizuno, author S. Mossa,   and author J.-L. Barrat,  10.1073/pnas.1409490111 journal journal Proc. Natl. Acad. Sci. U.S.A. volume
    111, pages 11949 (year 2014)NoStop
  [Kapteijns et al.(2020a)Kapteijns, Richard, Bouchbinder, and Lerner]scattering_prl_2020
    author author G. Kapteijns, author D. Richard,
    author E. Bouchbinder,  and  author E. Lerner, https://arxiv.org/abs/2008.08337 journal journal
    arXiv preprint arXiv:2008.08337  (year
    2020a)NoStop
  [Ozawa et al.(2018)Ozawa,
    Berthier, Biroli, Rosso,  and Tarjus]Ozawa6656
    author author M. Ozawa, author L. Berthier,
    author G. Biroli, author A. Rosso,  and author
    G. Tarjus, 10.1073/pnas.1806156115 journal journal Proc.
    Natl. Acad. Sci. U.S.A. volume 115, pages 6656 (year 2018)NoStop
  [Lerner et al.(2014)Lerner,
    DeGiuli, During, and Wyart]breakdown
    author author E. Lerner, author E. DeGiuli,
    author G. During,  and author M. Wyart, 10.1039/C4SM00311J journal journal Soft Matter  volume 10, pages 5085 (year
    2014)NoStop
  [Bouchbinder and Lerner(2018)]phonon_widths
    author author E. Bouchbinder and author E. Lerner, http://stacks.iop.org/1367-2630/20/i=7/a=073022
    journal journal New J. Phys. volume 20, pages 073022 (year
    2018)NoStop
  [Shimada et al.(2020)Shimada, Mizuno, Berthier, and  Ikeda]atsushi_high_dimension_dos
    author author M. Shimada, author H. Mizuno,
    author L. Berthier,  and  author A. Ikeda, 10.1103/PhysRevE.101.052906 journal journal Phys. Rev. E volume 101, pages 052906 (year 2020)NoStop
  [Shimada et al.(2018)Shimada, Mizuno, Wyart, and Ikeda]atsushi_core_size_pre
    author author M. Shimada, author H. Mizuno,
    author M. Wyart,  and author A. Ikeda, 10.1103/PhysRevE.98.060901 journal journal Phys.
    Rev. E volume 98, pages 060901
    (year 2018)NoStop
  [Kittel(2005)]kittel2005introduction
    author author C. Kittel, @noop  title Introduction to solid
    state physics (publisher Wiley, year
    2005)NoStop
  [Lerner and Bouchbinder(2017)]protocol_prerc
    author author E. Lerner and author E. Bouchbinder, 10.1103/PhysRevE.96.020104 journal journal Phys. Rev. E volume
    96, pages 020104 (year 2017)NoStop
  [Lerner(2020)]finite_size_effect_modes_2020
    author author E. Lerner, 10.1103/PhysRevE.101.032120 journal journal Phys. Rev. E volume
    101, pages 032120 (year 2020)NoStop
  [Hong et al.(2011)Hong,
    Novikov, and Sokolov]sokolov_boson_peak_scale
    author author L. Hong, author V. N. Novikov,   and author A. P. Sokolov,  10.1103/PhysRevE.83.061508 journal journal Phys. Rev. E volume 83, pages 061508 (year 2011)NoStop
  [foo(b)]footnote_ncal
    @noop  journal journal The precise value
    of the quasi-universal N depends on how QLMs' characteristic
    frequency ω_g is obtained. In <cit.> it is shown that
    ω_g estimated via the response to local force dipole (as explained in,
    e.g., <cit.>) agrees well — up to an overall numerical factor of
    order unity — with the ξ_g-based estimation as done here  NoStop
  [Mizuno et al.(2017)Mizuno,
    Shiba, and Ikeda]ikeda_pnas
    author author H. Mizuno, author H. Shiba,   and author A. Ikeda, 10.1073/pnas.1709015114 journal journal
    Proc. Natl. Acad. Sci. U.S.A. volume 114,  pages E9767 (year 2017)NoStop
  [Rossi et al.(2011)Rossi,
    Sacanna, Irvine, Chaikin,
    Pine, and Philipse]Rossi2011
    author author L. Rossi, author S. Sacanna,
    author W. T. Irvine, author P. M. Chaikin, author
    D. J. Pine,  and author
    A. P. Philipse, 10.1039/c0sm01246g journal journal Soft Matter  volume 7, pages 4139 (year
    2011)NoStop
  [Sun et al.(2013)Sun,
    Li, Zhang, Ma, and  Zhang]Sun2013
    author author X. Sun, author Y. Li, author T. H. Zhang, author
    Y. Q. Ma,  and author
    Z. Zhang, 10.1021/la304720h journal journal Langmuir  volume 29, pages 7216 (year
    2013)NoStop
  [Koeze and Tighe(2018)]brian_sticky_prl
    author author D. J. Koeze and author B. P. Tighe, 10.1103/PhysRevLett.121.188002 journal journal Phys. Rev. Lett. volume 121, pages 188002 (year
    2018)NoStop
  [Xu et al.(2007)Xu,
    Wyart, Liu, and Nagel]mw_prl_2007_kablj_unjamming
    author author N. Xu, author M. Wyart, author A. J. Liu,  and author S. R. Nagel, 10.1103/PhysRevLett.98.175502 journal journal
    Phys. Rev. Lett. volume 98, pages
    175502 (year 2007)NoStop
  [Mizuno et al.(2016)Mizuno,
    Saitoh, and Silbert]Silbert_pre_2016_jamming
    author author H. Mizuno, author K. Saitoh,   and author L. E. Silbert,  10.1103/PhysRevE.93.062905 journal journal Phys. Rev. E volume 93, pages 062905 (year 2016)NoStop
  [Lerner and Bouchbinder(2018b)]inst_note
    author author E. Lerner and author E. Bouchbinder, 10.1103/PhysRevE.97.032140 journal journal Phys. Rev. E volume
    97, pages 032140 (year
    2018b)NoStop
  [Müller and Wyart(2015)]mw_mm_marginal_stability_2015
    author author M. Müller and author M. Wyart, 10.1146/annurev-conmatphys-031214-014614
    journal journal Annu. Rev. Condens. Matter
    Phys. volume 6, pages 177 (year 2015)NoStop
  [foo(c)]footnote2
    @noop  journal journal In cases where the
    interaction-network topology is fixed, e.g. networks of Hookean springs as
    considered in <cit.>, the destabilizing contribution k_- due to
    negative stiffnesses might be absent NoStop
  [Kooij and Lerner(2017)]stefanz_pre
    author author S. Kooij and author E. Lerner,  10.1103/PhysRevE.95.062141 journal journal Phys. Rev. E volume 95, pages 062141 (year 2017)NoStop
  [Lutsko(1989)]lutsko
    author author J. F. Lutsko, 10.1063/1.342716 journal
    journal J. Appl. Phys. volume 65,  pages 2991 (year 1989)NoStop
  [Moriel et al.(2020)Moriel,
    Lubomirsky, Lerner, and Bouchbinder]avraham_arXiv
    author author A. Moriel, author Y. Lubomirsky,
    author E. Lerner,  and author E. Bouchbinder, https://arxiv.org/abs/2007.11421 journal journal
    arXiv:2007.11421  (year 2020)NoStop
  [Argon(1979)]argon_st
    author author A. Argon, http://dx.doi.org/10.1016/0001-6160(79)90055-5
    journal journal Acta Metallurgica volume 27, pages 47  (year
    1979)NoStop
  [Spaepen(1977)]spaepen_1977
    author author F. Spaepen, https://doi.org/10.1016/0001-6160(77)90232-2
    journal journal Acta Metallurgica volume 25, pages 407  (year
    1977)NoStop
  [Richard et al.(2020c)Richard, Kapteijns, Giannini, Manning, and  Lerner]phm_arXiv
    author author D. Richard, author G. Kapteijns,
    author J. A. Giannini, author M. L. Manning,  and author E. Lerner, https://arxiv.org/abs/2007.08181 journal journal
    arXiv preprint arXiv:2007.08181  (year
    2020c)NoStop
  [Pan et al.(2008)Pan,
    Inoue, Sakurai, and Chen]stz_experimental
    author author D. Pan, author A. Inoue, author T. Sakurai,  and author M. W. Chen, 10.1073/pnas.0806051105 journal journal Proc.
    Natl. Acad. Sci. U.S.A. volume 105, pages 14769 (year 2008)NoStop
  [Lewandowski et al.(2005)Lewandowski, Wang, and Greer]brittleness_BMG
    author author J. J. Lewandowski, author W. H. Wang,  and author A. L. Greer, 10.1080/09500830500080474 journal
    journal Philos Mag Lett. volume 85,  pages 77 (year 2005)NoStop
  [Stillinger and Weber(1985)]Stillinger_Weber
    author author F. H. Stillinger and author T. A. Weber, 10.1103/PhysRevB.31.5262 journal
    journal Phys. Rev. B volume 31,  pages 5262 (year 1985)NoStop
  [Rycroft and Bouchbinder(2012)]eran_fracture_toughness_prl_2012
    author author C. H. Rycroft and author E. Bouchbinder, 10.1103/PhysRevLett.109.194301 journal journal Phys. Rev. Lett. volume 109, pages 194301 (year
    2012)NoStop
  [Kumar et al.(2011)Kumar,
    Prades-Rodel, Blatter, and Schroers]KUMAR2011585
    author author G. Kumar, author S. Prades-Rodel,
    author A. Blatter,  and  author J. Schroers, https://doi.org/10.1016/j.scriptamat.2011.06.029 journal journal Scr. Mater. volume
    65, pages 585  (year 2011)NoStop
  [Shao et al.(2020)Shao,
    Ketkaew, Gong, Zhao,
    Sohn, Bordeenithikasem, Datye, Mota, Liu, Kube,
    Liu, Chen, Yao, Wu, and Schroers]jan_2020
    author author L. Shao, author J. Ketkaew,
    author P. Gong, author
    S. Zhao, author S. Sohn, author P. Bordeenithikasem, author A. Datye, author R. M. O. Mota,
    author N. Liu, author
    S. A. Kube, author
    Y. Liu, author W. Chen, author K. Yao, author S. Wu,  and  author J. Schroers, https://doi.org/10.1016/j.mtla.2020.100828 journal
    journal Materialia volume 12,  pages 100828 (year 2020)NoStop
  [Aihemaiti et al.(2020)Aihemaiti, Li, Li, Wang,
    Chang, and Ma]AIHEMAITI2020106834
    author author N. Aihemaiti, author Q. Li,
    author M. Li, author
    T. Wang, author C. Chang,  and author X. Ma, https://doi.org/10.1016/j.intermet.2020.106834 journal
    journal Intermetallics volume 123,  pages 106834 (year 2020)NoStop
  [Allen and Feldman(1989)]diffusons1
    author author P. B. Allen and author J. L. Feldman, 10.1103/PhysRevLett.62.645 journal journal Phys. Rev. Lett. volume 62, pages 645 (year 1989)NoStop
  [Allen and Feldman(1993)]diffusons2
    author author P. B. Allen and author J. L. Feldman, 10.1103/PhysRevB.48.12581 journal journal Phys. Rev. B volume
    48, pages 12581 (year 1993)NoStop
  [Rufflé et al.(2006)Rufflé, Guimbretière, Courtens,
    Vacher, and Monaco]Monaco_prl_2006_boson_peak_ioffe_regel
    author author B. Rufflé, author G. Guimbretière, author E. Courtens, author R. Vacher,   and author G. Monaco, 10.1103/PhysRevLett.96.045502 journal journal Phys. Rev. Lett. volume 96, pages 045502 (year 2006)NoStop
  [Beltukov et al.(2013)Beltukov, Kozub, and Parshin]Parshin_prb_2013_IR
    author author Y. M. Beltukov, author V. I. Kozub,
     and author D. A. Parshin,  10.1103/PhysRevB.87.134203 journal journal Phys. Rev. B volume 87, pages 134203 (year 2013)NoStop
  [Kapteijns et al.(2020b)Kapteijns, Richard, Bouchbinder, and Lerner]scattering_arXiv_2020
    author author G. Kapteijns, author D. Richard,
    author E. Bouchbinder,  and  author E. Lerner, https://arxiv.org/abs/2008.08337 journal journal
    arXiv preprint arXiv:2008.08337  (year
    2020b)NoStop
  [Azzalini(2013)]azzalini2013skew
    author author A. Azzalini, @noop  title The skew-normal and
    related families, Vol. volume 3 (publisher
    Cambridge University Press, year 2013)NoStop
  [Newville et al.(2014)Newville, Stensitzki, Allen, and  Ingargiola]LMFIT_python
    author author M. Newville, author T. Stensitzki, author D. B. Allen,  and author A. Ingargiola, 10.5281/zenodo.11813 title LMFIT: Non-Linear Least-Square Minimization and Curve-Fitting for
    Python,  (year 2014)NoStop
  [Gartner and Lerner(2016)]SciPost2016
    author author L. Gartner and author E. Lerner, 10.21468/SciPostPhys.1.2.016 journal journal SciPost Phys. volume
    1, pages 016 (year 2016)NoStop
  [Kapteijns et al.(2020c)Kapteijns, Richard, and Lerner]episode_1_2020
    author author G. Kapteijns, author D. Richard,
     and author E. Lerner, 10.1103/PhysRevE.101.032130 journal journal Phys. Rev. E volume 101, pages 032130 (year 2020c)NoStop
  
  
  