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The Dynamics Is Encoded in the Pair Correlation Function François Landes, Giulio Biroli, Olivier Dauchot, Andrea Liu, David Reichman

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The Dynamics Is Encoded in the Pair Correlation Function François Landes, Giulio Biroli, Olivier Dauchot, Andrea Liu, David Reichman Attractive versus truncated repulsive supercooled liquids: The dynamics is encoded in the pair correlation function François Landes, Giulio Biroli, Olivier Dauchot, Andrea Liu, David Reichman To cite this version: François Landes, Giulio Biroli, Olivier Dauchot, Andrea Liu, David Reichman. Attractive versus truncated repulsive supercooled liquids: The dynamics is encoded in the pair correlation function. Physical Review E , American Physical Society (APS), 2020, 101 (1), 10.1103/PhysRevE.101.010602. hal-02439295 HAL Id: hal-02439295 https://hal.inria.fr/hal-02439295 Submitted on 14 Jan 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Attractive versus truncated repulsive supercooled liquids: The dynamics is encoded in the pair correlation function Fran¸coisP. Landes,1, 2, 3, 4 Giulio Biroli,2 Olivier Dauchot,5 Andrea J. Liu,3 and David R. Reichman4 1LRI, AO team, B^at660 Universit´eParis Sud, Orsay 91405, France∗ 2Laboratoire de Physique de l'Ecole´ normale sup´erieure, ENS, Universit´ePSL, CNRS, Sorbonne Universit´e, Universit´eParis-Diderot, Sorbonne Paris Cit´e,Paris, France 3Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104 4Department of Chemistry, Columbia University, 3000 Broadway, New York, NY 10027, USA 5UMR Gulliver 7083 CNRS, ESPCI ParisTech, PSL Research University, 10 rue Vauquelin, 75005 Paris, France We compare glassy dynamics in two liquids that differ in the form of their interaction poten- tials. Both systems have the same repulsive interactions but one has also an attractive part in the potential. These two systems exhibit very different dynamics despite having nearly identical pair correlation functions. We demonstrate that a properly weighted integral of the pair correlation function, which amplifies the subtle differences between the two systems, correctly captures their dynamical differences. The weights are obtained from a standard machine learning algorithm. PACS numbers: The central tenet of liquid state theory is that the struc- hg(r)i does not contain the physical information relevant tural and thermodynamic properties of liquids are pri- for predicting the dynamical slowdown, at least not in a marily governed by the short-range repulsive part of way amenable to predictions, and that any theory of the their interaction potentials [1,2]. The extension of these glass transition based on the sole basis of hg(r)i is bound ideas to dynamical properties, however, is still in ques- to fail. tion. Studies of glass-forming liquids with purely repul- The view that hg(r)i alone cannot explain the dynamical sive potentials show that their relaxation times τα are well-described using a perturbative analysis around the difference between the WCA and LJ systems has recently hard-sphere limit [3,4]. However, a study of the Kob- been questioned. Combining it with the force-force cor- Andersen Lennard-Jones (LJ) mixture and its repulsive relation function and treating the attractive and slowly counterpart, the Weeks-Chandler-Andersen (WCA) mix- varying forces separately, Schweizer et. al. [13] proposed ture, found that the two systems have very different dy- an alternative microscopic theory for the dynamical differ- namics despite their structural similarities [5,6]. In par- ences between LJ and WCA systems. Modeling dynamics as that of a single-particle in an effective caging potential ticular, the relaxation time τα is much smaller in the WCA system than in the LJ one at the same supercooled and using the pair-correlations based configurational en- temperatures. While hidden potential scale invariances tropy, [14] points toward hg(r)i containing enough infor- [7], complex structural indicators such as the point-to- mation to predict the transition temperature. A compar- set length [8] and triplet correlation functions markedly ison of that theory with Schweizer's has recently shown differ in these two systems [9, 10], the pair correlation they produce comparable results, although the latter is functions are instead very similar. These findings indi- more precise [15]. Independently, approximate measures cate that long-ranged attractions strongly affect dynam- of the configurational entropy based on hg(r)i can be used ics while leaving simple structural features relatively un- to estimate the dynamics via the Adam-Gibbs relation, affected. and it was argued that the resulting difference of config- urational entropy can explain the dynamical difference This result is a challenge to theories directly based on between the two systems [16{18]. Although it is un- the exact (numerically computed) thermodynamically- clear why the configurational entropy should play a role averaged pair correlation functions hg(r)i [11]. Mode at the temperatures studied, which are above the mode- coupling theory is the most prominent example of such coupling temperature, these results [16{18] suggest that a theory, and indeed it was shown to largely underesti- the relationship between hg(r)i and dynamics should be mate the difference in dynamics of the WCA and LJ mix- revisited. tures [12]. From this negative result, it was argued that In this RRapid Communication we demonstrate that a properly weighted integral of the pair correlation func- tion obtained by machine-learning (ML) techniques [19{ ∗Corresponding author: [email protected] 24], the so-called "softness" designed to correlate strongly 2 (a) (b) ing place during a time interval W = [t1; t2], for a given particle i, may be captured by computing the quantity phop(i; t)[29{32]: r 1=2 1=2 D 2E D 2E phop(i; t) ≡ (~ri − h~riiU ) (~ri − h~riiV ) ; V U (1) FIG. 1: Structure and dynamics for WCA and LJ sys- for all t 2 W , where averages are performed over the time tems. (a) The AA pair correlation function hg (r)i at AA intervals surrounding time t, i.e. U = [t1; t], V = [t; t2]. T = 0:43, for the LJ liquid (blue), and the WCA one (red); ∗ t = argmaxt2W (phop(i; t)) is the time that best sepa- (vertical bars denote the standard deviation of Pr(g)). (b) rates the trajectory ~r ([t ; t ]) into two sub-trajectories Temperature dependence of the relaxation time τ (1=T ) for i 1 2 α ~r ([t ; t∗]) and ~r ([t∗; t ]). We record p∗ (i) = p (i; t∗) the two systems. Arrows indicate the onset and MCT temper- i 1 i 2 hop hop atures. The black dashed line outlines the initial Arrhenius and the process is iterated on each sub-trajectory. If behavior of the WCA liquid. the trajectory during the time interval W is contained ∗ within a cage then phop(i) is small and corresponds to the cage's size. Conversely, if W contains two distinct cages ∗ with instantaneous mobility, fully captures the dynamical then phop is large and corresponds to the jump's ampli- ∗ differences between the LJ and the WCA supercooled liq- tude. We thus interrupt the iteration when phop < pc, uids, despite their minute differences in average structure. where pc is chosen as the root-mean-squared displace- Our results offer a simple explanation of the relationship ment (RMSD) ∆r(t), computed at the time where the 3hr4i(t) between g(r) and relaxation time, and show that softness non-Gaussian parameter α = − 1 has a max- 2 5hr2i2(t) is an important structural fingerprint of dynamics. We imum. This choice is rather stringent: only displace- emphasize that in contrast to previous works on Softness, ments several times larger than the plateau of the RMSD here we are able to extract dynamics in systems with very are considered as jumps. Finally, as jumps are never in- similar hg(r)i, showing that fluctuations in g(r) are rel- stantaneous, we assign p (i; t) = p∗ (i) to all times evant to the dynamics. We conclude that, at least in hop hop t 2 [t∗ − t ; t∗ + t ], where t = 5 in LJ-time units, i.e. principle, one should be able to compute the relaxation f f f about five ballistic times. This procedure generates a la- time of these two liquids on the sole basis of the pair bel that classifies every particle at every instant as caged correlation. or jumping. This constitutes what is called in ML jargon Following Berthier and Tarjus [6], we compare two Kob- the ground truth: yGT (i; t) ≡ Θ(phop(i; t) − pc) 2 f0; 1g. Andersen [25, 26] binary mixtures of A and B particles, If dynamics is indeed related to local structure, then one with Lennard-Jones interactions and the other with there should be small but significant structural differ- WCA interactions [2]. The latter system shares the same ences between the local neighborhoods of caged and cage- repulsive short-range interaction with the LJ system, but jumping particles, respectively. A number of approaches possesses no attractive forces, i.e. the potential is zero have attempted to identify these subtle structural signa- at the minimum r = r = 21=6σ and beyond. The cut XY tures in supercooled liquids [33{35]. They all have in potentials are smoothed at the cutoff distance, so that common the fact that they are based on a deep knowl- their second derivatives are continuous everywhere. The edge of the physics of liquids. Here, in contrast, we use two liquids are first equilibrated in the NVT ensemble; a ML approach as introduced in Refs. [19, 20, 22].
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