Structural Covariance in the Hard Sphere Fluid
Benjamin M.G.D. Carter, Francesco Turci, Pierre Ronceray, C. Patrick Royall
SStructural Covariance in the Hard Sphere Fluid
Benjamin M.G.D. Carter, a) Francesco Turci, Pierre Ronceray, and C. Patrick Royall
1, 3, 4 HH Wills Physics Laboratory, Tyndall Avenue, Bristol, BS8 1TL, UK. Princeton Center for Theoretical Science, Princeton University, Princeton, NJ 08544,USA School of Chemistry, University of Bristol, Cantock Close, Bristol, BS8 1TS,UK. Centre for Nanoscience and Quantum Information, Tyndall Avenue, Bristol, BS8 1FD,UK.
We study the joint variability of structural information in a hard sphere fluid biased to avoid crystallisationand form fivefold symmetric geometric motifs. We show that the structural covariance matrix approach,originally proposed for on-lattice liquids [Ronceray and Harrowell, JCP 2016], can be meaningfully employedto understand structural relationships between different motifs and can predict , within the linear-responseregime, structural changes related to motifs distinct from that used to bias the system.
I. INTRODUCTION
Short-range local order is a distinctive feature of theliquid state. In models of simple liquids such as theLennard-Jones liquid or the hard sphere fluid, local struc-ture has been studied via the measurement of pair corre-lation functions (which define a characteristic correlationlength) or with higher order correlations, such as rings ofparticles and recurrent geometric motifs, since the earlytimes of the theory of liquids, with the pioneering workof Bernal and Finney in “ball-bearing” models.Since then, more sophisticated probing techniqueshave been developed to characterise the local struc-ture of disordered systems: projection of the nearestneighbours onto spherical harmonics ; the statistics ofVoronoi polyhedra and their facets ; the analysis of com-mon neighbours ; the match of local motifs with mini-mum energy clusters ; persistence homology of rings ofparticles are just a few examples.The idea underpinning these analyses is that theknowledge of the degree of local order may shed lighton interesting dynamical and thermodynamical proper-ties of disordered systems in general and of liquids inparticular. These include possible signatures of precur-sors to crystallisation in metastable liquids as wellas the eventual coupling between structural and dynami-cal heterogeneities in supercooled liquids and glasses (fora review on structure in dynamically arrested systemssee ).A major issue in this approach is the fact that differentdiagnostic and analysis tools of local structural proper-ties may lead to different conclusions on the role of localstructure in liquids. For example, the role of crystallineand icosahedral order in supercooled liquids has been ex-tensively debated and the metrics used to deter-mine each of those orders play a role in the interpretationof the results. Understanding how different types of localstructural motifs correlate would permit us to systemat-ically compare different metrics, and thus open the way a) Electronic mail: [email protected] towards a unified quantitative framework for local liquidorder.Here we consider the problem of the classification oflocal order in a canonical liquid from a simple statisticalpoint of view. Recent work on a toy model of a lat-tice liquid with a purely structural energy landscape, theFavoured Local Structures model , has demonstratedthe importance of correlations between different struc-tural geometric motifs present in the liquid. Indeed, thestatistics of high-temperature structural fluctuations pro-vides key information on the liquid entropy , while theircorrelations provides a quantitative metrics for the sta-bility/instability of the liquid towards crystal formation,being good predictors of crystallisation times and surfacetensions . Inspired by these results on a highly ide-alised system, we study here the structural statistics inthe hard sphere fluid at high packing fraction, a muchmore realistic liquid model. Following closely the ap-proach proposed by Ronceray and Harrowell , we mea-sure structural covariances and show how they encode,at the same time, geometric information on the classifi-cation itself and physical information on the propensityof the system to form crystalline or fivefold symmetricstructures.The article is organised as follows: in Section II weintroduce the studied model and the structural classifi-cation of reference; in Section III we discuss the struc-tural covariance formalism and its main results in thecase of hard spheres; in Section IV demonstrate that thecovariance framework allows us to predict quantitativelythe parameter dependence of the liquid structure; and inSection V we summarise our findings and propose furtherdirections of research. II. HARD SPHERES WITH STRUCTURAL BIAS
In the hard-sphere liquid, fivefold symmetry plays animportant role, frustrating the formation of crystallineorder . The degree of fivefold frustration is oftenquantified in terms of the number of fivefold symmetricstructures, identified through the pentagonal bipyramid, a r X i v : . [ c ond - m a t . d i s - nn ] A p r a geometrical arrangement which is formed by a bondedspindle pair of particles sharing exactly five neighbours.In order to study the local structure of the system,including fivefold symmetry, we employ the Topologi-cal Cluster Classification (TCC) . This algorithm hasbeen successfully used in the past to study the struc-ture of simple liquids , gels , glasses and ather-mal packings . It identifies a total of 33 structuresbased on minimum energy clusters of elementary pairpotentials, such as the Lennard-Jones and Morse liquids.Its labelling of different structures is inherited from thelabelling of minimum energy clusters of simple liquids(with Lennard-Jones, Morse or Dzugutov interactions) inthe Cambridge Cluster Database . Labels are typicallyformed by a number and a letter: the former refers to thenumber of particles in the motif, the latter indicates thenature of the potential the motif is a minimum of (lettersfrom A to F correspond to the Morse potential with in-creasing range, Z stands for the Dzugutov potential , Kand W stand for particular forms of the Lennard-Jonespotential and X for a BCC crystalline arrangement) .In Fig. 1 we illustrate the relationship between the sev-eral structures defined in the TCC. We differentiate theseveral families of structures present in the classification:three-fold (tetrahedral), four-fold and five-fold symmet-ric structures of different numbers of particles are defined.In particular, the pentagonal bipyramid is termed “7A”.Hence, we define the total number of pentagonal bipyra-mids as N A . In this classification, small structures canbe part of larger structures. Such a multiple countingcontributes to the total number N i of structures of agiven type i . In contrast with previous studies , N A does not correspond to the number of particles detectedin pentagonal bipyramids, but to the actual number of(possibly overlapping) bipyramids detected in the liquid,and similarly for all other structures i . The relation be-tween the number of bipyramids and the probability tofind a particle in a bipyramid is nontrivial, since particlescan be part of several overlapping bipyramids.In Reference , fivefold symmetry in hard spheres hasbeen studied through the addition of a many-body energyterm H fivefold = εN A to the Hamiltonian of the system,sampling via Monte-Carlo an extended two-dimensionalphase diagram in the packing fraction φ and bias en-ergy ε (with unit temperature), see Fig. 2. This modelexhibits a rich phase behaviour: biasing the system tomore negative/positive values of ε pushes the fluid-solidphase transition to higher/lower packing fraction; atstrong enough biases, the system spontaneously nucle-ates a quasi-crystalline phase rich in five-fold symmet-ric icosahedra. We refer the reader to Reference for amore complete discussion of the phase behaviour of the7A-biased hard-sphere fluid.In the present article, we extend this work andre-examine runs of N = 2048 hard spheres in theisothermal-isochoric ensemble for different values of φ ,with a specific interest in the influence of the fivefoldbias ε on the structure of the liquid. This parameter fully determines the Hamiltonian of the system, as thehard-sphere interaction has no other contribution thanforbidding configurations with overlaps. This model isthus entirely specified by a simple local energy landscape,making it ideally suited for a first study of structural co-variance in off-lattice systems. III. STRUCTURAL COVARIANCE FORMALISM
At any given time, the number N i ( t ) of structures oftype i in the system will exhibit some deviation to itsmean, reflecting the randomness of the configurations.The keystone of our statistical analysis of liquid structureis the covariance matrix C i,j between these numbers N i and N j of structures of types i and j , which reflects thecorrelations between these random variables. We nowexplain how we compute this matrix, before discussingits structure.We consider Monte-Carlo simulations of biased and un-biased hard sphere fluids analysed with the TopologicalCluster Classification. We retrieve time series of 1000Monte-Carlo Sweeps (MCs) of the number of particles N i ( t ) or n i ( t ) = N i ( t ) /N (the intensive concentration ofstructures of type i ) for all the structures defined in theclassification, an example of which is pictured in Fig.3.We note that, by definition, an individual particle mayparticipate in more than a single motif. For example,it may be a constituent of two or more distinct pentag-onal bypiramids. This is essential for the identificationof larger structures (for instance, the 10B structure) andimplies that the concentration n i ( t ) can in principle ex-ceed unity for some motifs.Comparing the evolution of, for example, the 6Z and6A structures with the 7A structure, we notice that whilethe former presents a very similar pattern to the pen-tagonal bipyramid ( n Z concentration increases as n A increases), the other shows the opposite behaviour, sug-gesting that some structures are positively while othersare negatively correlated to the five-fold symmetric struc-ture. The time average (cid:104) n i (cid:105) = (cid:104) N i (cid:105) /N for a selection ofstructures at packing fraction φ = 0 .
54 is plotted in Fig. 4and shows that the concentrations of different structuresdiffer of several orders of magnitude and have very dif-ferent responses according to the change in the bias ε . Amore complete picture for all the motifs with a significa-tive average concentration (cid:104) n i (cid:105) > − is presented inFig. 5. Unsurprisingly, small structures typically corre-spond to large concentrations while the opposite is true,in general, for structures composed of many particles.The largest structures such as the FCC, the HCP or 13A(i.e. icosahedral) motifs in the TCC comprise 13 particlesand all have relatively small concentrations n i ∼ − .For very negative values of the bias ε , 7A structures arestrongly favoured. This is clearly accompanied by the in-crease in the number of structures composed of 7A motifssuch as 10B (termed defective icosahedron ) or 13A (theicosahedron) [see Fig. 1 for three-dimensional rendering].
6Z 7K 8K HCP FCC9K 11A 11F 12E 10K13K12K9X 9A 10A
8A 8B 9B 10B 11B11C12D 10W13A 13B11E 11W 12A 12B sp5b
FIG. 1. Topological cluster classification. (Left) Structural motifs related to threefold symmetric (5 A ) or four-fold symmetric(6 A and sp4b) local order. The sp4b unit is smaller than the octahedron and three small motifs are derived from it; its subgroupis highlighted by a dashed line. Rings are represented by colored sticks connecting grey particles, spindle particles are in yellowand additional particles are in red, as in Reference . (Right) Structural motifs related to pentagonal (7 A or sp5b) local order.The sp5b unit is smaller than the pentagonal bi-pyramid and only one motif is derived from it, its subgroup highlighted by adashed line. Notice the presence of multiple interlaced pentagonal rings in the larger structures such as 10B or 11E. Correspondingly, the concentrations of structures relatedto four-fold symmetry, such as FCC or 11F, steadily dropat negative bias values.To obtain the covariances we directly evaluate cross-correlations of the time-series at specific values of thepacking fraction φ and bias ε . For any pair of structures i, j in the classification, we define the matrix element C i,j ( φ, ε ) = N Cov( n i ( φ, ε ) , n j ( φ, ε )) (1)= Nt max − t max (cid:88) k =1 ( n i ( k ) − (cid:104) n i (cid:105) )( n j ( k ) − (cid:104) n j (cid:105) )(2)With such a definition, the covariance matrix is an inten-sive property of the system. It should in principle dependon the packing fraction φ and the bias ε . However, as weshall see in Section IV, the knowledge of the covariancematrix in unbiased conditions C ( φ ) = C ( φ, ε = 0) is suf-ficient to quantitatively predict changes in the structuralproperties of the liquid. A. Structure of the covariance matrices
We now discuss the properties of the covariance ma-trix C ( φ, ε ), obtained using Eq. 2 over the set of K = 33structures defined in the Topological Cluster Classifica-tion that are composed of at least 5 particles. These structures include, for instance, the bi-tetrahedron (5A),the octahedron (6A), the 6-particle free energy minimumfor six colloids with depleted mediated attractions (6Z),the pentagonal bipyramid (7A) as well as much largerstructural motifs such as the defective icosahedron (10B),the icosahedron (13A) and crystalline motifs related toFCC (the 13-particle FCC motif) or HCP order (the 13-particle HCP cluster or the 11-particle 11F cluster). Weshow in Fig. 6 four instances of the covariance matrixfor different values of the packing fraction φ and thebias ε . These structures are sorted according to increas-ing covariance with the 7 A structure for a reference case( φ = 0 . ε = 0).While there are some slight variations in the value ofcovariances, the overall structure of these matrices is es-sentially independent of the values of (cid:15) and φ . A blockstructure with three groups of structures emerge: theleft and right parts sets of structures exhibit strong pos-itive correlation within each group and negative corre-lation to the opposite group, while the central part is a“no-man’s land” with essentially zero covariances to allstructures, including themselves. The rightmost group ofstructures contains 7A and all structures that correlatepositively to it. Employing the language of Ronceray andHarrowell , we term the structures j with C A,j > ag-onist to the pentagonal bipyramid 7 A while those in theleftmost group, with C A,j <
0, are antagonist to 7 A .Going into further details, we observe that the largestcovariances with 7A are C A, Z and C A, B . The 8B Quasicrystal φ / k ε T FCC -0.15-0.20-0.25-0.30 F l u i d Co-existence
FIG. 2. Phase diagram of biased hard spheres at high pack-ing fraction. We explore several state points: at zero bias (redsquares) with packing fractions φ ∈ [0 . , .
56] and t fixedpacking fraction and variable bias, φ = 0 . ε ∈ [ − . , . φ = 0 . ε ∈ [ − . ,
0] (blue crosses).Phase boundaries are reproduced from . structure is directly derived from the 7A bipyramid andhas larger concentrations for combinatorial reasons (itcorresponds to a 7A motif with an additional particleneighboring one of the two spindle particles, see Fig.1).The fact that the tetrahedral structure 6Z is a strong ago-nist is more surprising, as it does not contain any fivefoldmotif; we can rationalize its large magnitude by observ-ing that it has an entropic advantage compared to, forexample, the octahedron (6A) . The positive corre-lations revealed by the covariance analysis indicates thatthis structure overlaps well with 7A. These two examplesillustrate a feature of the agonist ( C A,j >
0) family:its members are either small structures with elementarytetrahedral order (5 A , 6 Z , 7 K ) or larger structures con-taining pentagonal rings (10 B , 11 C ,11 E , 12 B , 12 D andobviously 7 A itself). This fact demonstrates that thecovariance formalism is capable of detecting structuralrelationships between arbitrary motifs.Interestingly, the family of antagonist structures( C A,j <
0) displays positive mutual covariances C ij > i, j ∈ { antagonist } , so that the top-left corner of thecovariance matrix contains positive entries. Again, wecan identify in the TCC definitions the geometric originof these positive cross correlations: antagonist structuresinclude the octahedron (6 A ), combinations of 6 A suchas 9 K , structures with pairs of square rings such as 9 X and 9 A , or directly sections of crystalline cells such asthe 11 E , 11 F and 12 E motifs and finally the HCP andFCC structures. This indicates that, within the Topo- n ( t ) i FIG. 3. Time evolution in Monte-Carlo sweeps (MCs) of theconcentration n i for the four-fold symmetric 6 A , three-foldsymmetric 7 A and the five-fold symmetric 7 A . Concentration n i are rescaled and shifted to more visually highlight timecorrelations (and anticorrelations) between the different timesignals. B T10 n i FCC6A 11F9A 7A10B 6Z13A
FIG. 4. Average concentration of detected structures, (cid:104) n i (cid:105) , inthe system as a function of the bias ε towards the pentagonalbipyramid (7 A ) structure. The packing fraction is constantat φ = 0 . logical Cluster Classification, most of the antagonists tofivefold symmetry are of crystalline nature. The notableexception is provided by the 8 A cluster (composed of verydistorted pentagonal rings, strongly correlated with the6 Z tetrahedra and the 6 A octahedron), and the 13 B clus-ter (composed of two well aligned 7 A clusters and hencemismatching both crystalline and icosahedral order).We note that the while both the triangular bipyramid5 A and the octahedron 6 A are originally both in the min- A K A F F
CC HC P B A W A A D C B A E B A K B Z n i agonistantagonist FIG. 5. Effect of negative biases on the time-averaged concentrations (cid:104) n i (cid:105) at packing fraction φ = 0 .
54 for structures inthe Topological Cluster Classification with (cid:104) n i (cid:105) > − . Motifs that are agonist to 7 A (shaded area) show an increase inconcentration while the opposite occurs for the antagonist family of structures. imal energy structures of the HCP crystal in the case ofother simple liquids such as the Lennard-Jones model ,here they appear to play two different roles, the formercorrelating well with the emergence of pentagonal ringswhile the latter anticorrelates with it, promoting crys-talline order instead.Finally, a no-man’s land of structures of effectively zerocovariance separates the two families of agonist and an-tagonist structures. It includes structures such as 10 W or12 K which have been defined in the TCC from minimumenergy clusters of Lennard-Jones binary mixtures popu-lar in the literature of the glass transition (the Wahn-str¨om and the Kob-Andersen respectively). The co-variances for such clusters are null simply because theconcentrations n W and n K are close to zero in thehard sphere liquid. B. Dependence on packing fraction and bias
As the packing fraction or the bias vary, we move intodifferent regions of the phase diagram in Fig. 2. Takingthe high packing fraction unbiased point φ = 0 . , ε = 0(a metastable overcompressed liquid before nucleationoccurs), we show in Fig. 6 that the overall structure ofthe covariance matrix is broadly unchanged as we eitherreduce the packing fraction or bias the system to morenegative values of ε , suppressing crystallization. We ob-serve that, at the lower packing fraction, the antagonistfamily is restricted to a smaller number of structures,as large crystalline clusters such as 11 F , FCC or 12 E present small covariances, due to the smaller concentra-tions of n F , n F CC and n E respectively.In Fig. 7 we study the instructive case of the defectiveicosahedron (10B) structure and its covariances with no-table members of the agonist and antagonist families. This is an agonist structure to fivefold symmetry, as it iscomposed of three overlapping 7 A motifs. The averageconcentration of this motif increases both as the packingfraction is increased and as the bias is more negative (see,for instance, Fig.4). It is an important structure in hardsphere glasses as it dominates the free energy landscapein the metastable liquid branch at high densities . InFig. 7(a) we observe that increasing the packing frac-tion at zero bias leads to an increase in the magnitudeof the covariance coefficients, which become more nega-tive with the antagonist structures FCC, 6A, 11F and 9Aand more positive with other agonist structures such as7A, 6Z and the icosahedron 13A. This is an immediateconsequence of the increase in the concentration of 10Bat higher volume fractions compared to other structures,see Fig. 5.If we consider the dependence on the bias, Fig. 7(b),we observe an analogous behaviour at constant packingfraction φ = 0 .
54. We also note that covariances withrare structures, such as the FCC crystalline motif, arevery small and may flip sign with varying packing frac-tion/bias. This is the indication that more statistics (i.e.longer time series) are needed to more accurately esti-mate these covariances.
IV. LINEAR-RESPONSE PREDICTIONS
The knowledge of the covariance matrix does not onlyprovide insight on the geometrical relationship betweenstructures; it also allows us to make quantitative predic-tions on the parameter dependence of the liquid struc-ture. Indeed, we can apply to our system the fluctuation-response relation proposed by Ronceray and Harrowell Χ K A F E A HC P F CC A B A K K A W K B W A A K B D C B A E B A K Z B Χ K A F E A HC P F CC A B A K K A W K B W A A K B D C B A E B A K Z B Χ Χ ε φ φ = 0.54 ε = -0.05 φ = 0.52 ε = -0.05 φ = 0.52 ε = 0 φ = 0.54 ε = 0a bc d FIG. 6. Four examples of covariance matrices for different values of the bias ε and packing fraction φ : (a) ε = 0, φ = 0 . ε = 0, φ = 0 .
54; (c) ε = − . φ = 0 .
52; (d) ε = − . φ = 0 .
54. Very negative matrix elements are in blue while verypositive matrix elements are in yellow. Structures are sorted according to the ascending order of their respective covariancewith the pentagonal bipyramid 7 A at a unbiased fixed state point φ = 0 . , ε = 0 .
0. Notice the logarithmic color scale. in for on-lattice models, which reads (cid:104) n i ( ε ) (cid:105) = (cid:104) n i (cid:105) − (cid:88) structures j C i,j ε j , + O ( ε ) (3)where ε j is the vector of energy biases associated to eachstructure, i , such that the Hamiltonian is H = N (cid:80) i n i ε i .The derivation remains correct in our case, where theonly nonzero bias is for the pentagonal bipyramid i = 7 A .This results in a simple expression, (cid:104) n i ( φ, ε ) (cid:105) = (cid:104) n i ( φ ) (cid:105) − εC i, A + O ( ε ) (4) where n i ( φ ) is concentration of structure i for the un-biased system at packing fraction φ , and C A,i is thecovariance matrix element between i and 7 A at packingfraction φ .Equation 4 provides an exact prediction for the first-order dependence of the structural composition of theliquid on the applied structural bias. We demonstrate itsvalidity in Fig. 8, where we compare this linear-responseapproximation and the measured change in concentra-tions ∆ n i = (cid:104) n i ( ε ) (cid:105) − (cid:104) n i (cid:105) for four representative struc-tures at fixed packing fraction φ = 0 .
54: the 9 A , FCC ab FIG. 7. Example of the (a) packing fraction and (b) bias de-pendence of the covariance values between the agonist struc-ture 10 B and a selection of agonist and antagonist structures.In (a) the bias is ε = 0 and in (b) the packing fraction is φ = 0 . and 11 F (antagonist family), and 10 B (agonist family).The linear prediction quantitatively captures the bias de-pendency of the considered antagonist structures. Forthe agonist structure 10 B , we observe higher-order de-viations for large biases ε ≤ − .
10, with an acceleratedaccumulation of these structures that is not captured byour linear theory. Note that a similar trend is observedfor agonist structures in lattice models .Importantly, these results demonstrate that the ac-curate knowledge of the covariance coefficient at a zerobias is sufficient to infer with quantitative accuracy thestructural changes in the system for biases as large as ε ≈ ± .
1. This is not specific to structure 7A. In princi-ple, we could consider biasing the system towards any sin-gle structure, or any weighted combination of structuresas in Equation 3: our approach encompasses complex liq-uids described by an arbitrary set of biases (cid:15) i , providinga predictive tool to quantitatively assess the structure ofany liquid at reasonably low value of the biases, or equiv-alently at sufficiently high temperature. Beyond the lin-ear response regime, these results become quantitativelyinaccurate, but retain a qualitative pertinence: for in-stance, crystallization will become essentially impossibleif the concentrations of all four-fold crystalline structuresbecome too low. V. CONCLUSIONS
Through the analysis of structural covariances in thebiased hard sphere fluid we have shown that it is possi-ble to understand how fivefold local order affects other competing motifs, such as those with four-fold symmetrywhich are related to crystalline order. We have discussedhow covariances allow us to identify structural relation-ships between different motifs and we have illustratedhow this applies to the particular case of the TopologicalCluster Classification. Structural covariance reveals theexistence of two main families of structures in the classi-fication, pertaining to fivefold symmetric and crystal-likestructures respectively. An interesting line of researchwould be to extend the approach to other classifications(such as the Voronoi indexing) and to compare differentclassification strategies according to the metric providedby the covariances.In our study of the hard-sphere fluid we have foundthat the covariance approach is predictive in a wide rangeof bias values, estimating correctly, in the linear-responseregime, structural changes for any of the structures clas-sified in the Topological Cluster Classification.Our work demonstrates how an analysis based onstructural covariances can be employed to investigate off-lattice models, providing a first proof of principle in thecase of hard spheres. Other aspects of structural corre-lations in the fluid phase will deserve further study andcomparison with the original on-lattice results. For ex-ample, in Ref. it has been shown that the so-called crystal affinity Q X := ∂n X /∂ (1 /T ) can be expressed as Q X = − (cid:80) j C X,j ε X,j derived from the covariance co-efficients between the crystalline motif X and the re-maining motifs. Remarkably, in Ref. the affinity Q displays a characteristic anti-correlation with the crystal-lization times for the on-lattice systems. Understandinghow this relation holds in the case of off-lattice models -3 -2 FCC n i n i n i B T01 n i -2 FIG. 8. Tests of the linear response regime: the symbols rep-resent the variations in concentration ∆ n i = n i ( ε ) − n i withvertical bars corresponding to one single standard deviationas computed from the Monte-Carlo trajectory. The straightorange lines are the predictions of Eq. 4, with covariancesevaluated at ε = 0. For all the plots, the packing fraction is φ = 0 . and how it depends on the specific identification of crys-talline motifs (e.g. FCC, 11F or others such as bond orderparameters ) according to different structural descrip-tors will be the subject of further work.More generally, alternative routes to the calculationof the covariance matrix may provide efficient methodsto estimate structural changes for a given set of struc-tures: nonequilibrium protocols (such as shearing) are apotential avenue to measure structural couplings and co-variances quickly and at a lower computational cost thanbiased Monte-Carlo. On the experimental side, since theknowledge of the local motifs is key to our approach,colloidal experiments (where the individual particle co-ordinates can be resolved) are most suitable for a testin the laboratory of the predictive power of the struc-tural covariance analysis. However, since the covariancesare computed between concentrations of different struc-tures, spatial resolution is only necessary to identify cho-sen motifs. This means that as long as we are able to es-timate local concentrations of particular motifs and pre-serve sample to sample variations, it is possible to com-pute covariances between distinct motifs even withoutthe precise knowledge of all of the atomic positions. Ad-vanced scattering techniques on molecular liquids (suchas angstrom-beam electron diffraction ) may providethe route to measure such concentration and computecovariances between different sub-sampled regions of adense, or supercooled, liquid. ACKNOWLEDGEMENTS
The authors are grateful to Jade Taffs for providingsimulation data and Joshua Robinson for his advice.FT, BMGDC and CPR thank the European ResearchCouncil (ERC Consolidator Grant NANOPRS, projectnumber 617266) for financial support. PR thanks theBettencourt-Schueller Foundation for their support. J.-P. Hansen and I.R. Macdonald.
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