Maximum Entropy and the Stress Distribution in Soft Disk Packings Above Jamming

Transcription

1 Maximum Entroy and the Stress Distribution in Soft Disk Packings Above Jamming Yegang Wu and S. Teitel Deartment of Physics and Astronomy, University of ochester, ochester, New York 467, USA (Dated: August 3, 05) We show that the maximum entroy hyothesis can successfully exlain the distribution of stresses on comact clusters of articles within disordered mechanically stable ackings of soft, isotroically stressed, frictionless disks above the jamming transition. We show that, in our two dimensional case, it becomes necessary to consider not only the stress but also the Maxwell-Cremona force-tile area, as a constraining variable that determines the stress distribution. The imortance of the force-tile area had been suggested by earlier comutations on an idealized force-network ensemble. PACS numbers: n, g, Fg arxiv:40.463v [cond-mat.soft] 3 Jul 05 I. INTODUCTION As the density of granular articles increases to a critical acking fraction, φ J, the system undergoes a jamming transition from a liquid-like to a solid-like state [, ]. For large articles thermal fluctuations are irrelevant, and in the absence of mechanical agitation, the dense system relaxes into a mechanically stable rigid but disordered configuration. Given a set of macroscoic constrains there are in general a large number of such configurations that are accessible to the system. A long standing question is whether there is a convenient statistical descrition for the roerties of such quenched configurations. For hard-core, rough (i.e. frictional), articles, the jamming φ J (and hence the system volume at jamming) may san a range of values from random loose acked to random close acked. Edwards and co-workers [3] roosed a statistical descrition for the distribution of the Voronoi volume of such articles in terms of a maximum entroy hyothesis, assuming that all accessible states are equally likely. Henkes and co-workers [4, 5] extended these ideas to consider the distribution of stress on clusters of articles within ackings of frictionless soft articles, comressed above the jamming φ J. They denoted their formalism as the stress ensemble. Similar ideas were then roosed by Blumenfeld and Edwards [6]. Subsequently, Tighe and co-workers [7 9], using an idealized model called the force-network ensemble (FNE), argued that in two dimensions the Maxwell-Cremona force-tile area acts as an additional constraining variable, that must be taken into account in order to arrive at a correct maximum entroy descrition of the stress distribution. ecent exeriments [0 3] have sought to test such statistical models. The main goal of this work is to numerically investigate these statistical ensemble ideas as alied to the distribution of stress, and in articular to test if the analysis of Tighe and co-workers for the idealized FNE, continues to hold in a more realistic model of jammed soft-core articles. To this end we carry out detailed numerical simulations of a simle model of two dimensional, soft-core, bidiserse frictionless disks, to determine the distribution of stress and force-tile area on comact clusters of articles embedded in a larger, mechanically stable, acking at finite isotroic stress above the jamming transition. Measuring behavior as a function of both the cluster size and the total system stress, we find that the stress distribution is consistent with the maximum entroy hyothesis, rovided one takes both the cluster stress and the force-tile area as constraining variables that characterize the distribution. We find that it remains necessary to consider both variables even as the cluster size gets large, contrary to results reorted for the FNE [8]. The remainder of this aer is organized as follows. In Sec. II we rovide details of our numerical model and simulations, discussing our method to roduce jammed ackings with a secified isotroic total stress tensor, and defining the construction of our clusters and the quantities measured. In Sec. III we analyze our results in the context of the stress ensemble of Henkes et al. [4, 5]. We use a ratio of cluster stress distributions at different values of the total system stress to investigate the Boltzmann factor redicted for the distribution, and find that this Boltzmann factor includes a term quadratic in the cluster stress, rather than being linear in the stress as redicted by the stress ensemble. We comare our results against a simler Gaussian aroximation, and find that the quadratic Boltzmann factor gives a better descrition. We discuss the revious results by Henkes et al. [4, 5] and indicate why they may not have detected the quadratic term which we find here. In Sec. IV we define the Maxwell-Cremona force tile area, and consider the joint distribution of cluster stress and force-tile area. Using a ratio of this joint distribution at different values of the total system stress, we find results consistent with a Boltzmann factor that is linear in both stress and force-tile area, thus suorting the maximum entroy hyothesis. We make comarison between the temerature-like arameters resulting from this ratio analysis and those redicted from fluctuations via the covariance matrix of the constraining variables, and find reasonable, though not erfect, agreement. We then discuss the relation of our results to revious results of Tighe et al. [7 9] for the FNE, and discuss the relation between the Boltzmann factor of the joint distribution, and the quadratic Boltzmann factor of the stress distribution analyzed in the revious section. Finally, in Sec. V we summarize and discuss our conclusions.

2 II. MODEL as well as the box arameters L x, L y, γ, II.. Global ensemble Ũ U + Γ N (ln L x + ln L y ), U i<j V ij (r ij ). (3) Our system is a two-dimensional bidiserse mixture of equal numbers of big and small circular, frictionless, disks with diameters d b and d s in the ratio d b /d s =.4 []. Disks i and j interact only when they overla, in which case they reel with a soft-core harmonic interaction otential, V ij (r ij ) = { k e( r ij /d ij ), r ij < d ij 0, r ij d ij. Here r ij = r i r j is the center-to-center distance between the articles, and d ij = (d i + d j )/ is the sum of their radii. We will measure energy in units such that k e =, and length in units so that the small disk diameter d s =. The geometry of our system box is characterized by three arameters, L x, L y, γ, as illustrated in Fig.. L x and L y are the lengths of the box in the ˆx and ŷ directions, while γ is the skew ratio of the box. We use Lees-Edwards boundary conditions [4] to eriodically reeat this box throughout all sace. L y!l y L x () FIG.. Geometry of our system box. L x and L y are the lengths in the ˆx and ŷ directions, and γ is the skew ratio. Lees-Edwards boundary conditions are used. We consider here systems with a fixed total number N of disks, and study mechanically stable article ackings above the jamming transition, that have a secified isotroic total stress tensor Σ (N) αβ, Σ (N) αβ = Γ Nδ αβ, where Γ N = V, () is the system ressure, and V = L x L y is the total system volume (in two dimensions we will use volume as a synonym for area). Here α, β denote the satial coordinate directions x, y. To create our isotroic ackings, in which the shear stress vanishes, we use a scheme in which we vary the box arameters L x, L y and γ as we search for mechanically stable states [5]. We introduce [6] a modified energy function Ũ that deends on the article ositions {r i}, Noting that the interaction energy U deends imlicitly on the box arameters L x, L y, γ via the boundary conditions, we get the relations, U L x = Σ xx (N) + γσ (N) xy, L x U L y = Σ yy (N) γσ (N) xy. L y U γ = Σ(N) xy, We then start from an initial configuration of randomly ositioned articles in a square box (L x = L y, γ = 0) at acking fraction φ init = 0.84 (just slightly below the jamming transition φ J 0.84 [7]), and fixing a target value of Γ N, we minimize Ũ with resect to both article ositions and box arameters. The resulting local minimum of Ũ gives a mechanically stable configuration with force balance on each article and a total stress tensor that satisfies Σ (N) xx (4) = Σ (N) yy = Γ N, Σ (N) xy = 0. (5) For minimization we use the Polak-ibiere conjugate gradient algorithm [9]. We consider the minimization converged when we satisfy the condition (Ũi Ũ i+50 )/Ũi+50 < ε = 0 0, where Ũi is the value at the ith ste of the minimization. Tests that this rocedure gives numerically well minimized configurations, with the desired isotroic total stress tensor and force balance on articles, are discussed in the Aendix of ef. [8]. Our results are for a system with N = 89 disks, averaged over 0000 isotroic configurations, indeendently generated at each value of Γ N. We vary the total stress from Γ N = 6.4 to 8.4, in stes of 0.8. Since our simulations fix both N and Γ N, it is convenient to arameterize our results by the intensive, ressure-like, variable, Γ N /N = V/N, the total stress er article. Since our method varies the system volume L x L y so as to achieve the desired total stress Γ N, the acking fraction, φ = N π L x L y [ (ds ) + ( ) ] db, (6) at a fixed Γ N varies slightly from configuration to configuration. In Fig. 3a we lot the resulting average φ as a function of for the range of considered in this work. Error bars reresent the width of the distribution of φ; the relative width is roughly %. The Γ N values we consider here lace our systems moderately close above the jamming transition, which for our raid quench rotocol is φ J 0.84 [7, 8].

3 3 φ N = N FIG.. Average acking fraction φ vs total system stress er article Γ N /N. The error bars reresent the width of the distribution of φ, and not the statistical error of the average. Average number of articles N in a cluster of radius, at acking fraction φ = II.. Clusters of finite size In this work we are interested in the distribution of the stress on finite sized sub-clusters of the system. To define our article clusters, we ick a osition in the system at random and draw a circle of radius centered at that oint. All articles whose centers lie within this circle are considered art of the cluster, which we denote as C [0]. The total number of articles N in such a cluster will fluctuate from cluster to cluster, but the average N can be obtained from Eq. (6) using π rather than L x L y as the volume on the right hand side. In Fig. b we lot N vs radius for a system with acking fraction φ = We can then comute the stress tensor for the cluster C, Σ () αβ = sijα F ijβ, F ij = V(r ij )/ r j. (7) i C j The first sum is over all articles i in the cluster C. The second, rimed, sum is over all articles j in contact with i, where s ij is the dislacement from the center of article i to its oint of contact with j, and F ij is the force on j due to contact with i [4]. Although the total system stress is isotroic, the stress on any articular cluster Σ () αβ in general is not. However the stress averaged over many different clusters will be isotroic. If we define for each cluster then we will have Γ Tr[Σ() αβ ], (8) Σ () αβ = Γ δ αβ. (9) Here and henceforth, we will use... to indicate an average over different clusters. Our averages in this work are taken over different non-overlaing clusters within a given configuration, and then over the 0000 indeendently generated configurations at each Γ N. For a system with total stress er article = Γ N /N, we will denote the robability that a cluster of radius has a stress Γ by P(Γ ). In Fig. 3 we show these numerically comuted robability histograms P(Γ ) over the range of we study, for the articular case of clusters with radius = 5.4. We have chosen our sacing = Γ N /N = 0.8/89 so that the histograms at neighboring values of have substantial overla, as will be needed for our later analysis. Histograms are normalized so that Γ P(Γ ) Γ =, where Γ is our bin width; Γ is chosen small enough that P(Γ ) becomes indeendent of Γ. P(Γ ) x = 5.4 N = ! FIG. 3. (color online) Probability histograms of the stress Γ on a cluster of radius = 5.4 at different values of the total stress er article = Γ N /N. In this work we will consider a range of cluster sizes from =.8 to 8., corresonding to clusters with an average number of articles ranging roughly from 8 to 50. Our total system size of N = 89 articles was chosen so as to be large enough to exlore a moderate range of cluster sizes, while being small enough to generate a large number of indeendent configurations so as to get good recision for the histograms P(Γ ). The largest cluster size that we consider is chosen to be small enough that effects due to the finite size of the total system do not significantly effect the distributions P(Γ ). III. ESULTS: THE STESS ENSEMBLE In an effort to develo a statistical theory for the distribution of stress Γ on clusters within jammed ackings, Henkes et al. [4, 5] roosed the stress ensemble. Noting that the stress tensor Σ αβ is a conserved quantity, i.e. its global value for the total system is fixed and it is additive over disjoint subsystems, an analogy to the canonical ensemble of statistical mechanics can be made.

4 4 For isotroic systems, Γ lays the role of energy, and the distribution of Γ was roosed to be, P(Γ ) = Ω (Γ ) e α( )Γ Z ( ). (0) The angoricity [5, 6] /α is a temerature-like variable that is set by the total system stress er article. The number of available states Ω (Γ ) at a given value of Γ is resumed indeendent of. The normalizing constant Z, Z ( ) = dγ Ω (Γ )e α( )Γ, () is analogous to the artition function, and F ( ) ln Z ( ) () is analogous to the free energy. Alternatively, the distribution of Eq. (0) can also be viewed as resulting from a maximum entroy hyothesis [], in which all clusters with a given Γ are resumed equally likely, and the average is constrained to the known value Γ. Since the stress is conserved and additive, the average of Γ is constrained by, ( ) π Γ = Γ N, (3) V a result that we have reviously confirmed numerically [6]. The average ressure in the cluster is then equal to the global ressure in the total system, Γ π = Γ N V =. (4) Two articular consequences follow from the distribution of Eq. (0). The first relates to the fluctuation of stress on the cluster, var(γ ) Γ Γ. The second relates to the ratio of distributions at nearby values of. III.. Fluctuations As in an equilibrium thermodynamic system, one can use the free energy of Eq. () to write, and Γ α F α = Γ, (5) = F α = Γ Γ = var(γ ). (6) A change in the inverse angoritcity α therefore gives a change in the average cluster stress Γ, Γ = Γ α α = var(γ ) α. (7) By Eq. (4) we have Γ /(π ) = =, hence we conclude that a change in the total system ressure induces a change in the inverse angoricity α, given by, [ ] π α =. (8) var(γ ) Taking the limit 0 we then get, [ ] dα π d =. (9) var(γ ) In ef. [6] we showed that, for the range of cluster sizes and ressures considered here, the deendence of var(γ ) on cluster size was well fit by the form var(γ )/(π ) = c + c /. Thus from Eq. (9) we might exect to see / corrections to α( ) arising from the finite sizes of our clusters. III.. Histogram ratio The results of the revious subsection, in articular Eq. (9), hold if the distribution of stress Γ obeys the form of Eq. (0). However it is necessary to first demonstrate that this form does indeed hold. A direct test of whether or not the distributions P(Γ ) obey Eq. (0) is given by considering the ratio of numerically measured histograms at two neighboring values of []. Denoting quantities at a given or by the subscrit or, the log ratio of histograms at two neighboring values of < is given by, [ ] [ ] P Z, ln = ln + (α α )Γ = F + αγ, P Z, (0) where F F, F, and α α α. Execting that the right hand side of Eq. (0) scales roortional to the cluster area π, we define an intensive log ratio, [ ] π ln P = f + α. () P where Γ /(π ), and f F /(π ). The condition = 0 locates the oint of greatest overla between neighboring histograms, where P = P. In Fig. 4 we lot vs for several different airs of and = +, for cluster sizes =.8 to 8.. We find a fairly good looking collase of the data for different cluster radii. This suggests that, to leading order in /,, and hence f and α, are intensive quantities indeendent of the cluster size. However we find that the data for show a clear curvature, not the linear deendence on redicted by Eq. (). Instead of using Eq. () we may emirically fit our data in Fig. 4 to a quadratic form, = f + α + λ, ()

5 where f, α and λ vary with the stress, but are indeendent of the cluster radius. Such fits give the solid curves in Fig N = 89 = =.8 = 4. = = 5.8 = 7.0 = = 8. = = =! /" FIG. 4. (color online) Log ratio (/π ) ln[p /P ] of histograms P and P at total system stresses er article and = +, vs cluster ressure = Γ /π. Data for different cluster sizes (denoted by different symbol shaes) but the same, collase to a common curve that is well fit by a arabola (solid curves); dashed lines are the tangents at the oint of greatest overla between the histograms P and P, given by the condition = 0. We show results for stress er article = to eresentative error bars are shown at the tail ends of. Linear aroximation: If we for the moment ignore the curvature in the data of Fig. 4, we can aroximate by its tangent line at the value where = 0. This is the oint where the two distributions P and P have their largest overla. Such tangents are shown as the dashed lines in Fig. 4, and have sloes, ᾱ = α + λ. (3) In Fig. 5 we lot ᾱ/ vs ( + )/, where, =, (N/V, ) gives the corresonding total system ressure of the two overlaing histograms. We find an excellent fit to a ower-law, ᾱ/ Taking ᾱ/ as an aroximation to the derivative, we can integrate to get ᾱ() Given the rather limited range of our data, however, it is unclear how much significance should be given to the secific numerical value of this fitted exonent; the data is also well fit by the exression ᾱ/ (.9/ )( /). Viewing the stress ensemble of Eq. (0) as an aroximation to the true distribution, we can test whether ᾱ/ from the above linear aroximation to the histogram ratio is in agreement with the dα/d one would exect from the fluctuation exression of Eq. (9). We therefore also lot in Fig. 5 the quantity π /var(γ ) vs, showing results for several different cluster sizes. We see an excellent agreement. The agreement shown in Fig. 5 might naively be taken as evidence that the stress ensemble, while failing to give a strictly linear log ratio as redicted, is nevertheless not a bad aroximation to the stress distribution. _!"#/", $ /var(% ) 5x0 6 3x0 6 x0 6 8x0 5 6x0 5 N = 89.7! _!"#/" $ /var(% ) =.8 = 4. = 5.8 = 7.0 = FIG. 5. (color online) Comarison of (i) ᾱ/ vs = ( + )/ (solid circles) as comuted from the linear aroximation to the log histogram ratio, given by the sloes Eq. (3) of the tangent lines in Fig. 4, with (ii) π /var(γ ) vs, for several different cluster radii (oen symbols), which by the fluctuation exression of Eq. (9) gives dα/d in the stress ensemble aroximation. The solid line is a fit to an arbitrary ower-law, and finds ᾱ/ The dashed line is a fit to the the ower-law. However, as we will show in the next section, Eq. (9) also results from the assumtion that the distribution P(Γ ) is a simle Gaussian, rovided that the sacing between the overlaing distributions is not too great []. Moreover, such a Gaussian model also rovides a simle mechanism for roducing the curvature in that is evident in Fig. 4. We will discuss the extent to which a Gaussian aroximation can exlain the data of Fig. 4 in Sec. III.3. Quadratic fit: The quadratic form for the log ratio, given by Eq. (), clearly describes the data better than the linear exression of Eq. (). However, while the quadratic fits in Fig. 4 look reasonable, a quantitative test shows that they are not articularly accurate, given the high recision of our data. As a measure of the goodness of our fits we will use the chi squared er degree of freedom χ /ν, χ M d [ ] yi y(x i ) /ν, (4) M d M f δy i i= where M d is the number of data oints, M f the number of fit arameters, x i the indeendent variables, y i the measured deendent variable at x i, δy i the estimated statistical error in y i, and y(x i ) the fitting function. A good fit is usually indicated by χ /ν O(). In Fig. 6 we lot the χ /ν of the fit to using the quadratic form of Eq. (), where the fitting arameters f, α and λ are assumed to be indeendent of the cluster radius. Our results are lotted vs, the stress er article at the lower of the two stresses, used to define the histogram ratio. We show results (solid circles) for the fit to the entire data set including all cluster sizes, as well as the χ /ν (oen symbols) for the data set restricted to clusters of a given fixed radius (we kee f, α and λ the same, but sum Eq. (4) over only the

6 6 data for a given cluster size, with M d now being the number of data oints at radius, and M f the number of fit arameters divided by the number of different cluster radii). We see that the χ /ν becomes O() only as increases, and only for the larger cluster sizes; as decreases, the χ /ν steadily increases and becomes O(0) at our smallest, indicating a oor fit.! /" 00 0 N = 89 #$, #%, #f indeendent of all =.8 = 4. = 5.8 = 7.0 = FIG. 6. (color online) Chi squared er degree of freedom, χ /ν, for the fit of the histogram ratio to the quadratic form of Eq. (), where the fitting arameters f, α and λ are assumed to be indeendent of the cluster radius. esults are lotted vs, the stress er article at the lower of the two stresses, used to define (see Eq. ()). all denotes the χ /ν of the fit to the entire data set including all cluster sizes, while the other symbols denote the χ /ν of the same fit, but restricted to data at a given fixed cluster size. The fits discussed above in connection with Figs. 4-6 assumed that the fitting arameters f, α and λ were indeendent of the cluster radius. However the discussion at the end of Sec. III. leads one to susect that these arameters may have / corrections arising from the finite size of the clusters. We therefore extend our analysis to include this ossibility by using, α() = A( + a/), λ() = B( + b/), (5) f() = C( + c/), in the fit to Eq. (), where A, B, C, a, b and c are taken to be indeendent of. The values of A, B and C thus reresent the limiting values of α, λ and f. In Fig. 7 we lot the results of such fits with / corrections, showing in anels a,b,c α()/, λ()/ and f()/ vs the average histogram ressure = ( + )/ for several different cluster radii, as well as the limiting values A, B, and C. We see that as increases, all arameters are aroaching finite values. We also show in these figures the results from our earlier fit keeing α, λ, and f as constants indeendent of ; these are labeled in the figures as all. The ower-law behavior of the data for the largest is indicated in the figures, where we find α/, λ/.94, and f/ 0.7. Given the limited range of our data, it is unclear how much significance should be given to the secific numerical values of these exonents. We see that the / corrections are quite noticeable for our finite cluster sizes, and that the results we get when ignoring these corrections (the results labeled all ) tend to roughly agree with the values found for the smallest when the / corrections are included. The arameters a, b and c of Eq. (5) reresent length scales that determine the strength of the / corrections. We lot these vs in Fig. 7d and find that these are consistent with being constant, indeendent of the ressure. The lengths a.5, b.3 and c 7.0 are large enough comared to the range of our cluster sizes =.8 8., so as to exlain the noticeable finite size effects we see in Figs. 7a,b,c. The arameters α, λ and f, that describe the quadratic shae of the histogram ratio, thus show a clear deendence on the cluster size. However, if we use the α() and λ() from Fig. 7 in Eq. (3) to comute ᾱ(), the sloe of at the oint of maximum histogram overla, we find that this shows essentially no deendence on the cluster size. In Fig. 8 we lot this ᾱ()/ vs the average histogram ressure = ( + )/ for several different. For comarison we also lot the ᾱ/d, reviously shown in Fig. 5, obtained from fits assuming α, λ and f indeendent of. We see that there is essentially no difference between the two fits, nor between any of the cluster sizes, excet for the smallest size =.8. Since ᾱ is a measure of behavior at the oint of greatest overla of the two histograms, and this oint lies near the eaks of the distributions, the insensitivity of ᾱ to the cluster size illustrates, not surrisingly, that the deendence on the cluster size which is observed for the arameters α, λ and f in Fig. 7 is due to the deendence on of the tails of the distributions P(Γ ). It is interesting to note that, while the ᾱ() associated with the linear aroximation to at the oint of greatest histogram overla is ositive, the α() obtained from the quadratic fit to Eq. () is negative. We can see this from Fig. 5 where we find ᾱ/.9, and so ᾱ() 0.9, comared to Fig. 7a where we find that α/, and so α(). Finally, we test the accuracy of our model with / corrections by comuting the χ /ν of the fit. In Fig. 9 we show χ /ν as comuted for the entire set of data including all cluster sizes, as well as the χ /ν restricted to data for secific cluster sizes. We now see, in contrast to the results in Fig. 6, that in essentially all cases χ /ν O(). Including such / corrections to α, λ and f thus significantly imroves the quality of the fit. III.3. Gaussian aroximation In this section we consider an alternative ossibility, that the distribution of stress on clusters is given by a simle Gaussian distribution. We will show that such a Gaussian aroximation gives both (i) a simle mechanism for roducing a histogram ratio that is quadratic

7 7 $"%/" "#/" $ N = 89 $.94 =.8 = 4. = 5.8 = 7.0 = 8.!! all =.8 = 4. = 5.8 = 7.0 = 8.!! all _!"#/" 5x0 6 3x0 6 x0 6 8x0 5 6x0 5 N = 89 all =.8 = 4. = 5.8 = 7.0 = FIG. 8. (color online) Comarison of ᾱ/ vs = ( + )/ as comuted from Eq. (3) using the α() and λ() determined from the fits to the histogram ratio with the / corrections of Eq. (5) (oen symbols for different ), vs from fits to using α, λ and f taken to be indeendent of (solid circles, reviously shown in Fig. 5 and denoted here as all ). $"f/" a, b, c N = 89 (c) N = 89 (d) $ =.8 = 4. = 5.8 = 7.0 = 8.!! all c a b N = FIG. 7. (color online) α/, λ/, (c) f/ vs ressure = ( + )/ from quadratic fits to the histogram ratio with / corrections as in Eq. (5), for clusters of different radii. Also shown are the limiting values A, B, C of Eq. (5), as well as the values from fits keeing α, λ and f constant for all (labeled as all ). Solid lines are ower-law fits, with the ower indicated for the fit to the largest value of. (d) Length scale arameters a, b and c of Eq. (5) that determine the strength of the / corrections, vs ; the solid lines are the best fit to a constant, indicating no systematic deendence on ressure. in the cluster ressure, as in Eq. (), and (ii) a variation of an effective inverse angoricity (defined by the! /" 4 3 #$, #%, #f with / corrections N = 89 all =.8 = 4. = 5.8 = 7.0 = FIG. 9. (color online) Chi squared er degree of freedom, χ /ν, for the fit of the histogram ratio to the quadratic form of Eq. (), where the fitting arameters f, α and λ have the / corrections of Eq. (5). esults are lotted vs, the stress er article at the lower of the two stresses, used to define (see Eq. ()). all denotes the χ /ν of the fit to the entire data set including all cluster sizes, while the other symbols denote the χ /ν of the fit restricted to data at a given fixed cluster size. histogram ratio) with ressure, dα/d, that is the same as found in Eq. (9) for the Boltzmann distribution, rovided the sacing = between the histograms used in comuting is sufficiently small. Similar results have been resented earlier by McNamara et al. [] in the context of the volume distribution of granular ackings. However, we will show that this Gaussian aroximation gives a oorer descrition of our data than does the quadratic fit of the revious section. We will here assume that the distribution of stress Γ on a cluster of radius is given by the Gaussian, P(Γ ) = πσ e δγ /σ (6) where δγ Γ Γ is the fluctuation of Γ away from its ensemble average, and σ var(γ ) = δγ is

8 8 the variance of Γ. Both Γ and σ are functions of the total system stress er article, = Γ N /N. Using the above Gaussian distribution, it is straightforward to comute the histogram ratio at two neighboring values and. Doing so, one find a quadratic form as in Eq. (). We use the coefficients of this quadratic form to define effective arameters α g, λ g and f g, so that, π ln(p /P ) = f g + α g + λ g (7) where Γ /(π ), and ( ln f g = π α g = Γ σ [ σ σ Γ σ ( λ g = π σ σ ] + Γ σ Γ ) σ ), (8) where the subscrits, refer to values at,. Since we can easily comute averages and variances of Γ [6], the result of Eq. (8) involves no adjustable arameters, and we can directly see how well it agrees with our numerically comuted values for the histogram ratio. In Fig. 0 we lot our data together with the rediction of Eq. (8) (solid lines) for two different cluster radii, =.8 and = 4., at three different values of the total stress er article. We see that the agreement is not bad, although the rediction of Eq. (8) noticeably curves away from the data at both the high and low ends, articularly for the smaller value of. In Fig. we comare the values of α g, λ g and f g from the Gaussian aroximation of Eq. (8) with the values of α, λ and f obtained reviously by the quadratic fit to with / corrections. We see that the two sets of arameters are noticeably different. However, if we consider the sloe ᾱ of at the oint of greatest histogram overla, one can show that the Gaussian aroximation gives results essentially identical to that redicted for the Boltzmann distribution of Eq. (9) and so also identical to that found from the quadratic fit to, as shown in Figs. 5 and 8. Defining ᾱ g = α g + λ g, and assuming the oint of greatest overla between the two histograms is at = ( + )/, we find from Eq. (8), ᾱ g = π [ σ + σ ], (9) where =. For sufficiently small, we can take to leading order σ σ in Eq. (9) and hence the above becomes equal to Eq. (8) found for the Boltzmann distribution. Hence the agreement of ᾱ between the numerically comuted histogram ratio and the value found via the fluctuations of Γ as in Eq. (8) cannot in itself be taken as evidence for the correctness of the Boltzmann distribution of Eq. (0); the same relation holds ln(p /P )/! ln(p /P )/! = = = = =.8 = 4. = = FIG. 0. (color online) Histogram ratio at neighboring values of the total system stress er article and, vs cluster stress er area = Γ /(π ). Data are shown for three different values of, for clusters of radius =.8 and = 4.. Solid lines give the rediction of the Gaussian aroximation of Eqs. (7) and (8). just as well for a Gaussian distribution, rovided is not too big. The true test for the Boltzmann distribution of Eq. (0) is therfore the linearity of the histogram ratio in the cluster ressure. Finally, to check quantitatively how well the Gaussian aroximation is describing the histogram ratio data, we can comute the χ /ν of the fit of the Gaussian results of Eq. (8) to the measured data for. In Fig. we lot this χ /ν vs, the stress er article at the lower of the two stresses, used to define, for several different cluster radii. We see that the Gaussian aroximation is quite noticeably worse than the quadratic fits to with / corrections in the fitting arameters, as shown earlier in Fig. 9. Only for the largest cluster sizes is the Gaussian aroximation reasonable, with χ /ν O(). This is because as increases at fixed, our finite data samling for gets confined to an ever smaller region of about the oint of greatest histogram overla, and so the data is decreasingly sensitive to the curvature in. III.4. elation to revious work A similar analysis of the same bidiserse two dimensional model has reviously been carried out by Henkes et al. [4]. They used configurations quenched at constant acking fraction φ in a square box, rather than constant

9 9!"/! #!$/! #!f/! N = 89 N = 89 N = 89 solid symbols = Gaussian oen symbols = quadratic =.8 = 4. = 5.8 = 7.0 = solid symbols = Gaussian oen symbols = quadratic =.8 = 4. = 5.8 = 7.0 = solid symbols = Gaussian oen symbols = quadratic =.8 = 4. = 5.8 = 7.0 = FIG.. (color online) Comarison of arameters α/, λ/ and f/ from the Gaussian aroximation of Eq. (8) with those obtained from the quadratic fit with / corrections shown reviously in Figs. 7a,b,c. Solid symbols are for the Gaussian aroximation, while oen symbols are for the quadratic fit. esults are lotted vs the system ressure for several different cluster radii. isotroic stress Γ N. They also considered a somewhat different histogram ratio than that considered in the resent work. They used, [ P(Γ ) P(Γ H ln ] ) P(Γ ) P(Γ. (30) ) Plotting H vs Γ Γ, they found a linear relation, in agreement with exectations from the stress ensemble of Eq. (0). However, our result of Eq. () for leads to the conclusion that the ratio used by Henkes et al., when scaled by the cluster volume to be an intensive quantity, H H /(π ), should obey, H = α( ) + λ( ) (3) = [ α + λ( + )] ( ). (3)! /" 0 Gaussian aroximation =.8 = 4. = 5.8 = 7.0 = 8. N = FIG.. (color online) Chi squared er degree of freedom, χ /ν, of the fit of the histogram ratio to the Gaussian aroximation of Eqs. (7) and (8). esults are lotted vs, the stress er article at the lower of the two stresses, used to define (see Eq. ()), for several different cluster radii. To check the behavior of H, we consider the case with a stress er article = Generating a discrete set of evenly saced values of that san the range of the data for this in Fig. 4, and alying Eq. (3) using the values of α and λ obtained from the fit to Eq. () for this, we lot H vs in Fig. 3. H N = ! Ŕ = FIG. 3. (color online) Histogram ratio of Eq. (30) used by Henkes et al. in ef. [4], normalized by the cluster volume H = /(π ), vs. Data is comuted using Eq. (3) and reviously determined values of α and λ for the case of a stress er article of = Solid line is a linear fit to the data. At each of the discrete values of there is a range of values of H corresonding to the different ossible values of +, as seen from Eq. (3). But the average about these values is a straight line (solid line in Fig. 3) of sloe α + λ( + ), where, =, N/V, ; ( + )/ locates the ressure at the oint of greatest overla between the two distributions P, at and. This sloe is thus exactly equal to the sloe of the linear aroximation to our given by Eq. (3), and hence the results of Henkes et al. should be equivalent to the results shown in our Fig. 5. The straight line relation Henkes et al. observed between H and Γ Γ, as oosed to the quadratic relation we find for our simler

10 0, is therefore just an artifact of their having used the ratio of Eq. (30), which uon averaging data at fixed averages away the non-linear behavior. IV. ESULTS: THE STESS FOCE-TILE ENSEMBLE The results discussed in the revious section thus rovide no comelling evidence that the stress distribution P(Γ ) in our two dimensional system is indeed given by the simle stress ensemble form of Eq. (0). The Gaussian aroximation also seems to be a oor reresentation of the distribution. The good fit of the histogram ratio to the quadratic form of Eq. () suggests instead that the distribution P(Γ ) involves a Boltzmann factor with a quadratic term in the stress, [ ex αγ λ ] π Γ, (33) with /α and /λ as intensive temerature-like variables that vary with the total system ressure, and that aroach well defined values (with / corrections) as the cluster size increases. In this section we discuss and test one roosed mechanism for generating the above Boltzmann factor. As mentioned earlier, the stress ensemble of Eq. (0) may be viewed as resulting from a maximum entroy hyothesis, given that the average stress on the cluster Γ is constrained by the total system stress Γ N, according to Eq. (3). However, if the system ossesses other constrained observables, these too can effect the cluster stress distribution. As ointed out by the work of Tighe et al. [7 9], in two dimensions the Maxwell- Cremona force-tile area [3] is another such constraining quantity. Moreover, they showed that this force-tile area leads naturally to a stress distribution with a Boltmann factor such as in Eq. (33). IV.. The Maxwell-Cremona force-tile area The Maxwell-Cremona force-tiles were introduced by Maxwell in 864 [3]. We illustrate the construction of the force-tiles, a concet which alies only to two dimensional ackings, in Fig. 4. Panel a shows a sub cluster of articles within in a mechanically stable acking. The red lines indicate the elastic forces between articles in contact; the length of each line is roortional to the magnitude of the contact force. For our frictionless articles, these forces always oint normal to the surface at the oint of contact. In anel b, the force lines of anel a are rotated 90 so that they are now tangential to the article surface. In anel c, these rotated force lines are translated so as to lace the force lines from each article ti-to-tail going counterclockwise around each article. Since the net force on each article vanishes, the force (c) FIG. 4. (color online) Construction of the Maxwell-Cremona force-tiles for a sub cluster of our system: red lines reresent contact forces between the articles; the magnitude of the force is roortional to the length of the line; force lines are rotated by 90 ; (c) rotated force lines are translated to lie ti-to-tail forming closed loos that are the force tiles. In and (c), numbers denote articular articles and their corresonding force-tiles. lines for each article must form a closed loo [4]. The area of the loo for article i is the article s force-tile area A i. For frictionless articles, such as studied here, the force-tiles always have convex surfaces. In anels b and c we number the articles and their corresonding force-tiles. Because the contact force that defines a given edge of the force-tile of a article i must also be an edge of the force-tile of the article j that shares that contact, one can show that the force-tiles tile sace with no gas or overlas [8]. The force-tile area of a cluster of articles C is then just the sum of force-tile areas for each member article, A C = i C A i. For a acking with eriodic Lees-Edwards boundary conditions, the force-tiling is similarly eriodic, and the force-tile area for the total system A N is determined uniquely by the total system stress tensor, A N = ]/V [8]. For our system with isotroic stress det[σ (N) αβ Σ (N) αβ = Γ N δ αβ, and so A N = Γ N /V. (34) For finite clusters of radius, however, since the boundary is not fixed, A may take a distribution of values for each given value of Γ. We illustrate this in Fig. 5 where we show a scatter lot of the values of A and Γ found in individual clusters, for the articular cluster size = 5.4, at several different values of the total system stress er article. The distributions for neighboring values of overla each other, similar to the distributions of Γ in Fig. 3. Since the force-tile area is conserved (i.e. the total system value A N is fixed and A is additive over disjoint 3

11 and, we can again construct the log histogram ratio. From Eq. (38) we get, [ ] π ln P = f + α + λ a (40) P FIG. 5. (color online) Scatter lot of values of stress Γ and force-tile area A, for clusters of radius = 5.4, for different values of total system stress er article ranging from to The smaller the value of, the more comact is the distribution. subsystems) the average on clusters of radius is constrained by, ( ) π A = A N, (35) V a result which we have numerically confirmed elsewhere [6]. Combining the above with Eq. (3), and using the fixed relation between A N and Γ N given by Eq. (34), then yields the relation between the average cluster force-tile area and the average cluster stress, A = Γ π. (36) Defining an intensive force-tile area, a A /(π ), and recalling Γ /(π ), the above becomes simly, a = =. (37) Thus a maximum entroy formulation should consider the joint distribution of both Γ and A, treating both as constrained variables whose averages are known. Assuming that all configurations with a given air of (Γ, A ) are equally likely, one gets, e α( )Γ λ( )A P(Γ, A ) = Ω (Γ, A ), (38) Z ( ) with Z ( ) dγ da Ω (Γ, A )e α( )Γ λ( )A. IV.. Histogram ratio (39) Considering the joint distribution of Γ and A at two neighboring values of the total system stress er article, where f ln[z, /Z, ]/(π ), α α α and λ λ λ. If the arameters f, α and λ are intensive, with only a weak deendence on the cluster size, then lotting vs the intensive quantities and a, data for different cluster sizes should all collase to a single flat lane for a given air,. The sloes of the lane in directions and a determine the values of α and λ. Comuting from our numerically determined joint histograms, we find that our data for do indeed collase quite well onto a single flat lane for all. In Fig. 6 we show vs and a for several different cluster radii. Panels a and b show results for our lowest system stress, = Panel a shows a side view looking down uon this lane from the side; the data cluster into more comact regions as increases. Panel b shows a view looking edge on at the lane, thus confirming that the surface defined by our data is indeed a flat common lane for all. Panels c and d show similar results for our largest system stress, = = x0-7 x0-7 4x0 a x0-7 4x0-7 x0-7 a =.8 = = 5.4 = = x0-6 x0-6 3x0 a -6 4x0-6 (c) x (d) x x0 x0-6 x0-6 a 3x0-6 FIG. 6. (color online) Plot of log histogram ratio vs cluster ressure and force-tile area er volume a for different cluster radii at the total system stress er article = The data cluster into more comact regions as increases. Shaded region shows the best lanar fit to the data, where all fit arameters are taken to be indeendent of. Same as but looking edge on at the fitted lane, confirming that all data lies on a common flat lane. Panels (c) and (d) are the same as and but at the total system stress er article = To increase the clarity of the figure, in anels (c) and (d) error bars are shown on only a randomly selected 5% of the data oints. Fitting our data to the lanar form of Eq. (40), and taking the fit arameters f, α and λ as constants indeendent of the cluster radius, our fit gives the shaded

12 lanes shown in Fig. 6. In Fig. 7 we show the χ /ν of this fit (solid circles) to the entire data set of all cluster sizes ; we see that the fit is excellent with χ /ν for all stresses. We have also tried fits where we allow the arameters f, α and λ to have / corrections, as in Eq. (5). We find little change in our results, with χ /ν remaining for all, essentially no change in α and λ, and only a small shift in f. Finally, we have also done lanar fits to each cluster size indeendently, so that f, α and λ may deend on in any arbitrary way. The resulting χ /ν from such fits are shown in Fig. 7 for several different (oen symbols), and we see again that χ /ν everywhere.! /" fit to a lanar surface all =.8 = 4. = 5.8 = 7.0 = 8. N = FIG. 7. (color online) Chi squared er degree of freedom, χ /ν, of fits of the histogram ration to the lanar form of Eq. (40). esults labeled all (solid circles) are fits keeing the arameters f, α and λ the same for all cluster sizes. Other results are for fits secifically to the indicated cluster size alone. esults are lotted vs, the stress er article at the lower of the two stresses, used to define (see Eq. (40)). χ /ν indicates an excellent fit. In Fig. 8 we lot the resulting fit arameters as α/, λ/ and f/ vs the ressure = ( + )/. We show results for the case where we take α, λ and f to be the same for all cluster radii (solid circles), as well the case where we fit searately to clusters of a secific (oen symbols). For α/ and λ/ the results show little sensitivity to which case is used, or to the cluster size in the second case; f/ shows a somewhat greater sensitivity at the larger values of, suggesting that some -deendence does exist for f. IV.3. Fluctuations Similar to the discussion for the stress ensemble in Sec. III., in the stress force-tile ensemble we can relate the arameters α and λ to the fluctuations of stress Γ and force-tile area A. For the ensemble of Eq. (38), and with F ln Z, we have, ( ) F = Γ, α λ ( ) F = A, (4) λ α!"/! #!f/! #!$/! x0 3 8x0 6x0 4x0 x0 N = 89 N = 89 (c) N = 89 #.8 #3 #.9 =.8 = 4. = 5.8 = 7.0 = 8. all =.8 = 4. = 5.8 = 7.0 = 8. all =.8 = 4. = 5.8 = 7.0 = 8. all FIG. 8. (color online) Comarison of arameters α/, λ/ and (c) f/ obtained from fits of the histogram ratio to the lanar form of Eq. (40). esults labeled all (solid circles) are from fits where α, λ and f are taken the same for all cluster sizes. Other results (oen symbols) are from fits secifically to the indicated cluster size alone, with α, λ and f chosen indeendently for each. esults are lotted vs the ressure = ( + )/. Solid lines are fits to a ower-law, with the indicated ower-law being the result from the fit to the largest value of. and ( ) F α λ ( ) F λ = α ( ) F = α λ ( ) Γ = α λ ( ) A λ ( ) Γ λ α α = var(γ ) (4) = var(a ) (43) ( ) A = α λ = cov(γ, A ), (44) where cov(γ, A ) = Γ A Γ A is the covariance.

13 3 Defining the covariance matrix C [ var(γ ) cov(γ, A ) cov(γ, A ) var(a ) ], (45) the changes in the average cluster stress and average cluster force-tile area in resonse to changes α and λ in the arameters α and λ, are given by, [ ] [ ] Γ α = C. (46) A λ Consider now our global system with eriodic boundary conditions. If we vary the total system ressure an amount from = Γ N /V to = Γ N /V, then by Eq. (4) the average stress on the cluster will vary as Γ /(π ) = =. By Eq. (37), the average force-tile area of the cluster will vary as A /(π ) = a = = = ( + ). Taking the limit 0 and inverting Eq. (46) we then get, [ ] [ ] dα/d = π dλ/d C (47) where C is the inverse of the covariance matrix. Thus the use of a global system with eriodic boundary conditions, which by Eq. (37) restricts the average cluster behavior to lie on the secific curve a = in (, a ) sace, similarly requires that α and λ for the eriodic system can not be chosen as indeendent arameters, but must be related to each other arametrically via the global ressure so as to satisfy Eq. (47). Or to ut it another way, the use of a global system with eriodic boundary conditions restricts the Boltzmann distribution of Eq. (38) to arameters that lie on a secific arametric curve (α(), λ()) in the more general (α, λ) sace. Numerically comuting the covariance matrix as in ef. [6], in Fig. 9 we lot the dα/d and dλ/d redicted by Eq. (47) vs the system ressure, for several different cluster radii. For comarison, on the same lot we also show α/ and λ/ as obtained from our lanar fit to the histogram ratio, assuming constant fit arameters for all cluster sizes (as shown reviously in Fig. 8). We see good qualitative agreement, but quantitatively, the results from the histogram ratio are somewhat smaller than from the covariance matrix; α/ ranges from roughly 80% to 75% of dα/d, as ressure increases, while λ/ ranges from roughly 99% to 80% of dλ/d as increases. Given the very good degree to which our data for the histogram ratio is described by the flat lane of Eq. (40), it is not clear why the agreement is not better. We may seculate that additional macroscoic variables besides Γ and A might be needed for a more comlete descrition of the ensemble [6, 5, 6]. "$/", d$/d!"#/",!d#/d N = 89 N = 89!.7!.84!"#/" "$/" d$/d =.8 = 4. = 5.8 = 7.0 = !d#/d =.8 = 4. = 5.8 = 7.0 = FIG. 9. (color online) Comarison of α/ from the fit of the histogram ratio to Eq. (40) with dα/d redicted by the covariance matrix C in Eq. (47); α/ is comuted assuming constant fit arameters for all cluster sizes, while dα/d is comuted for the secific cluster sizes indicated. Similar comarison of λ/ from the histogram ratio to dλ/d from the covariance matrix. esults are lotted vs total system ressure. IV.4. Gaussian aroximation As we did in Sec. III.3 for the distribution P(Γ ), we can consider a Gaussian aroximation to our joint distribution P(Γ, A ). Defining the two dimensional vector of observables X (Γ, A ), we have, P(Γ, A ) = π det[c] e δx C δx, (48) where C is the covariance matrix of Eq. (45), and δx X X is the fluctuation of the observables from their average. The histogram ratio in this Gaussian aroximation is then given by, π ln [P /P ] = π [ ln ( det[c ]/det[c ] ) (49) + δx C δx δx C δx ]. The quadratic forms in the above exression result in a arabolic surface rather than the flat lane exected for

14 4 the Boltzmann distribution of Eq. (38). To comare this surface against our numerical results, in Fig. 0 we show our data for, together with the surface redicted by Eq. (49), for a cluster of radius = 4. at our smallest = , and a cluster of radius =.8 at our largest = In both cases we see that the surface of the Gaussian aroximation shows a clear curvature away from the comuted numerically from our overlaing histograms. Unlike our results in Sec. III.3, where the curvature of the Gaussian aroximation gave a better descrition of the histogram ratio than did the straight line of the stress ensemble, here the Gaussian aroximation is yielding a curvature that is absent from the data. The Boltzmann distribution of Eq. (38) is therefore clearly a better descrition of our data than the Gaussian aroximation of Eq. (48). ln(p /P )/" ln(p /P )/" !0.!0.4 = = 4. 5x0-7 4x0-7 a 3x0-7 x !0. = =.8!0.4!0.6 4x0-6 3x0-6 x0-6 x0-6 a FIG. 0. (color online) Log histogram ratio at neighboring values of the total system stress er article and, vs = Γ /(π ) and a = A /(π ). esults are shown for = , cluster radius = 4. and = 0.005, cluster radius =.8. ed oints with error bars are our data for the numerically comuted, while the curved surfaces shown are the redictions of the Gaussian aroximation of Eq. (49). IV.5. elation to revious work The ideal that the Maxwell-Cremona force-tile area should lay an imortant role in determining the stress distribution in two dimensional jammed ackings was first ut forward by Tighe and co-workers [7 9]. They, however, considered an idealized model known as the force-network ensemble (FNE) [7 9] rather than the more realistic satially disordered ackings considered here. The FNE is defined by noting that a mechanically stable acking above the jamming transition has an average article contact number z that is larger than the isostatic value z iso [, ]. For a fixed set of article ositions, when z > z iso, the constraint of force balance on each article under-determines the set of contact forces, and so there are many ossible contact force configurations that can lead to a mechanically stable state, consistent with a given global stress tensor. In the FNE one assumes that all such mechanically stable contact force configurations are equally likely, and osits that it is such contact force fluctuations, decouled from fluctuations in the article ositions, that is the rimary factor determining the distribution of stresses in the jammed acking. The FNE thus considers only such contact force fluctuations for a given fixed set of article ositions. Unlike the jammed ackings considered in the resent work, the FNE ossesses no fluctuations in article density nor system volume. In most of their comutations for frictionless articles, Tighe and co-workers [7 9] emloyed an FNE where the articles are constrained to sit at the sites of a regular triangular lattice, with forces acting between articles that share nearest neighbor bonds of the lattice. In such a network each article has a contact number z = 6, well above the isostatic value z iso = 4 that characterizes the jamming transition for frictionless circular disks in two dimensions [, ] (the configurations in the resent work have z ranging from 4.5 to 4.5 as increases). In their original work [7] Tighe et al. focused on the distribution of the ressure on an individual single article. Execting such a single article roerty to obey a maximum entroy distribution is in effect making an ideal-gaslike assumtion, where correlations between neighboring articles are ignored [9]. While they argue that this is reasonable for their triangular FNE, it is likely to be too simlistic for our disordered jammed ackings, where the length scales measured in Fig. 7d suggest that correlations may extend over at least a few article diameters for the range of stress considered here. In ef. [8], however, Tighe and Vlugt consider the distribution of total stress within a canonical ensemble on finite triangular clusters of N articles with non-eriodic boundaries, comuting the stress arameter α and the force-tile arameter λ (this is denoted as γ in their work) as a function of cluster size N. They use a similar range of N as the N we consider here. Several clear differences exist between their results on finite clusters for the triangular FNE and our results for satially disordered ackings. They find that α and λ are both ositive. In our work, where we can only comute the discrete derivatives with resect to global ressure, we find (see Fig. 8) α/.8 and λ/ 3. Integrating, and assuming that α, λ 0 as, we conclude that λ() > 0, but α() < 0. Furthermore, they found numerically that both α and λ vary significantly with the cluster size, and that λ vanishes as the cluster size increases. We, however, find that both α and λ aroach non-zero constants as the cluster size increases.

15 5 Tighe and Vlugt [8] argue that λ 0 as the cluster grows large because then fluctuations in the cluster forcetile area A C decay to zero, and hence A C and Γ C should no longer be regarded as indeendent observables that need to be indeendently constrained with searate Lagrange multiliers. However, as we exlain below, this argument does not aear to hold for our disordered soft disk ackings. Consider first the extreme limit where the cluster force-tile area is comletely slaved to the cluster stress, i.e. A = Γ /(π ) holds for each cluster configuration. To lowest order in the fluctuations we then have δa A A = Γ δγ /(π ) = δγ. The covariance matrix of Eq. (45) then becomes, [ ] C = var(γ ) C, C 4. (50) C has eigenvalues ρ = 0 and ρ = + 4. The eigenvector for ρ in the two dimensional sace of (Γ, A ) lies tangential to the curve A = Γ /(π ), while the eigenvector for ρ lies orthogonal to the curve. Eq. (46) then yields the constraint, dα d + dλ d = π var(γ ). (5) This result may also be obtained by taking the derivative with resect to ressure of Eq. (5) in Tighe and Vlugt [8]. This constraint is well satisfied for our clusters, as we show in Fig. a by lotting both the left hand and right hand sides of Eq. (5) vs ressure. For our smallest cluster size with =.8, the two quantities are fairly close, but for our biggest cluster size = 8. they are essentially equal. However the constraint of Eq. (5) is not sufficient to uniquely determine α and λ. Because ρ = 0, α and λ ossess a degree of freedom such that we are free to shift to a new α = α+g() and λ = λ+h() for any functions g and h that satisfy dg/d = dh/d. One may use this freedom to choose dα/d = π /var(γ ) and λ = 0, which is just the stress ensemble result of Eq. (9), or one can choose dλ/d = π /[ var(γ )] and α = 0. Indeed, Tighe and Vlugt [8] exlicitly show that, for a eriodic FNE (where A N is slaved to Γ N as in Eq. (34)) in the canonical ensemble, either of these choices gives the same single article ressure distribution if the system size N is sufficiently large. We may note that the constraint of Eq. (5) is the same as was found in Sec. III., if we take α and λ as the arameters describing the distribution P(Γ ) via Eq. (). In that case, we defined ᾱ in Eq. (3) such that in effect, dᾱ/d dα/d + dλ/d, and we found in Fig. 5 excellent agreement between this and π /var(γ ), just as found now in Fig. a from the distribution P(Γ, A ). One can show that the constraint of Eq. (5) just ensures that the location and width of the eak in the stress distribution P(Γ ) behaves correctly in a Gaussian aroximation (which becomes more exact as increases), when the Boltzmann factor is a quadratic form as in Eq. (33). For a finite cluster with non-eriodic boundaries, however, fluctuations in A C away from the average value at fixed Γ may be small, but they are finite. Consequently ρ > 0 is small but finite, the covariance matrix C is invertible, the above freedom to vary α and λ is broken, and a unique α() and λ() result. Where these unique α() and λ() lie in the sace of ossibilities allowed by Eq. (5) is determined in detail by such finite size effects. To investigate this for the case of our soft disk ackings, we exlicitly comute the two eigenvalues ρ and ρ of the scaled covariance matrix C C/var(Γ ) as a function of cluster radius and total system ressure. In Fig. b we lot ρ / vs. The data for different collase to a common curve that is very well fit by the form (c /)( c /), and thus ρ aears to vanish as gets large. We find c = 0.93 ± 0.0 and c = 0.58 ± 0.04, where the errors here and in the following aragrah reresent the variation in fit arameters found as varies. Next we consider ρ. Anticiating that ρ should aroach +4 at large, we lot in Fig. c (ρ )/4 vs. Again we find a fairly good collase of the data for different to a common curve that is well fit by c 0 ( c / + c / ), with c 0 = ± 0.00, c = 0.60 ± 0.05, and c = 0.0 ± Thus ρ indeed aroaches + 4 as increases. Finally, we consider the orientation of the eigenvector ê for ρ (the eigenvector ê for ρ is necessarily orthogonal to this). Defining θ as the angle between ê and the tangent to the curve A = Γ /(π ), in Fig. d we lot θ/ vs. Again we find a good collase of the data for different to a common curve that is well fit by the form (c /)( c /), showing that ê aligns arallel to the tangent to the curve as gets large; we find c = 0.6 ± 0.0 and c = 0.0 ± 0.0. Thus fluctuations in the direction orthogonal to the curve A = Γ /(π ) vanish a factor of / faster with increasing than do the fluctuations in the tangential direction. We can now use the results of Fig. to write dα/d and dλ/d in terms of the eigenvalues and eigenvectors of the scaled covariance matrix C of Eq. (50). Projecting the vector (, ) onto the eigenvectors ê and ê of C and alying Eq. (47) we have, dα d = π var(γ ) dλ d = π var(γ ) [ ( ρ sin θ cos θ sin θ ) +ρ ( cos θ + cos θ sin θ )] (5) [ ( ρ sin θ + cos θ sin θ ) +ρ ( cos θ cos θ sin θ )]. (53) To leading order, our results in Fig. give ρ /, ρ, and θ /. Inserting these into the above, we find that as increases, both dα/d and dλ/d aroach non-zero constants, with / corrections that vanish as

16 6!" /var(# ), d$/d + d%/d (! " )/ =.8 (c) = 8.!" /var(# ) d$/d+d%/d !" /var(# ) d$/d+d%/d ! /!/ (radians) (d) A 0.0 ê Γ ê FIG.. (color online) Comarison of dα/d + dλ/d with π /var(γ ) vs ressure, so as to check the constraint of Eq. (5). esults are shown for our smallest cluster size, =.8 and our largest cluster, = 8.. and (c) Eigenvalues ρ and ρ of the scaled covariance matrix C of Eq. (50). esults for different collase to a common curve when lotted as ρ / and (ρ )/4 vs cluster radius. (d) Angle θ between the eigenvector ê corresonding to the non-vanishing eigenvalue ρ, and the tangent to the curve A = Γ /(π ), as illustrated in the inset. Data for different collase to a common curve when lotted as θ/ vs. Solid lines in and (d) are fits to the form (c /)( c /), while those in (c) are fits to the form c 0( + c/ + c / ). gets large. For dα/d, the contribution from the rojection onto ê is negative while the contribution from the rojection onto ê is ositive, and both are O() in magnitude. Although the rojection onto ê becomes vanishingly small as gets large (i.e. θ 0), the refactor ρ is diverging so that the contribution from this term remains finite. Thus dα/d is determined by a balance between the two terms. For dλ/d the contribution from the rojection onto ê becomes O(/) as gets large, while the contribution from the rojection onto ê becomes O(). Hence it is the rojection onto ê that dominates dλ/d for small aroaching the jamming transition. Thus, although the fluctuations in direction ê are decaying more raidly as a function of cluster size than are the fluctuations in the direction ê, nevertheless the fluctuations along ê continue to give significant, non-vanishing, contributions to both dα/d and dλ/d even as the cluster size gets large. This conclusion is contrary to the qualitative argument of Tighe and Vlugt. The analysis of Tighe and Vlugt for the triangular FNE roceeds differently from our own aroach here. ather than analyze the stress distribution on a finite cluster embedded within a larger microcanonical (i.e. fixed Γ N ) θ system as we do, they consider a finite cluster on its own within a canonical ensemble, and determine α and λ so as to get the desired Γ and A for the cluster. It is ossible that the differences they observe, as comared to our own work, might be a consequence of the differing ensembles used; equivalence of ensembles is only exected in the thermodynamic limit. Or it may be that fluctuations in the FNE are sufficiently different from soft disk ackings so as to yield a different balance between the contributions from ê vs ê, and so select qualitatively different values for α and λ from among the family of choices allowed by Eq. (5). We note, in this regard, that the behavior of var(γ) aears to be different in the two models. In ef. [9], Tighe and Vlugt show that, for a cluster of N articles in the canonical FNE, var(γ) = Γ /( zn). Here z = z z iso, which for the harmonic soft-core interaction used here is believed to scale with system ressure as z / [8, 30]. Taking Γ = V, we get for the FNE, var(γ)/v 3/. In contrast, for clusters of radius embedded in our soft disk ackings, we have reviously found from numerical simulations [6] that var(γ )/(π ).9, for the range of ressure and cluster sizes considered here; it is of course ossible that the ower-law.9 is only an effective value that could change if we robed closer to the jamming transition. To clarify the difference between the FNE and soft disk ackings, it would be interesting to comute the covariance matrix of stress and force-tile area for the FNE and do a similar analysis as in Fig., however such a comutation lies outside the scoe of the resent work. Finally, it is interesting to note that if we define our clusters by a fixed number of articles M, rather than a fixed radius [6], then we find that both eigenvalues ρ and ρ go to finite non-zero constants as M increases, hence fluctuations remain comarable in all directions in the (Γ M, A M ) lane. Our results are shown in Fig.. However, we find that dα/d and dλ/d, as comuted from the covariance matrix for such fixed M clusters, behave qualitatively the same as found for the fixed clusters; although we find a somewhat stronger deendence on the cluster size M than we do for clusters of fixed radius, both dα/d and dλ/d aroach limiting non-zero values as M increases and dislay similar ower-law behaviors with ressure as found in Fig. 9 for the clusters of fixed. IV.6. elation to the stress ensemble Our analysis of the histogram ratio of the joint distribution P(Γ, A ) thus clearly suggests that P(Γ, A ) has the form of Eq. (38), with a Boltzmann factor ex[ αγ λa ]. In this section we exlore how this form may give rise to the marginalized distribution P(Γ ) = da P(Γ, A ), which was found in Sec. III to have the quadratic Boltzmann factor of Eq. (33), ex[ αγ λγ /(π )]. We consider