arXiv:2003.03476v1 [cond-mat.stat-mech] 7 Mar 2020 ett tcatclydie olsoa eteigulti- weathering collisional driven stochastically to ject on facets 3,600 to up with average. polyhedra yielding slices, nar 00soe,wihacmlt smn s2 as least many at as of accumulate trajectories which shape stones, the 1000 sta- follow Carlo we Monte error, statistical accumulate To tistical away. fractures worn stochastic is volume as and shapes rocks the of of indi- ensembles evolution how the of the over of characterize distributed we terms is facets, area in vidual surface smoothness total the and evenly oblate- prolateness, overall the and shape, ness spherical perfect a from viation the from material frac- away planar stones. of carve parent sequence successively a small which to a subject tures with are proto-clasts) faces of primordial number (i.e. which rocks in polyhedral simulations convex Carlo with Monte concerns removed scale these is large investigate mass We which yield at ex- weathering. fractures rate through the the random as and accumulating well profiles, the as spherical be- which scale, surfaces to macroscopic rock a tent which on to vol- neighbor- smooth degree more with come the and interactions include random more stones by ing as removed shapes is questions emerging ume Salient the frag- to induced stochastic. collision related volume. inherently as their are such away mentation, processes, eroding these by rocks of Many of shapes the form sn w-rne prah efidta tnssub- stones that find we approach, two-pronged a Using ihacmiaino uniaiemaue ftede- the of measures quantitative of combination a With trans- or weather to act mechanisms of types Various 1 tt efidapwrlwdcyo eitosi ogns par toughness a in deviations variet of a decay With law events. power fracture a find in provi we material slice state of prospective removal a by the exposed d for shap area and spherical) the orientation in (i.e. random round probability and of smooth slices yielding ultimately planar with weathering to reua rget rae ycevn eua oi man solid ini regular stones a for cleaving shapes by of created evolution fragments time irregular the examine We shape. ie o h civmn fsletsrcua milestones structural salient stones, of irregular achievement resp initially with the of smooth for case are times observables the relevant of In and shi loss transitions, a the stages. struc involves solids, in order irregular transition second occurs of second of cohorts the mono-dispersed sets while the for two involves solid, first find The parent we behavior. the former, critical of the hallmarks of usual case the In rsetv e ae nti anr ecluaetm scal numbers: time PACS simulatio calculate direct We bound probabilit manner, which acceptance this expressions fracture approximate In the form which face. in new with scenarios case prospective erosion the of ori as variety the such a of fraction for the variables on dependent variables of dependence universal eateto hsc srnm,Yugtw tt Univer State Youngstown Astronomy, & Physics of Department ieSae o onigo ok hog tcatcChippi Stochastic through Rocks of Rounding for Scales Time o he iesoa emtis ecnie tns(mode stones consider we geometries, dimensional three For .INTRODUCTION I. × Dtd ac 0 2020) 10, March (Dated: 10 .J ror Jr Priour, J. D. 6 pla- oreeso h hpsa qiiru ie h mean the (i.e. equilibrium at area parameter shapes sense relative the a a in of tuning thus coarseness By and values constant monotonically observablesequilibrate. at decreases stabilize where which time) scenario volume, in hand, state the one from steady the (apart On a consider shapes. we spherical to converge mately ies,w osdrpoolssi h omo irregu- of form the in protoclasts consider we structurally be may diverse, weathering to prior stones primordial bounds upper as serving results. point while simulation for times a finite expressions diverge, yielding form sce- not closed also erosion approximate do with specific but underscore we the consideration on under that depending finding nario of vary volume, removal primordial times the the such as of such fraction scales milestones specific time a of for attainment results quantitative the mass. glean for their cur- we of way, a independent this speed by In fixed constant impelled mean the a rocks of at for rent context envisaged di- the scheme, with in validate velocity simulation we Carlo technique to Monte a rect latter non- scenarios, of the range state which use broad steady a We in for quantities scenario corresponding equilibrate. the obtain eventually in observables latter of the time) for proxy clsas scales edsseaial ozr ihincreasing with zero surface to smooth perfectly systematically a tend and/or shapes spherical from ihtenme fssandslices (i.e. sustained dependence of time number the the calculate also with we hand, other htteeulbimma number mean equilibrium the that aigit osdrto hti rciechrsof cohorts practice in that consideration into Taking rmrilfct sntsmlaeu but simultaneous not is facets primordial s nepnnilydcyn acceptance decaying exponentially An es. iutnosls ffct neie from inherited facets of loss simultaneous ia ouermiigt aclt time calculate to remaining volume ginal 1 e tcatclydie hsclbasis physical driven stochastically a des c otm.Mr ral,w n that find we broadly, More time. to ect eut rmasnl eteigscheme weathering single a from results iodrosue niiulstructural individual obscures disorder eut rmabove. from results n tt peia rfie Nevertheless, profile. spherical a to ft ie ln admfatr planes. fracture random along times y cl udaial in quadratically scale eed nterltv rao the of area relative the on depends y fqatttv esrs nsteady in measures, quantitative of y γ phscesvl eoigmaterial, removing successively epth ilyi h omo ue swl as well as cubes of form the in tially so neetadas banclosed obtain also and interest of es hl uniaiemaue ftedeviation the of measures quantitative while iy onson H455 USA 44555, OH Youngstown, sity, e scne oyer)subject polyhedra) convex as led ameter h ua hs rniin ihthe with transitions phase tural ∆ A i /A γ Σ rmapretspherical perfect a from fnwyfre aes,w find we facets), formed newly of γ euethe use We . h N n ng γ i sust fpaa faces planar of otoln the controlling evn sa as serving γ nthe On . 2 larly shaped polyhedra to account for aggressive frag- approximately 50% of the original volume has been re- mentation events prior to the collision induced weather- moved, and declining and tending to zero thereafter. ing. To mimic structures resulting from dramatic frag- As a novel feature of the calculations in this work, no mentation processes, initial cohorts of structurally dis- restrictions are imposed as to how many vertices and ordered shapes are generated with a sequence of random facets are truncated or removed by a sustained slice, and fracturing planes, invariably accepted independent of the our calculation is compatible with the elimination of an orientation and depth of the prospective slice. With pro- arbitrary number of vertices per fracture. For a plausi- toclast cohorts generated in this manner, we find relevant ble physical basis for fracture events, measures such as observables to vary smoothly with time. As companion the area of a newly exposed face are considered to deter- calculations, we also examine structurally homogeneous mine if a candidate fracture plane is accepted. To set the primordial stones, including regular cubes, which ulti- scale of new facet areas, we implement an energy based mately are smoothed into spherical shapes just as occurs criterion for accepting prospective slices, taking the ki- in the case of irregular protoclasts. However in contrast netic energy input needed to cleave away a slice to be to structurally disordered stones, observables in the case proportional to the area of the new face. In this vein, in of cubes exhibit two continuous phase transitions with the spirit of statistical mechanical treatments, we assume all of the hallmarks of a second order phase transition, the fracture probability to be exponentially suppressed in including power law scaling for singularities in relevant the exposed area ∆A, and given by P (∆A) = e−γ∆A/θ observables as we show explicitly in this work. The first where γ is a constant related to the toughness of the transition is a structural transformation in which the pri- material comprising the rocks and θ(V ) is a volume de- mordial cube facets are sheared away, while in a subse- pendent measure of the mean kinetic energy or “temper- quent transition the stones derived from regular solids re- ature”, with volume serving as a proxy for the mass in vert to spherical shapes with concomitant singularities in the case of stones of uniform composition as assumed in quantities sensitive to deviations from a perfect spherical this manuscript. In addition, since we consider initial profile. A structural phase transition of the former type cohorts of stones to be mono-dispersed with respect to has been proposed in previous studies based on labora- mass (though structurally diverse in the case of irregular tory experiments and numerical studies [1, 2]. We apply protoclasts), we operate in terms of the relative volume the tools of finite size scaling to locate the transition and v˜ V/V0 with V0 the volume prior to the erosion process. calculate associated critical exponents. With≡ analytical arguments and direct simulation results, In addition to regular shapes, we consider a homoge- we show that characteristic times scale quadratically in neous ensemble of geometrically irregular solids finding the toughness parameter γ. asynchronous transitions for the erosion of primordial In this work to be definite we assume a power law α facets instead of a single structural transition for the solid dependence θ = θ0v˜ (i.e. with α > 0), reflecting ki- as a whole, though again we find a single second order netic energy dependence on volume as a rock is borne along (e.g. by water currents in a river or stream as col- transition to a round spherical shape at a later time af- ′ ter all of the primordial facets have been removed. Thus, lisions chip away material). Defining γ γ/θ0 for the only in the case of highly symmetric shapes is there a sake of brevity, the acceptance probability≡ for a prospec- ′ −α simultaneous elimination of the facets inherited from the tive fracture plane is P (∆A) = e−γ ∆Av˜ . Assuming ′ parent solid. γ ∆A v˜−α 1 for the mean area ∆A of a new facet, In spite of the stochastic nature of the erosion scheme, h i ∼ h i one would argue on dimensional grounds′ that the mean we find in the regime that fragments cleaved away number of faces is n A / ∆A = γ v˜−αA , where h i ∼ Σ h i Σ through random fracture events are small in volume rela- AΣ is the total polyhedron surface area. For the 3D 2/3 tive to that of the stone as a whole that shapes evolve de- case, taking the total area to scale asv ˜ , αc marks terministically, with reproducible changes on scales small a boundary between stones which in principle could be- in comparison with the overall size of the stone but large come smoother over time (α > αc) and shapes which be- relative to freshly exposed faces. As in a recent study, come coarser (α < αc) as fractures accumulate; whereas we find a universal dependence of observables on the re- relative areas ∆A /AΣ of new facets decrease for the maining volume fraction [3]. We take advantage of this former, newly exposedh i facets encompass an increasingly characteristic to calculate time dependent observables for large share of the surface in the case of the latter. The a variety of erosion schemes using results from the sce- boundary case α = αc thus offers the possibility for the nario of fixed relative area for typical fresh facets created attainment of a steady state for the mean relative area by fracture events. of new faces, as well as other variables of interest. Aside from measures of deviation from a perfectly Another merit of obtaining steady state or relative area spherical shape, we also examine more specific macro- results is the fact that the universal dependence of ob- scopic structural characteristics, including the degree to servables on volume fraction permits one to map remain- which stones are oblate or prolate as volume is chipped ing stone volumes onto time for a scheme distinct from away. While both the former and the latter tend to the relative area case, an technique bolstered with results zero as rocks are rounded, the oblateness measure is non- from large-scale Monte Carlo simulations. We show in monotonic in time, initially rising and then peaking after this manuscript that both the former and latter are iden- 3 tical up to random Monte Carlo statistical error. Time faces (i.e. for smooth shapes), but converges to a finite scales obtained in this manner are then used to validate a value if a small number of facets contain a macroscopic closed form expression serving as an approximation and fraction of AΣ (i.e for rough stones), and thus serves as strict upper bound for times associated with the chipping a quantitative measure of smoothness. Appealing to the away of a specific volume fraction or the structural phase IPR in this manner is analogous to its usage in the con- transition for cohorts of regular protoclasts in which pri- text of charge transport discussions to distinguish among mordial facets disappear. extended and localized carrier states [5]. In Section II, we discuss principal quantitative mea- In this work, we simulate collisional weathering with sures of deviation from sphericity as well as key method- large scale Monte Carlo simulations involving sequences ological elements. Section III contains results for rocks of randomly oriented planar fractures, where rocks begin in the case of the steady state scenario, while Section IV as regular or irregular protoclasts with a small number broadens the discussion to non-steady state situations. of faces [i.e. 6 and a mean of 9.03(1) sides for cubic In Section V, we calculate time scales for a range of ero- and irregular polyhedra respectively], ultimately yield- sion schemes, with a closed form expression obtained as ing polyhedra with as many as 3,600 facets. Although an upper bound for time scales for salient structural mile- there are qualitative differences for the time dependence stones. Conclusions are found in Section VI. of observables in the cases of cube shaped and irregu- lar protoclasts (e.g. the structural phase transition for cube shaped protoclasts not seen in ensembles of irregu- II. QUANTITATIVE MEASURES AND lar shapes), both cohorts ultimately converge to spheri- METHODOLOGY cal shapes in the long time regime. Moreover, we obtain an approximate analytical expression for time scales for Two salient quantities which permit one to address in salient stages in the attainment of a perfectly spherical a direct manner the degree to which a shape departs from shape. The fact that these times are finite, bounding a perfect spherical profile include the configuration aver- corresponding numerical results from above is compati- min ble with the eventual conversion of rocks into spherical aged sphericity φ3 and the ratio rmax of the minimum and maximum distance of surface points for the center shapes for a wide range of erosion scenarios. of mass. Both measures distinguish among a spherical In stochastic chipping sequences, each sustained slice shape and a stone which is not perfectly round. The exposes a new face while truncating one or more ver- sphericity benefits from the fact that surface area to vol- tices. The likelihood of accepting a prospective slice ume ratio of a solid attains a global minimum for spheres, hinges solely on the area ∆A and volume of the par- 1 2 and is given by φ [6π 2 V ] 3 /A [4], where A is the ent solid with no restriction on the number of truncated 3 ≡ Σ Σ surface area and V the volume. Since φ3 < 1 (except in vertices, raising the possibility of the elimination of an the case of a perfect sphere where φ3 = 1), the comple- entire face or faces if each of the constituent vertices are ment 1 φ3 serves as a measure of the departure from a cleaved away. For the sake of an efficient implementa- perfectly− spherical shape. An alternative measure, sen- tion, updates of lists of vertices and planes making up min sitive to local features, is rmax dmin/dmax, the ratio the polyhedron include the addition of new features as of the minimum and maximum≡ distances, respectively, well as the deletion of those eliminated by the most re- of points on the rock surface from its center of mass. cent fracture event. Indeed, in the steady state scenario, In the context of our calculations involving faceted ob- as many vertices and facets are removed on average as jects, dmin is the distance to the closest planar facet and accumulate with each sustained slice. Previous studies min dmax is the distance to the farthest vertex. Again, rmax in two and three dimensional geometries have involved peaks at unity only in the case of a perfect sphere, with the truncation of a finite number of vertices or facets per min 1 rmax serving as a measure of the deviation from a fracture event [1, 3, 6, 7]. A calculation involving two di- perfect− spherical shape. mensional shapes allowed for the removal of an arbitrary Related to whether there is a systematic convergence number of vertices [8] To our knowledge, our calculations to a spherical shape is if erosion through the stochastic are the first to examine erosion phenomena due to a se- chipping mechanism yields a stone with a smooth surface. quence of planar fractures allowing for the elimination of Although sphericity implies smoothness, the converse is an arbitrary number of vertices and/or facets for three not true. In fact, in this work we find an example of a dimensional geometries. perfectly smooth non-spherical shape at the phase transi- To facilitate the examination of polyhedra with a large tion for structurally homogeneous cohorts where primor- number of faces, a variety of measures are employed to dial facets disappear. Hence, as a measure of smoothness optimize the efficiency of locating vertices of prospective not anchored to a specific geometry, we use the Inverse facets. While one could in principle examine all possible − Participation Ratio (IPR), where IPR = A 2 N A2 intersections of a fracture plane with the faces compris- h Σ i=1 i i with AΣ being the total surface area, Ai the area of the ing the stone, in general only a small subset of vertices ith of N facets, and angular brackets indicatedP a config- identified in this manner populate the new face, compris- urational average. The IPR tends to zero with increasing ing a share of the total which diminishes as the number N for an even distribution of the area of the polyhedron of planar faces increases. We avoid considering spurious 4 vertices by only examining faces which contain vertices both above and below the prospective fracture plane, as facets not meeting this condition are either sheared away altogether or are not truncated by slice at all and hence do not yield any new constituent vertices. As a fur- ther constraint, we only seek intersections of the slicing plane with pairs of faces which have an edge in com- mon. Finally, we avoid computational costs associated with slices unlikely to be accepted by considering a sphere inscribed in the polyhedron and concentric with its center of mass, for which circular regions exposed by a fracture plane bound the area of a prospective face from below, with the corresponding acceptance probability overesti- mating that of the prospective new facet. In this manner, we avoid consideration of fracture planes for which the acceptance probability is no greater than 10−8, events with negligible incidence over the course of an erosion sequence. Operating in this manner, the computational FIG. 1: (Color online) Sixty four sample irregular protoclasts burden of a sustained slice scales no more rapidly than the number of facets of the solids we consider. The areas of individual facets and the total volume a tetrahedron of uniform composition is the arithmetic are key quantities in finding the probability of a prospec- mean of its four vertices to find the center of mass of the tive fracture event as well as many of the observables we entire shape. In terms of semi-major axes a,b,c of the calculate. For individual faces, the arithmetic mean of equivalent ellipsoid (i.e. with a>b>c{), the} oblate- each of the member vertex coordinates provides a con- ness and prolateness measures ψO and ψP are taken to venient interior point for subdividing the polygonal facet be ψO = (b c)/a and ψP = (a b)/a, respectively. into triangles, whose areas are calculated and summed − − to yield the combined face area. On the other hand, to find the total volume of the stone one operates in an A. Regular and Irregular Protoclasts analogous fashion, partitioning the polyhedron into con- stituent tetrahedra whose combined volume is the shape In selecting initial shapes, our simulation program bi- total volume. To this end, we again subdivide faces into furcates, including both regular and irregular polyhedra triangles; the three vertices of the latter along with an as primordial forms. In the case of the former, we ex- interior point (given by the mean of all of the polyhedron amine cubes, and the benefit of beginning with regular member vertices) define the component tetrahedron. shapes is two-fold, with one useful feature being that Complementary to the sphericity measures and the equilibration in steady state scenarios is more rapid for IPR are quantities which bear more specifically on the regular polyhedra than for irregular counterparts. In ad- shape of the stone, namely parameters which characterize dition, we find in the case of cube shaped protoclasts the degree to which a stone is oblate or prolate. Measures two structural phase transitions, in the first of which all of the latter are gleaned from the principle moments of remnants of the primordial facets are cleaved away si- inertia of the equivalent ellipsoid, obtained by diagonal- multaneously. Subsequently, an additional second order izing the moment of inertia tensor of the stone. As in transition signals a shift to a spherical shape. the calculation of total volume, the polyhedron is par- Cohorts of initially irregular shapes, on the other hand, titioned into constituent tetrahedra with elements I of ij are envisaged as a closer geometric match to protoclasts the moment of inertia tensor given by [9] formed through dramatic fragmentation events. To gen- 3 erate irregular polyhedra in this very aggressive man- V I = tet δ (s2 + t ) s s t (1) ner, a cube is subject to a sequence of slices, with each ij 20 ij k kk − i j − ij " k=1 # prospective slice accepted independent of the area ∆A of X the new face. Images of sample polyhedra generated in where Vtet is the volume of the constituent tetrahedron, this fashion are shown in Fig. 1. As illustrated in panel 4 4 while si l=1 xli and tij l=1 xlixlj with, e.g., xli (a) of Fig. 2, with on the order of 15 sustained slices, the being the≡ith component of≡ the lth vertex of the tetra- mean number of sides quickly converges to 9.03(1) facets hedron. CombiningP moments forP each component tetra- on average. The traces in the main graph, plotted with hedron yields the moment of inertia tensor for the shape respect to Nsust, were obtained with cubes and tetrahe- as a whole. To find the moment of inertia relative to the dra as initial forms. The frequency distribution of sides, center of mass, as appropriate for a bona fide ellipsoid, shown in the graph inset, converges with similar rapid- we appeal to the Steiner parallel axis theorem [10] while ity; in the inset graph, symbols indicating Monte Carlo taking advantage of the fact that the center of mass for results and the solid line being a fit to a log-normal distri- 5

10.0 faces n converging to a fixed value since ∆A AΣ/γ (a) Initially tetrahedra h i h i∼ Initially cubes per the acceptance probability relation. Images of stones 9.0 sampled from the steady state ensemble appear in Fig. 3, 0.2 showing the emergence of smooth spherical shapes for 8.0 toughness parameters ranging from γ = 10 to γ = 2000. 0.2 A salient question related to the attainment of equi- librium is how long is needed (in terms of the number of 7.0 0.1 sustained slices Nsust) for an ensemble of stones to reach steady state. In the context of numerical simulations, 6.0 0.1 we insist that all observables of interest remain constant Fractional Occurrence with respect to doubling of Nsust. For γ = 100, 3,000 0.0 sustained slices satisfies this criterion. As we now argue 5 10 15 20 25

Mean Number of Facets 5.0 Number of Facets from geometric considerations, for γ 1 (i.e. certainly for γ 100) time scales for the removal≫ of equivalent frac- 0 10 20 30 40 tions≥ of the original volume of an ensemble of rocks (and Sustained Slices hence salient stages such as the achievement of steady 1.0 0.0 (b) Initially tetrahedra (c) Initially tetrahedra state) scale quadratically with the toughness parameter Initially cubes Initially cubes 0.8 -1.0 γ. Although the acceptance probability sets the scale 0.6 -2.0 ∆A for the typical area of a new facet, fractures leav- h i 0.4 -3.0 ing behind faces with the same area need not cleave away the same volume. One anticipates more volume to be re-

0.2 (Facet Survival Probability) -4.0 moved from sharply peaked features than from regions 10 Facet Survival Probability with less curvature by the same number of sustained

0.0 Log -5.0 0 10 20 30 40 0 500 1000 1500 slices [11]. Thus the edges and vertices of the primordial 2 Sustained Slices (Sustained Slices) polyhedron, are quickly worn down in the initial stages of the erosion process. FIG. 2: (Color online) Mean number of facets versus sus- In the γ 1 regime, fracture planes are constrained tained slices versus sustained slices for cubic and tetrahedral ≫ protoclasts (square and triangular symbols respectively), with to be shallow, unable to penetrate deeply due to the ex- solid curves a guide to the eye. Shown in the inset is the ponential penalty on ∆A in the acceptance probability. frequency facet number distribution; circles are Monte Carlo We introduce the cleaving plane, just below the rock sur- results while the solid line is an optimized fit to a log-normal face, as well as the parallel plane of tangency with a distribution. Panels (b) and (c) show the fraction of original single point of contact with the surface, while superim- facets remaining with solid lines a fit to a Gaussian decay in posing Cartesian axes with the xy plane coinciding with Nsust. the tangent plane and the z axis directed toward the in- terior of the stone. Hence, for the rock surface one would generically have have f(x, y) Ax2 +2Bxy + Cy2 near bution with σ =0.233 and µ =2.189. We also exhibit in its minimum at the point of tangency,≈ where A, B, and panels (b) and (c) of Fig. 2 the survival probability fsur C are locally determined constants. Seeking to eliminate or mean likelihood of the persistence of a facet or por- the diagonal term, we have in matrix form tion of a facet original to the parent solid. As indicated in panel (b) of Fig. 2, fsur is strongly suppressed after 25 A B x x f(x, y)= x y ; ~v (2) sustained slices. As the semilogarithmic plot with respect B C y plane ≡ y      to the square of the number of sustained slices in panel   (c) of Fig. 2 suggests, the fsur decay at an approximately 2 where ~vplane is the component in the tangent plane. The −(Nsust/N0) 1 Gaussian rate with fsur e where N0 =9.98 eigenvalues of the symmetric matrix are: λ = [(A + ≈ 1,2 2 for cube shaped initial forms and N = 11.4 for tetra- 2 2 2 0 C) (A C) +4B ] with λ1,2 > 0 if AC > B , as hedron shaped protoclasts. In simulations involving ir- must± be the− case for the convex objects we consider. In regular protoclasts, 100 sustained slices are imposed to p terms of the orthonormal eigenvectorsu ˆ1 andu ˆ2, one remove any vestige of the seed form. may write ~vplane = α1uˆ1 + α2uˆ2 and thus

2 2 f(x, y) g(α1, α2)= λ1α1 + λ2α2 (3) III. STEADY STATE SCENARIO → The cleaving plane hence exposes an elliptical region For the α = α case where the acceptance probability for which z = λ α2 +λ α2 for a fixed depth z beneath the c 1 2 1 1 2 2 −γ∆A/As for a prospective slice is e (As = [6π 2 V ] 3 be- point of tangency, where the area is A(z) = πab = πz/l −1/2 ing the area of the volume equivalent sphere), a steady with l (λ1λ2) being a locally defined length scale. state may in principle develop with the mean number of Thus,≡ the depth of a typical fracture plane (i.e. with 6 γ 4.0 3.0

2.0

3.0 1.0 (mean number of facets) 10 0.0

Log 0.0 1.0 2.0 3.0 (γ) 2.0 Log 10 γ = 10 γ = 20

(mean number of facets)/ 1.0 0.0 1.0 2.0 3.0 (γ) Log10

FIG. 4: (Color online) Mean number of facets with respect to γ. Symbols are Monte Carlo results, while the solid line in the main graph and inset is an analytical curve. γ = 40 γ = 80 this Independent Facet Model (IFM), characteristics of polyhedron faces are considered to statistically uncorre- lated; nevertheless, we find good quantitative agreement with salient observables from Monte Carlo simulations. For a spherical geometry, the relative area of a fresh facet is ∆A/A = (2˜u u˜2)/4 whereu ˜ = u/R, with R being the mean radius− of the stone and u the distance of the planar slice below the surface of the rock. With the probabilities of new faces being exponentially suppressed in ∆A, mean facet variables f(˜v) are given by h i γ = 400 γ = 2000 2 1 f(˜u)e−γ(2˜u−u˜ )/4du˜ f(˜u) = 0 (4) h i 1 −γ(2˜u−u˜2)/4 FIG. 3: (Color online) Steady state shapes for various γ values R 0 e du˜ where the denominatorR plays a role analogous to that an area A ) is D = ∆A (πl)−1, and the typical vol- of the partition function in statistical mechanics; for the h i h i D 2 sake of obtaining closed form relations accurate to leading ume ∆V removed is ∆V = 0 A(z)dz = ∆A /(2πl) − − h i ∼ or next to leading order in γ in for γ 1, Eq. 4 reduces γ 2A /(2πl) since ∆A A γ 1 for the relative area ≫ Σ R Σ to scenario. Thus, timeh scalesi ∼ for the removal of volume 2 ∞ scale as γ in the steady state scenario and more broadly 1 − f(˜u) γ f(˜u)e γu/˜ 2du˜ (5) as the square of the toughness parameter in non-steady h i≈ 2 0 state schemes as well. Z Fig. 4 shows the mean facet number n with respect to γ, with symbols representing Monte Carloh i results; an- A. The Independent Facet Model alytical results indicated by the solid curve are in good agreement with the latter. The main graph is a plot of In the steady state calculations, we obtain configura- the ratio n /γ with respect to log10(γ), which converges tion averaged observables (i.e. for at least 1000 distinct to 1.82(1)h ini the large γ regime, while the inset graph realizations) for two and a half decades of the mean num- is a log-log plot of n versus γ. A salient feature is the ber n of facets, finding power law decays in γ and n comparatively gradualh i increase in n for γ 10 with a for γh i 1 for all measures of the deviation from perfecth i more rapid asymptotically linear riseh i thereafter.≤ spherical≫ shapes. As a model for quantities of interest, The analytical curve is obtained assuming a truncation we assume quasi-spherical shapes for cohorts of stones over time of newly exposed faces, where in terms of the in steady state with properties (e.g. areas) of polyhe- mean area A0 of newly created faces, the average area dron faces assumed to be statistically uncorrelated; in of A overh itsi lifetime is A = η A where η < 1. h i h i h 0i 7

-0.5 0.0 1 − r min -1.0 max -1.0 -1.5

(IPR) 0.6 1 − min 0.15 rmax 10 -2.0 1 − φ -2.0 3 0.10 0.4 Log IPR -2.5 0.05 0.2 1 − φ 3

0.00 0.00 0.02 0.04 0.06 0.08 0.10 -3.0 Sphericity Measure Complement 0.0 -3.0 −1 0.00 0.02 0.04 0.06 0.08 (mean number of facets) (mean number of facets) −1 10

0.0 1.0 2.0 3.0 Log (Sphericity Measure Complement) 0.0 1.0 2.0 3.0 (γ) (γ) Log10 Log 10

FIG. 5: (Color online) Inverse Participation Ratio (IPR) log- FIG. 6: (Color online) Spherical measure complements 1 − min log plot (main graph) and IPR with respect to the reciprocal rmax (open circles) 1 − φ3 (filled circles) log log plot (main of the mean facet number (inset). Symbols represent Monte graph) and results plotted with respect to the reciprocal of Carlo data, while the solid curve is from an analytical model. the mean facet number (inset). The solid blue curve in the main graph is an analytical curve from the Independent Facet min Model, while the magenta line fitted to the 1−rmax is a power − The mean solid angle subtended by a facet, modeled as law decay scaling as γ ζ where ζ = 0.81. a circular shape (with the reduction in area taken into 2 account) is fΩ(˜u)=2π[1 1 η(2˜u u˜ )] where the − − − Sphericity complement measures are shown in Fig. 6 mean number of facets is hence n =4π/ fΩ(˜u) . p h i h i for 1 φ (filled circles) and 1 rmin (open circles) on One fixes η by appealing to the large γ limit whereu ˜ − 3 − max 1 due to the shallowness of typical sustained slices. In this≪ a log-log scale with respect to γ in the main graph and versus n −1 in the inset graph. From the main graph, regime, one may expand the radical in fΩ(˜u) expression; h i using Equation 5, we find n γ/η, η = 0.549(3) and one see that both complements shift to power law decays n =1.82(1)γ for γ 1. h i ≈ after a plateau qualitatively similar to that other IPR in h i ≫ Fig. 5, with 1 φ decreasing more rapidly than 1 rmin The Inverse Participation Ratio (IPR) is plotted in − 3 − max Fig. 5 with the solid curve obtained in the Independent with the asymptotic downward slope in the log-log graph Facet Model framework and symbols representing Monte appearing to be greater for the former than for the latter. Carlo results in both the inset and the main graph with The trend to zero of departures from a perfect spherical good quantitative agreement among the latter and the shape with increasing number of facets is also evident in the graph inset which shows 1 φ and 1 rmin with former. In the case of the inset, the IPR is plotted with − 3 − max respect to n −1, suggesting an extrapolation to zero as respect to the reciprocal of the mean number of facets, h i n −1. the number of facets becomes infinitely large and thus h i smooth surfaces in the large γ limit. On the other hand, In the IFM framework, we obtain the correct asymp- the linear decrease of IPR in the log-log curve for γ 10 totic behavior in the case of the sphericity complement in the main graph is compatible with an asymptotically≥ 1 φ3. In this vein, one begins by considering a sphere −1 −1 with− a volume of 4 πR3, and n lenticular slices are power law decrease (specifically as γ or n ) for the 3 h i Inverse Participation Ratio. h i sheered away by the fracture planes; one calculates in n the IFM context the mean of the ratio of the surface area In the IFM framework, we posit that i=1 Ai may be replaced with A n and n A2 h with Ai2 n , of the volume equivalent sphere to the area of this solid, i=1 i 1 2 yielding h ih i h i P h ih i namely [(2/s)(1 s)(1 √1 s)] 3 where s η(2˜u u˜ ). P The resulth is the− solid− curve− in thei main graph≡ of 6.− Al- 2 χ f(2˜u u˜ ) though depressed relative to the 1 φ3 curve, the IFM re- IPR h − i (6) − ≈ n f([2˜u u˜2]2) sult and Monte Carlo simulation results nonetheless have h ih − i in common a plateau-like region for γ 10 which gives where χ =1.32(1) is a dimensionless fitting parameter on way to a linear slope signaling a power≤ law decay. In the order of unity. The analytical IPR curve obtained in fact, from Eq. 5, the steady state sphericity complement 4 −1 this manner with the mild rescaling by χ is in excellent 1 φ3 scales as 3 ηγ , asymptotically correct apart from quantitative agreement with Monte Carlo simulation re- a− dimensionless prefactor. min sults as may be seen in Fig. 6. By appealing to Equation 5 Whereas 1 rmax is likely sensitive to correlations in the γ 1 regime, we find that the inverse participa- among facet sizes− and positions, and thus less amenable tion ratio≫ tends to: IPR = (2χη)γ−1 = (2χ) n −1. to the IFM treatment, a statistical analysis of the time h i 8 0.0 prolateness oblateness -0.5

0% volume removed 2% volume removed 5% volume removed -1.0 0.5

0.4

0.3 (ellipticity measure) -1.5 10 0.2

0.1 Log 10% volume removed 25% volume removed 44.9% volume removed 0.0 0.00 0.02 0.04 0.06 0.08 -2.0 ellipticity measure (mean number of facets)−1 FIG. 8: (Color online) Images of stones derived from a cube 0.0 1.0 2.0 3.0 shaped protoclast at various stages of erosion for γ = 2000, (γ) with facets original to the parent cube shown in blue. The Log10 structural transition appears at the extreme lower right cor- ner. FIG. 7: (Color online) Log-log plot of ellipticity measures, in- cluding oblateness and prolateness with respect to the mean − facet number hni 1 Simulation results are indicated with sym- the variables we calculate apart from measures of devi- bols, while the red line on the right side corresponds to a − ation from a spherical shape. The latter exhibit critical power law scaling of γ 1. In the inset, the ellipticity mea- −1 behavior at a subsequent phase transition in which the sures are plotted with respect to hni . stones revert to spherical shapes. Selected images from the erosion trajectory of cube

min shaped protoclasts appear in Fig. 8 with the image at dependence of the 1 rmax suggests that it saturates − −ζ the lower right coinciding with the structural transition at values proportional to γ where ζ = 0.81(5). This in which all primordial facets are removed; blue areas are power law decay is indicated by the magenta line in the facets or portions of facets original to the parent solid. main graph of Fig. 6. On the other hand, due to the inherent strong structural Measures of ellipticity, the oblateness (open symbols) disorder of the irregular protoclasts, the disappearance and prolateness (filled symbols), are shown in Fig. 7 in of primordial facets occurs at different times for differ- a log-log plot. Both the oblateness and the prolateness ent shapes, with a concomitant loss of any well defined decrease gradually with γ, for γ 10, with the former ≤ phase transition for ensemble averaged observables for initially flat as indicated by the blue line. Asymptotically the cohort as a whole. In fact, by considering mono- linear on a log-log scale, the prolateness and oblateness dispersed cohorts made up of a single irregular shape, converge in the large γ regime. The red line highlights we see that the elimination of protoclast faces is in gen- the power law decay of both ellipticity measures, corre- eral asynchronous even for an ensemble of initially identi- sponding to a 1/γ dependence. cal shapes, with a simultaneous elimination of primordial facets limited to highly symmetric shapes and being the exception rather than the rule. B. Time Evolution of Shapes In statistical mechanical analyses of phase transitions, a standard practice is to specify an order parameter, a 1. Structural Phase Transitions in Mono-dispersed variable which is finite when the phase in question is Protoclasts present, and zero otherwise if one is in the bulk or ther- modynamic limit. For this purpose, we use the survival Although steady state shapes are smooth and spherical probability (denoted in this work as fsur), or the ensem- for γ 1, the initial cohort of protoclasts are polyhedral ble averaged fraction of primordial facets remaining. with a≫ relatively small number of facets. We consider reg- The survival probability for cube shaped protoclasts is ular cubes as protoclasts as well as the highly irregular shown in in Fig. 9 for γ values ranging from γ = 250 to initial shapes mentioned earlier. In the case of the for- γ = 2000. With characteristic times scaling as γ2, we mer, we observe critical behavior as primordial facets are plot the survival probability and other salient variables 2 eroded away, and stones become smoother and rounder as with respect to the scaled time, Nsust/γ , also denoted as additional volume is chipped away; this transition from τ˜. The mean fraction of surviving original facets initially shapes which possess facets form the parent solid and exhibits a plateau, decreasing to zero as all of the primor- those which do not is abrupt with all of the hallmarks of dial cube facets are eroded away. That the descent of the a second order phase transition and is reflected in all of survival index sharpens as γ increases [with curves cross- 9

1.00 0.90 (a) (a) -1.0 γ = 250 0.80 γ = 500 γ = 500 0.70 γ = 250 γ = 1000 0.80 γ = 2000 0.60 γ = 1000 (IPR) -2.0 10 0.50 γ = 2000 Survival Probability 0.092 0.094 0.096 0.098 0.100 0.60 Log /γ 2 Nsust -3.0

0.40 0.00 0.05 0.10 0.15 0.20 γ = 500 2 Nsust /γ 0.20 3.0

Survival Probability γ = 1000 γ = 250 γ = 250 (b) γ = 2000 ] γ = 500 0.00 2.0 γ = 1000 γ = 2000 0.00 0.05 0.10 0.15 0.20 (IPR)

2 γ Nsust /γ [ 1.00 10 1.0

(b) Log 0.80 0.0 0.00 0.05 0.10 0.15 0.20 ~τ 0.60 2.5 γ = 250 (c) ] 2.0 γ = 500 0.40 γ = 1000

(IPR) 1.5

γ γ = 2000 γ = 500 [ γ = 1000 10 1.0

Survival Probability 0.20

γ = 2000 Log 0.5 0.00 0.0 -1.00 0.00 1.00 -1.0 -0.5 0.0 0.5 1.0 ε γ Ξ ~ ~ γ ~ ~ (τ − τc) (τ − τc)

FIG. 10: (Color online) Inverse Participation Ratios (IPR) FIG. 9: (Color online) Facet survival probabilities fsur for cube shaped protoclasts are shown in panel (a) with respect for cube shaped protoclasts in panel (a) for various γ values. 2 Panel (b) is a plot of γ × IPR with respect toτ ˜ on a semi- toτ ˜ = Nsust/γ ;a closer view of the fsur curves in the inset. Panel (b) shows an optimized data collapse for cube shaped logarithmic scale with a splaying of the curves forτ ˜ < 0.097. Panel (c) is a data collapse plot with γ × IPR plotted with protoclasts with ε = 0.76(5). Ξ respect to γ (˜τ − τ˜c) where Ξ = 0.30(5). ing at a common point as shown in the inset of panel (a) of Fig. 9] is compatible with the original facet survival late critical indices. Although in general one would plot β ε probability being a viable order parameter (i.e. non-zero γ fsur with respect to γ (˜τ τ˜c), the crossing of the fsur − only when vestiges of the original protoclast faces are still curves for different γ values suggests that fsur is of zero present) in the context of a second order phase transition scaling dimension, as is true for the Binder Cumulant [12] at a criticalτ ˜ value,τ ˜c. used in connection with thermally driven magnetic phase Singularities in observables or their derivatives in the transitions, for which β = 0. Accordingly, fsur is on the vicinity ofτ ˜ are hallmarks of a second order phase tran- ordinate axis in panel (b), with the optimal data collapse sition, which we use to show that salient variables are occurring for τc =0.0968(2) and ε =0.76(5). influenced by critical behavior, and to calculate indices The Inverse Participation Ratio (IPR), a measure of of interest such asτ ˜c. In the context of our simulations, the smoothness of stones, is shown in panel (a) of Fig. 10 the thermodynamic limit corresponds to γ 1, where on a semi-logarithmic scale for cube shaped protoclasts. ≫ the number of facets also is large, (e.g. as high as 3,600 The regular spacing on the logarithmic scale of the for γ = 2000). asymptotically flat IPR curves for large enoughτ ˜ is com- As is customary in the analysis of second order phase patible with the Participation Ratio scaling as γ−1 at transition, we elucidate critical behavior in the structural steady state, also evident in panel (b) of Fig. 10 where phase transition of eroding cubes by appealing to single γ IPR is plotted with respect toτ ˜. Forτ ˜ > τ˜c, the parameter finite size scaling theory, where one uses the curves× for various γ values merge onto a single flat line, data collapse phenomenon as a quantitative tool to calcu- splaying outward forτ ˜ < τ˜c, suggesting singular behav- 10

0.0 2.0 (a)

1.5 1.85 (1 − min γ = 250 rmax ); γ = 2000 -1.0 γ = 1000 (1 − r min ); γ = 500 max γ = 500 (1 − min γ = 1000 rmax ); γ = 250 /γ (1 − min γ = 2000 1.0 r ); /γ max n 1.80 (1 − φ γ = 250 n -2.0 3); (1 − φ γ = 500 3); 0.5 γ = 2000 (1 − φ ); γ = 1000 γ = 1000 3 1.75 (Sphericity Measure Complement) γ = 500 (1 − φ γ = 2000 0.085 0.090 0.095 0.100 0.105 0.110 10 3); γ = 250 N /γ 2 -3.0 sust Log 0.00 0.05 0.10 0.15 0.20 0.0 /γ 2 0.00 0.10 0.20 0.30 0.40 Nsust /γ 2 Nsust

] (b) ) 3 2.0 γ = 250 FIG. 12: (Color online) Normalized mean facet numbers, plot- 2 γ = 500 ted versusτ ˜ = Nsust/γ are shown for cube shaped protoclasts (broken lines) and irregular protoclasts (solid lines). The in-

γ(1 − φ γ = 1000 [ 1.0 set shows a magnified view of the gray rectangular region with 10 γ = 2000 symbols indicating Monte Carlo simulation data points. Log 0.0 -0.8 -0.2 0.4 in which primordial facets disappear. As in the case of γσ1 ∼ ∼ (τ − τc) the IPR, both complements are asymptotically flat and

] evenly spaced on the logarithmic ordinate scale, and both ) (c) measures exhibit singular behavior indicating a struc-

min 2.0 max tural phase transition which does not coincide with the r structural transformation in which faces original to the γ = 250 parent cube vanish, but occurs at a later time indicated

(1 − γ = 500 ζ 1.0 by the vertical blue line in panel (a) of Fig. 11. γ

[ γ = 1000 In a spirit similar to the case of the IPR, panel (b) 10 γ = 2000 and panel (c) of Fig. 11 show the collapse onto universal min

Log 0.0 curves of of 1 φ3 and 1 rmax respectively. In the -3.0 -2.0 -1.0 0.0 1.0 − − σ ∼ ∼ case of the former, γ(1 φ3) is plotted with respect to γ 2 ) σ1 − (τ − τc γ (˜τ τ˜c) where σ1 = 0.23(2) andτ ˜c = .137(5). On the other− hand, for the latter we achieve a collapse− of the min FIG. 11: (Color online) Sphericity complement measures are rmax complement measure onto a universal scaling curve shown for various γ values in panel (a) in the case of cube by plotting γζ(1 rmin ) with respect to γσ2 (˜τ τ˜ ) where − max − c shaped protoclasts, with continuous and broken curves repre- the collapse is optimal for ζ =0.81(5) and σ2 =0.36(5). min senting 1 − rmax and 1 − φ3 respectively. Vertical gray and In this manner, we see that the stones become smooth blue lines indicate structural phase transitions involving the even as their primordial facets vanish. removal of primordial facets for the former and a reversion of In Fig. 12, normalized mean facet numbers ( n /γ) are stones to spherical shapes for the latter. Panel (b) and Panel h i min juxtaposed for cube shaped and regular protoclasts (bro- (c) display collapses of 1 − φ3 and 1 − rmax onto universal curves. ken and solid lines respectively), and are well converged with respect to γ in both cases. In addition, n /γ in- creases monotonically with increasingτ ˜ towardh thei same ior forτ ˜ = 0.0967. In panel (c) of Fig. 10, the quan- asymptotic value of 1.82, though the characteristic time tity γ IPR is plotted with respect to γΞ(˜τ τ˜ ) where constant in the case of irregular protoclasts exceeds that × − c Ξ=0.30(5) [andτ ˜ = 0.097(1) as in the case of fsur] for of the cubic counterparts. On a qualitative level, while various γ values, with the overlap of the curves indicating the convergence to a normalized facet number of 1.82 is a collapse of the entire gamut of the IPR onto a universal gradual for initially irregular fragments, the mean facet curve. number curves for cube shapes protoclasts saturate for a min In Fig. 11 the complements 1 φ3 and 1 rmax are finiteτ ˜ value, quickly becoming level thereafter. The in- shown for the case of cube shaped− protoclasts,− graphed set is a magnified view illustrating the increasing abrupt- on a semilogarithmic scale in panel (a); the gray line in ness of change in slope nearτ ˜c with increasing γ, singular the plot corresponds to the structural phase transition behavior signaling second order phase transition. 11

0.0 -2.0 (a) (b) (c) (a) (b) -0.1 -3.0 τ ∼ d

) ∼ ∼ ∼ )/ τ = 0.0 τ = 0.0124 τ = 0.0248 v ~ -0.2 -4.0 v ~ ( ( (d) (e) (f) 10 10 -0.3 -5.0 Log Log

γ = 250 d γ = 250 γ = 500 γ = 500 -6.0 ∼ ∼ ∼ -0.4 γ = 1000 γ = 1000 τ = 0.0571 τ = 0.0893 τ = 0.1134 γ = 2000 γ = 2000 (g) (h) (i) -0.5 -7.0 0.00 0.05 0.10 0.15 0.20 0.00 0.05 0.10∼ 0.15 0.20 ∼τ τ 0.0 -2.0 (c) (d) ∼τ = 0.1375 ∼τ = 0.206 ∼τ = 0.275

-3.0 τ ∼

-0.5 d FIG. 14: (Color online) Sequences of structures for an irreg- ) )/ v

~ -4.0 v ~

( ular protoclast where γ = 2000. Images withτ ˜ values in red ( 10 10 correspond to transitions involving the loss of one or more -5.0 Log

-1.0 Log primordial facets. Red areas the latter or portions of the lat-

γ = 250 d γ = 250 γ = 500 ter. -6.0 γ = 500 γ = 1000 γ = 1000 γ = 2000 γ = 2000 -1.5 -7.0 0.0 0.1 0.2 0.3 0.4 0.5 0.00 0.10 0.20∼ 0.30 0.40 0.50 ∼τ τ in the case of irregular protoclasts. The latter, in which the slope abruptly becomes constant, signals pure expo- FIG. 13: (Color online) Semilogarithmic plots of volume frac- nential decay of a quasi-spherical shape and also coincides tion remainingv ˜ with respect toτ ˜ as well as the derivatives with the facet elimination phase transition indicated by of the latter are show in panel (a) and panel (b) respectively the vertical gray line in panel (b) of Fig. 13. for cube shaped protoclasts; the vertical gray line indicates A sharply defined structural phase transition in the the structural transition for which primordial facets vanish. case of regular protoclasts such as cubes, even for mono- Similarly, panel (c) and panel (d) showv ˜ on a semilogarith- dispersed solids, is the exception rather than the rule, as mic scale as well as the slope of log10 v˜ versus for cohorts of irregular protoclasts. may be seen in the case of cohorts of identical irregular shapes. In this vein, we consider an irregular six sided protoclasts for which the elimination of primordial facets is not simultaneous. Instead, the facets of the original In Fig. 13, volume fraction remainingv ˜ also reflects protoclast vanish at distinct times τ. The parent solid singular behavior at the structural transition forτ ˜c = is shown in the upper left corner of Fig. 14, illustrat- 0.0968(2). The main graphs in panel (a) and panel (b) ing the erosion trajectory with intermediate stages (with of Fig. 13 showv ˜ on a semilogarithmic scale, while the τ˜ values in black) interspersed with images correspond- inset plots display the sloe of the log10(˜v) curves with ing to transitions in which one or more primordial facets 2 respect toτ ˜ = Nsust/γ . In general, as discussed previ- are removed (withτ ˜ values in red). Crimson regions are ously, one anticipates the small decrease of volume frac- facets original to the parent solid, with the expanding 2 tion per fracture event to be dv˜ = ∆A /l with l a length gold areas being material exposed by stochastic fracture 1/3 − scale on the order ofv ˜ . In the regime that stones are events. worn down to quasi-spherical shapes, one would expect In spite of the absence of a single structural phase tran- the volume fraction decrement for a fracture event to be sition, the loss of individual primordial facets is accom- ∆˜v = ∆A2/(12πR), exact for the case of a truncated − panied by singular behavior in salient variables encoun- sphere in the limit that r/R (i.e. with r the radius of the tered in the vicinity of a second order phase transition. new circular facet and R the sphere radius) tends to zero. The ensemble averaged facet survival probability shown 2 Taking ∆A to be on the order of 4πR /γ, we see that in Fig. 15 for various γ values ranging from γ = 250 to the typical volume sheared away per normalized time in- γ = 2000 exhibits abrupt transitions signaling the loss −2 crement dτ˜ = γ ∆Nsust is dv˜ vd˜ τ˜, which when of individual facets of the original protoclast, with com- ∼ − −Γ˜τ integrated leads to an exponential decay inτ ˜,v ˜ = e mon crossings of the curves for each transition. While where Γ is a dimensionless constant on the order of unity. some of the latter involve the loss of a single facet, in For both cube shaped and irregular protoclasts, the some cases multiple surfaces from the original protoclast mean remaining volume fraction curves appear to become vanish at the same time. The first and third transitions asymptotically linear, behavior highlighted in panel (b) [forτ ˜ = 0.0248(1) andτ ˜ = 0.1375(5)] are of the former and panel (d) for cubic and irregular parent solids respec- variety, while the second and fourth steps downward [for tively. Common to both cases, slopes of log10 v˜ level out τ˜ =0.0893(1) andτ ˜ =0.275(1)] involve the simultaneous at a common value, which is well converged with respect disappearance of two primordial facets. The inset graphs to the toughness parameter γ in both instances. A salient are data collapse plots corresponding to the four facet ε distinction is a discontinuity in the slope of the volume transitions, with the horizontal scale being γ (˜τ τ˜c) as fraction curves for cube shaped protoclasts not replicated in the case of panel (b) of Fig. 9. However, while the− data 12

1.00 2.0 1.00 γ = 250 γ = 500 0.90 0.95 γ = 1000 0.90 γ = 2000 0.80 /γ Survival Probability 0.85 1.0 n -1.0 -0.5 0.0 0.5 1.0 γ ε (τ~ − ~τ 0.70 c) 0.8 (a) 0.60 0.7 0.0 0.0 0.1∼ 0.2 0.3 0.50 0.6 τ

Survival Probability 0.5 -1.0 0.0 1.0 -2.0 0.5 γε ~ ~ 0.40 (τ − τc) ∼ 0.5 τ d

Survival Probability 0.30

0.4 )/ -4.0 v ~

0.20 ( 0.4 10 Survival Probability -2.0 0.0 2.0 γε(τ~ − ~τ -6.0 0.10 c) Survival Probability Log ε ~ ~

γ d (τ − τc) 0.00 (b) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 -8.0 N /γ 2 0.0 0.1 0.2 0.3 sust τ∼ FIG. 15: (Color online) Survival probability for the irregular FIG. 16: (Color online) Normalized facet numbers for mono- solid in Fig. 14. Inset graphs are data collapses at structural dispersed irregular protoclasts and the slope of the log-log transitions in which one or more facets original to the parent volume curves are depicted in panel (a) and panel (b) respec- solid are cleaved away. tively. collapses are optimized for ε =0.76(5) in the case of the 0.0 1 − min (a) first three facet elimination transitions, we find a depar- rmax ture for the transition involving the removal of the last two facets of the parent solid where instead ε =0.45(5). Signatures of the asynchronous facet removal transi- -1.0 1 − φ tions are evident in variables other than fsur, such as the 3 mean facet number n and the slope the logarithm of γ = 250 the mean remaining volumeh i fractionv ˜ displayed in panel -2.0 γ = 500 (a) and panel (b) of Fig. 16 for various γ values. In both γ = 1000

cases, shift in slopes of the curves are subtler than those (Sphericity Complement) γ = 2000 marking the disappearance of facets original to the parent 10 solid for cubic protoclasts, with slope changes at phase -3.0 boundaries most pronounced following the fourth tran- Log 0.00 0.10∼ 0.20 0.30 sition in which the final two primordial faces are worn τ 3.0 2.6 away. γ = 250 ] (b) ) 2.4 (c)

min ] 2.5 γ = 500 Spherical deviation measures 1 φ3 and 1 r are ) max min max − − 3 γ = 1000 2.2 shown in the main graph of Fig. 17 on a semilogarith- 2.0 r γ = 2000 2.0 mic scale. In both cases, the curves plotted for differ- 1.5 1.8 (1 − ζ ent γ values begin to diverge, in a manner qualitatively γ(1 − φ γ = 250 [ γ 1.6

1.0 [ similar to the case of cube shaped protoclasts, though 10 γ = 500

10 1.4 γ = 1000 asymptotically flat regions do not appear on the domain 0.5 Log 1.2 γ = 2000 of time scalesτ ˜ accessed in the simulation. Data collapse Log 0.0 1.0 plots onto universal scaling curves for the sphericity and 0.5 1.0 1.5 2.0 0.0 2.0 4.0 6.0 8.0 σ ∼ σ2 ∼ ∼ min γ 1(τ∼ − τ ) γ (τ − τ ) rmax complements are displayed in panels (b) and (c) of c c Fig. 17, respectively. In the case of the former, γ(1 φ3) is plotted with respect to γσ1 (˜τ τ˜ ) where σ =0.17(1)− and FIG. 17: (Color online) Sphericity complement measures are − c 1 τ˜c = 0.57(5). On the other hand, for the latter we plot show for various γ values in the case of mono-dispersed ir- ζ min σ2 regular protoclasts, with continuous and broken curves repre- γ (1 rmax) with respect to γ (˜τ τ˜c) where ζ =0.81, min − − senting 1 − rmax and 1 − φsph respectively in panel (a). Panel σ2 =0.36(5), andτ ˜c =0.52(5), compatible with the tran- (b) and panel (c) show collapses of complements 1 − phi3 and sition time obtained for the φ3 complement. As for cube min 1 − rmax respectively onto universal scaling curves. shaped parent solids, the transition timesτ ˜c occur later 13 0.0 min (a) 1 − r max

γ = 500 γ = 250 -1.0 γ = 1000 1 − φ 3 γ = 250 γ = 2000

-2.0 γ = 500 0% volume removed 10% volume removed γ = 2000 (Sphericity Measure Complement)

10 γ = 1000

Log -3.0 0.0 0.2 0.4 2 Nsust /γ 1.0 -0.5 (b) γ = 250 (c) γ = 500 -1.0 0.8 γ = 1000 γ = 2000 -1.5 0.6

(IPR) -2.0 γ = 250 10 0.4 γ = 500 Log -2.5 25% volume removed 50% volume removed γ = 1000

Survival Probability γ = 2000 0.2 -3.0

0.0 -3.5 0.0 0.2 0.4 0.6 0.0 0.1 0.2 0.3 0.4 0.5 /γ 2 2 Nsust Nsust /γ

FIG. 19: (Color online) Sphericity complements 1 − φ3 and min 1 − rmax are shown in panel (a) for irregular protoclast co- horts, while fsur and the IPR (on a semi-logarithmic scale) are displayed in panel (b) and panel (c) respectively.

75% volume removed 90% volume removed ratio are shown in panels (a), (b), and (c) of Fig. 19 re- spectively for irregular protoclasts. In the case of the FIG. 18: (Color online) Twenty five irregular protoclasts at measures of deviation from a spherical shape in panel various erosion stages (a), though the curves diverge, the separation occurs at later and later times with increasing γ, suggesting con- vergence with respect to the toughness parameter. The than for any of the structural transitions in which one survival probability shown in panel (b) of Fig. 19 evi- or more primordial facets are removed for the irregular dently converges rapidly with increasing γ when plotted 2 mono-dispersed cohort. with respect to Nsust/γ with little difference among the curves for all γ values shown. In contrast to the case of cube shaped parent stones, IPR curves for irregularly 2. Structural Evolution of Poly-dispersed Cohorts shaped protoclasts converge with increasing γ, and there is no finite value ofτ ˜ where the participation ratio de- Sample stones at various stages of their erosion trajec- cays to zero in the large γ limit. This characteristic is tories are shown in Fig. 11, with the irregular protoclasts compatible with primordial facet removal events being in the upper left panel. As in the case of cube shaped spread over the entireτ ˜ domain due to the strong struc- protoclasts, individual stones are found to shed primor- tural disorder in the cohort of poly-dispersed irregular dial facets, but in an asynchronous manner due to the protoclasts. structural disorder inherent in the irregular protoclasts, Oblateness and prolateness measures ψP and ψO, as eliminating the possibility of a well defined structural described in Section II, are displayed in Fig. 20 for a phase transition. Thus, variables exhibit no singular be- range of γ values. The prolateness measure initially ex- havior, converging for γ 1 as in the survival indices ceeds the oblateness measure, but eventually decreases ≫ plotted in Fig. 14, which overlap closely for γ ranging sharply and falls below ψO with a continued monotonic from γ = 250 to γ = 2000. decrease thereafter. A study of strongly disordered frag- min Results for the complements 1 φ3 and 1 fmax, the ments [13] generated stochastically found a prevalence of facet survival probability, and the− Inverse Participation− prolate fragments in cohorts generated with a variety of 14

0.5

Oblateness Prolateness γ = 250 γ = 250 0.4 γ = 500 γ = 500 γ = 1000 γ = 1000 γ = 2000 γ = 2000 0.3

0.2

Oblateness Measure 0.1 Ellipticity Measure

0.0 0.0 0.5 1.0 1.5 6 2 Nsust /10 Nsust /γ

FIG. 21: (Color online) Oblateness measure ψO for an individ- FIG. 20: (Color online) Ellipticity measures plotted versusτ ˜ ual case showing non-monotonic behavior in the oblateness. for various γ values, with solid and broken lines representing The solid curve is the oblateness measure plotted with re- oblateness and prolateness respectively. Gray regions about spect to time, and rectangular gray boxes enclose images of γ = 2000 curves indicate Monte Carlo statistical error. the stones at specific stages. The lower image in each case is an edge-on view of the same stone.

IV. NON STEADY STATE EROSION SCENARIOS techniques. On the other hand, the disorder averaged Although we discuss a variety of non-steady state oblateness measure actually increases initially, peaking schemes, choose the constant velocity scenario (e.g. for when approximately half of the stone’s original volume stones borne along at constant speed in a stream or river) has been chipped away. The variance in the ψO curves, for direct Monte Carlo simulations. With kinetic energy while greater than for the ψ traces, is comparable to the 1 2 P scaling as 2 mv , and m = ρV in the case of stones of uni- Monte Carlo statistical error shade in gray in the case of form composition, typical energy would be proportional the γ = 2000 curve, and thus both the oblateness and to the volume. That the α = 1 exponent for the fixed ve- prolateness measures may be considered to be converged locity scenario exceeds the critical αc =2/3 for the rela- with respect to the toughness parameter γ. That ψO tive area scheme implies an ever diminishing relative area peaks even as ψP continues to decrease could predispose with time for typical sustained slices and a concomitant stones to long term oblate shapes, though much of the rise in the mean number of facets. Nevertheless, although enhancement and maintenance of an oblate profile, not the relative area and fixed velocity scenarios are distinct, addressed in this study, may be due to specific nature of the universal dependence of stone profiles on fractional fluvial motion along a stream or river bed [14]. remaining volume [3] dictates that key structural mile- stones occur at finite values of the reduced timeτ ˜, borne out in the case of the fixed velocity scheme. To gain an understanding on an intuitive level as Fixed velocity erosion trajectories are calculated for to why the time dependent oblateness measure is non- irregular protoclasts with 2000 distinct realizations of monotonic in some cases, we exhibit ψO as well as images disorder.′ A′ range of toughness parameter values from of the stone at various stages in the erosion trajectory. γ = 20to γ = 160 are considered with the mean number The protoclast for the example shown in Fig. 21 begins of facets n exceeding 4000 for the latter. Representa- as a blade-like shape, narrow but also thin. As fractures tive examplesh i of relevant variables are shown in Fig. 22 min begin to carve away volume, the stone becomes thicker with the sphericity complements 1 φ3 and 1 rmax (as may be seen in the edge-on views), but appears to and the IPR shown in panel (a) and (b)− while oblateness− lose material from the extremes along the longer axis at and prolateness measures are shown in panel (c) with an even greater rate. In combination, these trends con- the primordial facet survival probability fsur plotted in tribute to an initial rise in oblateness (with the shape panel (d). As in the case of the time dependent relative remaining relatively thin while becoming less oblong), area results, fsur and′ ellipticity measures are converged ultimately peaking an declining as the stone ultimately with respect to the γ parameter with minor discrepan- is rounded into a spherical shape. cies for the oblateness measure near its maximum likely 15

0.0 2.0 -0.5 γ = 20 γ = 80 (a) 1 − φ -1.0 γ = 40 γ = 160 3 1.5 -1.0 -1.5 eff /γ 1 − 1.0 (IPR) -2.0 10 r min n max -2.5 -2.0 Log 0.5 -3.0 (a) (b) 0.0 -3.5 Sphericity Measure Complement -3.0 0 500 1000 1500 2000 0 500 1000 1500 2000 0 10 20 30 40 50 γ γ /γ 2 eff eff Nsust -0.5

-1.0 γ = 20 γ = 80 (b) ) (c)

3 -1.0 γ = 40 γ = 160

(IPR) -2.0 -1.5 10 (1 − φ -2.0 10 Log -3.0 -2.5 0 10 20 30 40 50 Log 2 Nsust /γ -3.0 0.6 0 500 1000γ 1500 2000 γ = 20 γ = 80 (c) eff γ = 40 γ = 160 0.0

0.4 ) (d)

min -0.5 max r 0.2 -1.0 (1 − Ellipticity Measure 0.0 10 -1.5 0 10 20 30 40 50 2 Log Nsust /γ -2.0 1.0 0 500 1000γ 1500 2000 γ = 20 γ = 80 (d) eff γ = 40 γ = 160 0.5 FIG. 23: (color online) Effective Gamma plots in the fixed velocity scenario are displayed for the mean facet number hni 0.0 and IPR in panel (a) and panel (b), and spherical comple-

Survival Probability min 0 10 20 30 40 50 ments 1 − φ3 and 1 − rmax in panel (c) and panel (d) respec- 2 Nsust /γ tively. Symbols represent equilibrated steady state systems, while broken and solid lines are simulation data in the fixed FIG. 22: (Color online) Observables plotted with respect to velocity scenario. ′ 2 min Nsust/(γ ) for 1 − rmax and 1 − φ3 in panel (a), the IPR in panel (b), the oblateness and prolateness in panel (c), and the primordial facet survival probability in panel (d). the characteristic area exposed by a fracture event ob- tained by setting the exponent in the prospective slice acceptance probability to 1 and solving for the area. In due to Monte Carlo statistical error. Quantities such as the case of the fixed′ velocity erosion scheme, one has − 1 1 −1 measures of deviation from a spherical shape and the γeff =v ˜ 3 (6π 2 ) γ , which increases with decreasing IPR do diverge on the semilogarithmic scale, but at later remaining volume fraction. For representative variables times with increasing γ suggesting convergence with re- of interest in Fig. 23, we show on the same plots equi- spect to the latter. In a qualitative sense, in compari- librated observables from steady state calculations and son to results in the case of the relative area scenario, fixed velocity simulation results (solid or broken curves) fixed velocity variables evolve more rapidly initially with plotted with respect to γeff . The mean number of sites structural change slowing as sustained slices accumulate. normalized with respect to γeff , shown in panel (a), ulti- This trend is compatible with the typical area of freshly mately saturates at the large γ equilibrium value of 1.82. exposed facets decreasing relative to the total area with min The IPR and complements 1 φ3 and 1 rmax obtained decreasing remaining volume fractionv ˜. from fixed velocity scheme simulations− − decrease mono- An additional merit of discussing the steady state sce- tonically with time in panels (b), panel (c), and panel nario in the equilibrated regime is its relevance to non- (d) of Fig. 23, converging with steady state equilibrium steady state scenarios after a sufficient number of frac- results (open or filled circles). The merging of the fixed tures have accumulated. We define an instantaneous velocity simulation and equilibrium results suggests that or “effective” γ with γeff = AΣ/Achar where Achar is even in non-steady state situations, stones reach a condi- 16

0.8 mean facet survival probability and the Inverse Partici- min (1 −r max ) Rel. Area pation Ratio also show very close agreement among the min 0.6 (1 −r max ) Fixed v (a) relative area and constant velocity results when plotted versus the volume fraction remaining. Finally, measures 0.4 of ellipticity are displayed in panel (d) of Fig. 24, with (1 −φ 3 )Rel. Area (1 −φ 3 )Fixed v close agreement for both the prolateness and oblateness 0.2 measures. This quantitative agreement of result for dis- tinct scenarios when plotted with respect to the remain- 0.0 ing volume fraction has been mentioned previously [3]. Sphericity Measure Complement 0.0 0.2 0.4~ 0.6 0.8 1.0 v The relative area scenario (i.e. α = αc = 2/3) con- 1.0 0.20 Rel. Area Rel. Area sidered in the preceding section provides an avenue for (b) Fixed v (c) 0.8 Fixed v 0.15 predicting the time evolution of salient observables in a 0.6 broad set of cases in which the number of facets does not 0.10

IPR ultimately saturate at a finite value, but continues to rise 0.4 monotonically. The universal dependence of variables on 0.05 Survival Probability 0.2 the mean volume fractionv ˜ for steady state and non- 0.0 0.00 steady state scenarios (e.g. the constant velocity scheme) 0.0 0.5 1.0 0.0 0.5 1.0 ~v ~v would permit a prediction of the time dependence of vari- ables in a wide variety of non-steady state scenarios if one Oblateness (d) knew the relationship of Nsust andv ˜ for the scheme un- Rel. Area der consideration. We argue here and validate with large Fixed v scale simulations involving irregular protoclasts that this Prolateness mapping ofv ˜ onto Nsust in greater generality may be Rel. Area determined from steady state results as long as γ 1. Fixed v ≫ With the characteristic mean facet area Achar being −1 2/3 2 1/3 on the order of Achar = κγ v˜ , where κ (36π ) , the volume cleaved away by a sustained slice≡ scales as 2 Ellipticity Measures Achar/r˜, wherer ˜ is as discussed earlier a length scale related to the local radii of curvature, which we take to be on the order ofv ˜1/3, the cube root of the volume ~v fraction. We obtain an equality with a volume dependent function f(˜v), yielding for the mean volume decrement 2 −2 FIG. 24: (Color online) Variables for relative area and fixed ∆˜v = κ γ f(˜v)˜v∆Nsust where ∆Nsust is a series of velocity scenarios plotted with respect to volume fraction v˜. hsustainedi − slices small in the sense that ∆˜v v˜. In h i ≪ the large γ limit, one may replace ∆˜v and ∆Nsust with corresponding differential quantities,h andi in this regime tion of quasi-equilibrium, in the sense that variables cor- one has respond closely with the steady-state counterparts calcu- γ2 dv˜ lated for γeff after enough time has elapsed. f(˜v)= (7) −κ2v˜ dN  sust steady state which may be extracted numerically from data obtained A. General Scenario Variables from Relative Area in the context of the relative area scenario for γ 1 (i.e. Results for γ = 2000 in this work). ≫ The characteristic function f(˜v) in Eq.7 in principle Equivalent milestones occurring at the same volume allows one access of Nsust(˜v) for a broad range of erosion fractions remaining independent of the model (e.g. the scenarios distinct from the steady state case by exploiting oblateness maxima near the halfway point in the erosion the fact that dv˜ = f(˜v)˜v−1/3A (˜v)2dN Solving − char sust of volume) is an important aspect of the deterministic for dNsust and integrating yields time evolution of structures in spite of the stochastic nature of the collisional erosion process; in more suc- 1 dv˜ N (˜v)= v˜1/3 (8) cinct terms, graphs of relevant observables should coin- sust 2 v˜ Acharf(˜v) cide when plotted versusv ˜ even for distinct models, a Z phenomenon which we see in Figure 24 for sample observ- Characteristic functions for cubic and irregular proto- ables of interest. In panel (a), global measures of depar- clasts are shown in Fig. 25, with solid lines representing ture from a perfect spherical shape are plotted with re- the former and broken lines the latter. The inset graph spect tov ˜, with the solid curve gleaned from relative area is a magnified view of the region indicated by the pink calculations and the broken trace corresponding to the box in the main graph for cubic protoclasts. Common to fixed velocity scenario. Panel (b) and (c), depicting the both cases is the decline from an elevated value forv ˜ near 17

0.24 0.8 γ = 250 γ = 500 (a) γ = 1000 0.6 γ = 2000 0.23 0.6 v ~ () f 0.4 0.22 1 − r min max 0.4 1 − φ Calculated Data 3 0.2 Calculated Data Simulation Data 0.21 Simulation Data v ~ 0.45 0.50 0.55 0.60 0.65 Sphericity Measure () ~ f v 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.2 cubic protoclasts irregular protoclasts 6 Nsust/10 γ = 250 γ = 250 γ = 500 γ = 500 0.2 γ = 1000 γ = 1000 Calculated Data (b) γ = 2000 γ = 2000 Simulation Data 0.0 0.1 0.0 0.5 1.0 IPR ~ v 0.0 0.0 0.2 0.4 0.6 0.8 1.0 FIG. 25: (Color online) Characteristic function f(˜v) for cube 6 Nsust /10 shaped protoclasts (broken lines) and irregular protoclasts (solid lines). the inset is a magnified view of the regions boxed (c) in pink for cubic protoclasts. Oblateness Prolateness Rel. Area Rel. Area Thermal Thermal unity at the beginning of the erosion process, with f(˜v) leveling out at a finite value forv ˜ 1 (i.e. for later times after a large portion of the original≪ volume has eroded). This asymptotic value, indicated in the main graph and Ellipticity Measure inset plots with a solid gray line, is 0.216(1) for both cube 6 shaped and irregular protoclasts. The elevated portion Nsust /10 1.0 of f(˜v) for small times is compatible with an initially ac- (d) celerated loss of volume as corners and edges give way to Calculated Data Simulation Data regions of high local curvature which shed volume com- ~ v 0.5 paratively rapidly. Although f(˜v) for the case of irregular and cube shaped 0.0 counterparts appears to converge to a common asymp- 0.0 0.2 0.4 0.6 0.8 1.0 6 totic value, the characteristic function for the case of cu- Nsust /10 bic protoclasts is distinct in abruptly reaching 0.216 for 1.0 a finite volume fractionv ˜ =0.5732, where the second or- Calculated Data (e) Simulation Data der phase transition to shapes lacking primordial facets 0.5 occurs. The singular behavior is highlighted in the inset, which shows a magnified view of f(˜v) for cube shaped 0.0 protoclasts with a slope discontinuity signaling the at- Survival Probability 0.0 0.2 0.4 0.6 0.8 1.0 tainment of the long time value, and coinciding with the 6 Nsust /10 loss of primordial facets transitions indicated with the solid vertical line. FIG. 26: (Color online) Primordial facet survival probabilities The flattening of f(˜v) following an initial rapid de- are show in panel (a) and panel (b) respectively; non-steady ′ crease provides an avenue for obtaining a closed form state simulation data (broken lines) for γ = 160 and calcu- expression for time scales for the attainment of specific lated curves based on γ = 2000 steady state results (solid line) structural milestones at particular volume fractionsv ˜; are plotted versus Nsust with corresponding semilogarithmic given the structure of f(˜v), the relationship we obtain graphs in the insets of panel (a) and panel (b). serves as an approximation as well as a rigorous upper bound for the time needed to wear stones down to a par- ticular volume fraction. V. SALIENT TIME SCALES Curves predicted based on relative area results (solid lines) and results from direct Monte Carlo calculations Due to the universal dependence of relevant variables (broken traces) appear in panel (a) and panel (b) of on volume fractionv ˜, to find time scales for a salient event Fig. 25, respectively with log-log graphs in the inset. such as a structural phase transition, one need only de- Broadly there is excellent agreement among the predicted termine the volume fraction (e.g. in the context of the and directly calculated variables. relative area scenario) for the case of interest and calcu- 18

3.0 (a) tionsv ˜ = 0.57 andv ˜ = 0.37, respectively. On the other hand, in panel (b) of Fig. 27, for cohorts of irregular ~ v = 0.10 (upper bound) protoclasts, the three time scales shown correspond to ~v = 0.10 (calculated) ~ v˜ = 0.50,v ˜ = 0.25, andv ˜ = 0.10. In spite of the mono- v = 0.25 (upper bound) tonic rise inτ ˜ for the attainment of various volume frac- 2.0 ~v = 0.25 (calculated) 2 ~v = 0.50 (upper bound) tion milestones, panel (a) and panel (b) of Fig. 24 do

/γ ~v = 0.50 (calculated) not diverge for a finite value of α. Moreover, the upper bound in Eq. (i.e. the same upper bound for the cases sust

N of cubic and irregular protoclasts), albeit increasing with 1.0 increasing α, nevertheless remains finite for finite α.

0.0 VI. CONCLUSIONS 0.8 1.0 1.2 α In conclusion, with large scale Monte Carlo simula-

(b) ~v = 0.327 (upper bound) tions, we have examined the erosion of rocks through 0.5 stochastic chipping, considering polyhedral stones with 0.4 as many as 3,600 facets and averaging over at least 1000 realizations of disorder. Using an energy based criterion ~ 2 v = 0.327 (calculated) for accepting a randomly chosen slicing plane, our cal-

/γ 0.3 culation is unique in placing no restrictions on the num- sust

N 0.2 ~v = 0.551 (upper bound) ber of vertices and faces sheared away with each fracture event. We have argued on theoretical grounds and veri- 0.1 fied by direct simulation that time scales for the removal ~v = 0.551 (calculated) of a specific amount of volume or the attainment of a 0.0 particular structural milestone scale quadratically in the 0.8 1.0 1.2 α toughness parameter γ which specifies the amount of en- ergy per area associated with a fracture; the largest time FIG. 27: (Color online) Time scales for selected volume frac- scales examined in our calculations exceed a million sus- tionsv ˜ for cubic and irregular protoclasts in panel (a) and tained slices per erosion sequence. panel (b) respectively. Solid lines are calculated from di- Cohorts of stochastically chipped protoclasts in the rect Monte Carlo simulation results, while broken lines are form of regular polyhedra undergo a structural phase closed form results bounding time scales from above. Open transition in which all primordial facets are abruptly lost and filled symbols represent direct simulation results for the with concomitant singularities in other relevant observ- relative area and fixed velocity scenarios, respectively. ables, marking a genuine second order phase transition; in a subsequent structural transformation evident in mea- sures of sphericity, the stones revert to spherical shapes. late the elapsed time in terms of sustained slices using More broadly, however, ensemble averaged observables in Equation 8. Since f(˜v) decreases monotonically, ulti- the case of initially identical irregular shapes are punctu- mately converting to the asymptotic value f 0.216, 0 ated by an elimination of facets in multiple stages instead one has f(˜v) f and thus time scales gleaned≡ from 0 of a single event. On the other hand, cohorts of irregu- Equation 8 are≤ approximately given and bounded from 1 1 lar protoclasts exhibit no structural phase transitions as above by f A2 v˜ 3 dv˜, often amenable to exact cal- 0 v˜ char an aggregate, with individual eliminations of primordial culation. For an power law scaling of the kinetic energy surfaces being blurred by structural disorder. where A =R (κ/γ)˜vα, we obtain char We find direct measures of deviation from a spherical 3 − profile (i.e. the sphericity φ3 and ratio of minimum and 4/3 2α min τ˜ = 2 [˜v 1] (9) maximum center of mass distances r ) decrease mono- κ f0(6α 4) − max − tonically with accumulating sustained slices. However, Fig. 27 shows salient reduced time scalesτ ˜ for a range the oblateness measure actually rises at the initial stages of values of the exponent α; solid curves are obtained of the erosion trajectory, reaching a peak when half of from Monte Carlo simulation results in the case of the the original volume has been chipped away and then de- relative area scenario, while broken lines are an upper creasing. This non-monotonic behavior of the oblateness bound for time scales from Equation 9. In panel (a) of is attributed to initially blade-like protoclasts which lose Fig. 27, pertaining to cube shaped protoclasts, the two material more rapidly from the ends than from the edges, time scales shown include the structural phase transi- briefly becoming more oblate in shape before eventually tion involving the loss of primordial facets as well as the being rounded into a spherical profile. attainment of spherical shapes (corresponding to equi- That data from distinct erosion scenarios (i.e. relative libration in the relative area scenario) for volume frac- area and fixed velocity schemes) collapse onto common 19 curves when plotted with respect to the remaining vol- form approximate expressions which bound these time ume fraction confirms the universal dependence of ob- scales from above, and which are in good quantitative servables on the latter [3]. This phenomenon provides an agreement with the latter. avenue for using results in the relative area case to cal- culate time dependent observables for arbitrary erosion schemes, a technique we validate by reproducing results gleaned in the context of the fixed velocity scenario by Acknowledgments direct Monte Carlo calculation. In addition, one also is afforded an efficient approach for calculating character- We acknowledge helpful discussions with Michael istic times (e.g. for specific remaining volume fractions) Crescimanno. Calculations in this work have benefited for a range of erosion scenarios. We have obtained closed from use of the Ohio Supercomputer facility (OSC) [15].

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