ENAbstract. We study the origin of the scaling behavior in frictionless granular media above the jamming transition by analyzing their linear response. The response to local forcing is non-self-averaging① and fluctuates over a length scale that diverges at the jamming transition. The response to global forcing becomes increasingly nonaffine near the jamming transition. This is due to the proximity of floppy modes②, the influence of which we characterize by the local linear response. We show that the local response also governs the anomalous scaling of elastic constants and contact number.

中译摘要。作者通过分析无摩擦颗粒介质在堵塞转变之上的线性响应，研究标度行为的起源。局域外力所激发的响应不具自平均性①，其涨落跨越一个在堵塞转变处发散的长度尺度。整体外力所激发的响应在临近堵塞转变时越来越非仿射；作者将其归因于松弛模（floppy modes）的邻近效应②，并以局域线性响应刻画这一影响。研究还表明，局域响应与弹性常数及接触数的异常标度密切相关。

ENThe general picture of jamming which was advanced for systems [1–4] that form a shear-resistent solid phase at high densities is bringing a new perspective to the deformations of granular and disordered media. A good model for studying such media are packings of polydisperse weakly compressible spheres [3,4]. If we measure pressure in units of the elastic constants and characteristic radius of the balls (as we will do below), the relevant limit for granulates is the small-deformation or, equivalently, the small-pressure limit in the absence of thermal fluctuations. This limit is also relevant for weakly compressed emulsions [5]. We will focus on the case of frictionless, deformable spherical particles, and introduce a simple, experimentally accessible and local measure to characterize the nature of their deformations [6].

中译对于在高密度下形成抗剪固相的系统 [1–4]，堵塞的一般图像正在为颗粒与无序介质的形变研究带来新视角。研究这类介质的一个好模型是多分散、弱可压缩球的堆积 [3,4]。如果以颗粒的弹性常数与特征半径为单位来度量压强（下文即如此），则颗粒介质的相关极限是无热涨落条件下的小形变极限，亦即小压强极限。这一极限同样适用于弱压缩的乳液 [5]。我们将聚焦于无摩擦、可变形球形颗粒的情形，并引入一个简单、实验上可实现且局域的度量，来刻画其形变的本质 [6]。

ENDeformable particles form a stiff jammed phase when the pressure becomes larger than zero. At the zero pressure jamming point J, packings form a “marginal solid” and are isostatic, i.e., the average number of contacts per particle z reaches the minimum z_iso⁰ = 2d③, needed for a frictionless packing to remain stable in d dimensions. When the point J is approached by decreasing the pressure, several surprising scaling relations emerge: the excess contact number Δz ≡ z − z_iso⁰ scales as √δ, with δ the typical dimensionless compression of the particles, while the ratio G/K of the shear modulus G to the compression modulus K scales as Δz④. In addition, a characteristic frequency ω* ∼ Δz has been identified in the density of states of vibrational modes; equivalently, the associated time scale τ* ∼ 1/ω* diverges. The jamming point J thus exhibits features of a critical point [2–4].

中译当压强大于零时，可变形颗粒形成具有有限刚度的堵塞态。在零压强堵塞点 J，堆积形成“边际稳定固体”（marginal solid）并处于等静定状态：每颗粒平均接触数 z 达到最小值 z_iso⁰ = 2d③——这是 d 维无摩擦堆积维持稳定所需的最少接触数。随压强降低并逼近 J 点，出现若干标度关系：过剩接触数 Δz ≡ z − z_iso⁰ 按 √δ 标度（δ 为典型无量纲压缩量），而剪切模量 G 与体积压缩模量 K 之比 G/K 按 Δz 标度④。振动态密度中还存在特征频率 ω* ∼ Δz；等价地，相应时间尺度 τ* ∼ 1/ω* 随 Δz → 0 而发散。因此，堵塞点 J 呈现临界点特征 [2–4]。

ENSince packings at the jamming point are marginal, every additional broken contact generates a global zero-energy displacement mode, a so-called floppy mode⑤ [3,8–10]. Wyart and co-workers [9–11] have shown that the scaling near J is related to those floppy modes, by creating trial modes for the deformations of weakly jammed solids. These modes are based on the floppy modes that would occur when bonds along the faces of cubes of linear size ℓ were cut; the largest such scale obeys ℓ* ∼ 1/Δz⑥. Even though truly floppy modes do not occur in the stable p > 0 packings studied here, their proximity governs the scaling just above the jamming point.

中译堵塞点处的堆积处于边际稳定状态，每多断开一个接触，都会产生一个全局零能位移模，即松弛模（floppy mode）⑤ [3,8–10]。Wyart 及合作者 [9–11] 通过为弱堵塞固体构造试探形变模，表明 J 点附近的标度与这些松弛模有关。其计数构造是在边长为 ℓ 的区域表面切断接触；允许此类自由形变的最大尺度满足 ℓ* ∼ 1/Δz⑥。在本文研究的 p > 0 稳定堵塞网络中并不存在严格零能松弛模，但其邻近效应控制着堵塞点上方的标度。

ENIn this Letter we uncover that this proximity of floppy modes causes an increasingly nonaffine response when approaching point J, and that this response is intimately related to the (anomalous) scalings of the shear modulus G, the excess contact number Δz, and the length scale ℓ*. We numerically study the linear, quasistatic response of systems near the jamming transition. The response of granular media has been widely studied [12–18], but not, we believe, systematically as a function of the distance to the jamming point J. Nor does it seem to have been fully appreciated that the scaling behavior can essentially be captured within linear response⑦.

中译在本快报中，作者提出：松弛模的邻近效应使响应在逼近 J 点时越来越非仿射，并与剪切模量 G、过剩接触数 Δz 和长度尺度 ℓ* 的异常标度密切相关。本文数值研究堵塞转变附近系统的线性、准静态响应。颗粒介质响应虽已被广泛研究 [12–18]，但作者认为，此前尚未系统考察其对“距堵塞点 J 多远”的依赖；同样未被充分认识的是，这些标度行为在很大程度上可由线性响应框架捕捉⑦。

ENWe represent the linear response by relative displacements and changes in contact forces, and find significant changes with the distance to point J. (i) Fig. 1 illustrates that the response to the loading of a single grain becomes increasingly disordered over an increasingly large scale⑧ when the jamming transition is approached, this leads to a direct observation of the diverging length scale ℓ* ∼ 1/Δz, shown below. We will show that such a local force response is not self-averaging, even though it is smooth upon ensemble-averaging and then quantitatively agrees with continuum elastic behavior. (ii) The response to a uniformly applied compression or shear also varies with the distance to jamming. We introduce the distribution P(α) of angles α between the bonds and the local deformations as an indicator of the nonaffine nature of the response⑨. Near J, P(α) becomes strongly peaked around α = π/2, with the width and height of the peak scaling with the distance to the jamming point. Grains then predominantly slide past each other, which signals an increasingly nonaffine response of the material caused by the proximity of floppy modes, for which P(α) = δ_D(α − π/2). (iii) The component of the relative displacements perpendicular to the bond vector diverges upon approaching the jamming point. (iv) Finally, the Δz ∼ √δ scaling [3] is identified to originate precisely from this increasingly “sliding” response⑩. Hence a simple picture emerges: the influence of floppy modes can be quantified by the local linear response of the material, which becomes increasingly nonaffine near jamming, in turn causing anomalous scaling.

中译作者以相对位移和接触力变化表征线性响应，并发现它们随到 J 点的距离显著变化。(i) 图 1 表明，逼近堵塞转变时，单颗粒加载的响应在不断扩大的空间范围内变得更加无序⑧，这引出下文对发散交越尺度 ℓ* ∼ 1/Δz 的观测。此类局域力响应并不自平均；但经系综平均后会变得平滑，并在定量上符合连续介质弹性。(ii) 均匀压缩或剪切的响应同样依赖于距堵塞点的距离。作者引入接触键与局域相对位移之间夹角 α 的分布 P(α)⑨，用以衡量非仿射性。临近 J 时，P(α) 在 α = π/2 附近形成尖峰，峰宽和峰高随到堵塞点的距离标度；颗粒相对运动因而越来越以横向滑移为主。理想松弛模对应 P(α) = δ_D(α − π/2)，其中 δ_D 为 Dirac δ 函数。(iii) 相对位移的横向分量在逼近堵塞点时发散。(iv) 最后，作者提出越来越显著的横向、非仿射响应为 Δz ∼ √δ 标度提供微观机制⑩。由此，松弛模邻近效应可由局域线性响应量化，并与异常标度联系起来。

ENLinear response.—The response of a jammed granular medium to external loads has been studied mostly by full scale molecular dynamics [14,15]. We calculate it here in linear order from expansion of energy to second order

中译线性响应。——堵塞颗粒介质对外加载荷的响应，此前大多用全尺度分子动力学研究 [14,15]。我们这里按线性阶计算它，依据是能量展开到二阶：

ENWe study 2D packings of N frictionless Hertzian spheres for which f_ij ∝ δ_ij^3/2⑬, where δ_ij is the overlap between neighboring particles. The confining pressure ranges from p ∼ 10⁻⁶ to p ∼ 10⁻¹, in units of the effective Young modulus of the constituent particles. See Ref. [19] for details. For each packing, the expansion (1) yields the dynamical matrix M. Instead of studying the vibrational dynamics [7,9,10,20] we obtain here the quasistatic response to external forces f^ext [21] by solving the linear equation M_ij;αβ u_j;β = f^ext_i;α, for the u_j and, through the force law, the forces f_ij. Here i, j label the particles and α, β the coordinate axes.

中译我们研究 N 个无摩擦 Hertzian 球的二维堆积，其 f_ij ∝ δ_ij^3/2⑬（δ_ij 为相邻颗粒间的重叠量）。围压范围从 p ∼ 10⁻⁶ 到 p ∼ 10⁻¹，以组成颗粒的有效杨氏模量为单位。细节见文献 [19]。对每个堆积，展开式 (1) 给出动力学矩阵 M。我们不研究振动动力学 [7,9,10,20]，而是通过求解线性方程 M_ij;αβ u_j;β = f^ext_i;α 得到对外力 f^ext 的准静态响应 [21]，解出 u_j，再经由力律得到力 f_ij。其中 i、j 标记颗粒，α、β 标记坐标轴。

ENElastic moduli.—We have calculated the elastic moduli from the linear response by applying an overall compression or shear and by point-loading a single particle, for packings with N = 10³ and N = 10⁴, respectively. The resulting force fields are translated into local stress fields [18] which are then ensemble averaged. From fits of the point response [22], we determine both K and G and compare these to the values obtained from the response to global shear and compression. Figure 2(a) shows that these two methods agree very well quantitatively⑭, and that the elastic moduli scale with pressure as K ∝ p^1/3, G ∝ p^2/3, in agreement with earlier results [3,23].

中译弹性模量。——我们对 N = 10³ 的堆积施加整体压缩/剪切、对 N = 10⁴ 的堆积做单颗粒点加载，从线性响应计算弹性模量。所得力场被转换为局域应力场 [18] 再作系综平均。通过拟合点响应 [22]，我们同时确定 K 和 G，并与整体剪切/压缩响应所得的值比较。图 2(a) 表明两种方法在定量上非常一致⑭，且弹性模量随压强标度为 K ∝ p^1/3、G ∝ p^2/3，与早期结果 [3,23] 一致。

ENNonaffinity.—The typical bond stiffness k_ij is proportional to p^1/3 for Hertzian contacts. Hence, a simple estimate for the elastic moduli scaling as p^1/3 follows under the affinity assumption that the bond deformations are of order of the applied deformation. This estimate fails for the shear modulus G, which vanishes faster than K when approaching the jamming point: G/K ∼ Δz⑮. This has been thought to be caused by strongly nonaffine behavior of the system under shear [3] and the proximity of the floppy modes [11]. We will elucidate now the cause of the scaling and the influence of the floppy modes via the local deformations u_∥,ij and u_⊥,ij.

中译非仿射性。——对 Hertzian 接触，典型接触刚度 k_ij 正比于 p^1/3。若假设接触形变与外加形变同阶，即采用仿射估计，则弹性模量也应按 p^1/3 标度。但该估计对剪切模量 G 失效：逼近堵塞点时 G 比 K 更快趋零，满足 G/K ∼ Δz⑮。已有工作将这一现象与剪切下的强非仿射行为 [3] 及松弛模邻近效应 [11] 联系起来。下面通过局域纵向位移 u_∥,ij 与横向位移 u_⊥,ij 分析其标度来源。

ENAs the eigenmodes or snapshots of the response look very disordered [11,19–21], it has turned out to be difficult to find a simple measure to characterize the nonaffinity and the overall floppy-mode character. We characterize the local angle by tan α_ij = u_⊥,ij/u_∥,ij, with the quadrant fixed by the displacement direction, and study its distribution P(α). In a disordered, isotropic system, one expects P(α) = δ_D(α − π) for a purely homogeneous compression, and P(α) = 1/π for a purely affine shear. In contrast, for a floppy mode, P(α) = δ_D(α − π/2)⑯. This is because in floppy modes the relative angles between particles change while the relative distances r_ij remain unchanged, as if all the bonds are replaced by inextensible sticks [8]. Hence, for a true floppy mode u_∥,ij = −u²_⊥,ij/(2 r_ij) + O(u⁴_⊥,ij/r³_ij)⑰ [24].

中译本征模或响应快照高度无序 [11,19–21]，因此需要一个简洁统计量来刻画非仿射性与松弛模特征。这里以 tan α_ij = u_⊥,ij/u_∥,ij 定义局域夹角，并由位移方向确定象限，再考察其分布 P(α)。对无序、各向同性体系，纯均匀压缩应有 P(α) = δ_D(α − π)，纯仿射剪切应有 P(α) = 1/π；松弛模则对应 P(α) = δ_D(α − π/2)⑯。这是因为松弛模改变颗粒间相对角度而保持接触距离 r_ij 不变，相当于以不可伸长刚杆约束接触 [8]。因此，真正的松弛模满足 u_∥,ij = −u²_⊥,ij/(2r_ij) + O(u⁴_⊥,ij/r³_ij)⑰ [24]。

ENAs the pressure is lowered, P(α) and P(u_⊥) evidence that the local deformations evolve from near-affine to extremely nonaffine, floppy-modelike [Figs. 2(b)–2(d)]. Indeed, for a sheared system, P(α) evolves from a flat distribution at large pressures to a sharply peaked distribution for lower pressures [Fig. 2(c)]. This peak is located around π/2 and its weight approaches 1: locally the response becomes more and more transverse for p → 0 and P(α) approaches that of a floppy mode. For a compressed system at large pressures, P(α) has the “affine” peak around α = π, while for lower pressures P(α) again develops a sharp peak around α = π/2 [Fig. 2(d)].

中译随着压强降低，P(α) 与 P(u_⊥) 表明局域形变由近仿射逐步演化为强非仿射、类松弛模状态 [图 2(b)–2(d)]。对剪切体系，P(α) 从高压下的平坦分布演化为低压下的尖锐峰 [图 2(c)]；峰位于 π/2 附近且权重趋于 1，说明 p → 0 时局域响应越来越横向化。对压缩体系，高压时 P(α) 在 α = π 附近出现仿射峰，而低压时同样在 α = π/2 附近形成尖峰 [图 2(d)]。

ENEven though the response is far from affine for both compression and shear, the affine prediction for K holds true while it fails for G. The reason is that for compression, only a finite fraction of the displacements is essentially transverse and P(α) remains nonzero away from the peak at π/2. Since according to the energy expression (1) the compression of bonds given by u_∥,ij gives the dominant contribution to ΔE, this is consistent with the fact that the compression modulus scales with the bond stiffness k: K ∼ k ∼ p^1/3. For shear deformations, however, fewer and fewer bonds contribute to leading order to the energy, and the weight outside the peak vanishes as Δz ∼ p^1/3, consistent with the scaling G/K ∼ Δz⑲.

中译尽管压缩与剪切响应都明显非仿射，仿射预测对 K 成立、对 G 却失效。压缩时，P(α) 在 π/2 峰之外仍保留有限权重；按式 (1)，由 u_∥,ij 表示的接触纵向形变对 ΔE 作主导贡献。因此体积压缩模量沿接触刚度标度：K ∼ k ∼ p^1/3。剪切时，对能量作主导阶贡献的峰外权重随 Δz ∼ p^1/3 消失，与 G/K ∼ Δz 一致⑲。

ENScaling of P(α) and P(u_⊥).—We can understand the development of the peak in P(α) from the balance of terms in the energy expansion (1). Focusing on typical values, ΔE ∼ k(u²_∥ − δu²_⊥). Since k ∼ p^1/3 and δ ∼ p^2/3, balancing the terms we find that u_∥/u_⊥ ∼ √δ ∼ p^1/3 ∼ Δz⑳,  (2) so that for small p, P(α) develops a peak around α = π/2, and the width of this peak should scale as p^1/3. This is what we find; see the insets of Figs. 2(c) and 2(d).

中译P(α) 与 P(u_⊥) 的标度。——P(α) 中尖峰的发展可由式 (1) 两项的量级平衡理解。对典型值，ΔE ∼ k(u²_∥ − δu²_⊥)。由于 k ∼ p^1/3、δ ∼ p^2/3，平衡两项得到 u_∥/u_⊥ ∼ √δ ∼ p^1/3 ∼ Δz⑳，(2) 因此对小 p，P(α) 在 α = π/2 处发展出峰，且峰宽应按 p^1/3 标度。这正是我们发现的；见图 2(c)、2(d) 的插图。

ENHow do the typical values u_∥ and u_⊥ scale when we impose a global shear or compression of order γ on the system? Equating the elastic energy densities for compression and shear, Kγ² and Gγ², to the energy expansion, and knowing that the elastic moduli scale as G ∝ p^2/3 and K ∝ p^1/3, we can predict the scaling of u_∥ and u_⊥:

中译若对系统施加量级为 γ 的整体剪切或压缩，典型纵向位移 u_∥ 与横向位移 u_⊥ 如何标度？将压缩和剪切的弹性能密度 Kγ²、Gγ² 与能量展开相匹配，并利用 G ∝ p^2/3、K ∝ p^1/3，可预测：

ENΔz scaling.—The nonaffine response also provides a microscopic explanation for the anomalous Δz ∼ √δ scaling under compression; theories assuming affine deformations give Δz ∼ δ. Consider a small compression of the packing with typical bond compression u_∥. The resulting infinitesimal change in contact number is (dΔz/dδ)u_∥. (5) Upon lowering the pressure, global compression excites distorted floppy modes, so for many bonds u_⊥,ij is of order u_∥/√δ [see Eq. (2) and Fig. 2(d)]. Moreover, the chance that one particle encounters a nearby, previously noncontacting particle during an almost perpendicular displacement is proportional to this motion, u_∥/√δ. Equating this contact-number change to (5) yields dΔz/dδ ∼ 1/√δ and hence Δz ∼ √δ㉓. Thus, transverse displacements—and the probability of creating new contacts—grow as pressure decreases.

中译Δz 标度。——非仿射响应还为压缩下的异常标度 Δz ∼ √δ 提供了一种微观解释；仿射形变理论则给出 Δz ∼ δ。考虑一次典型纵向接触压缩为 u_∥ 的小压缩，由此产生的无穷小接触数变化为 (dΔz/dδ)u_∥（5）。降压时，整体压缩会激发畸变的松弛模，因此许多接触上的 u_⊥,ij 达到 u_∥/√δ 的量级 [见式 (2) 与图 2(d)]。作者进一步假设，颗粒在近乎横向位移中遇到一个此前未接触近邻的概率正比于位移幅度 u_∥/√δ。将这一接触数变化与式 (5) 对应，得到 dΔz/dδ ∼ 1/√δ，进而得到 Δz ∼ √δ㉓。因此，压强越低，横向位移以及形成新接触的概率越大。

ENIn earlier papers, it was noted [3,25,26] that the Δz ∼ √δ scaling with compression was consistent with a square root divergent term in the correlation function g(r), if it was assumed that compression would be essentially affine. As we have seen, however, distortions are not at all affine near jamming. Our analysis turns this around: it suggests that the natural coordinates for the floppy-mode-like distortions are the perpendicular displacements, not the radial ones, and that these generate the square root behavior of g(r) in the radial direction㉔.

中译早期论文指出 [3,25,26]：若假设压缩近似仿射，则 Δz ∼ √δ 与近接触径向分布 g(r) 中的平方根奇异性相容。然而堵塞附近的形变高度非仿射。本文由此反向提出：类松弛模畸变的自然坐标是横向位移而非径向位移，而径向 g(r) 的平方根行为可能由这些横向位移生成㉔。

ENPoint response and diverging length scale.—We finally return to the point response. Figure 3(a) illustrates that the ensemble average of such a response conforms to elasticity; the stress fields fit well and the fitted elastic constants agree with those obtained from bulk response [Fig. 2(a)]. On the other hand, individual responses become very disordered and suggest the occurrence of a large length scale ℓ* when approaching point J (see Fig. 1).

中译点响应与发散长度尺度。——最后回到点响应。图 3(a) 表明，这类响应的系综平均符合弹性理论：应力场拟合良好，拟合得到的弹性常数也与整体响应所得结果一致 [图 2(a)]。另一方面，单个样本的响应高度无序，并提示逼近 J 点时出现不断增大的交越尺度 ℓ*（见图 1）。

ENTo extract this length scale, we characterize the response to an infinitesimal inflation of a single central grain, since the response to a local directional force, as shown in Figs. 1 and 3(a), is highly anisotropic. We normalize the forces by fitting the radial stress to the elastic response. We focus on the radial component of the change in contact force df_r, calculate ⟨df_r(r)⟩ by averaging df_r over concentric rings, and study the rms fluctuation h(r) = √⟨[df_r(r) − ⟨df_r(r)⟩]²⟩. The range over which these fluctuations are felt grows when approaching the jamming transition [Fig. 3(b)]. When plotted as a function of rΔz, the data for h collapse [Fig. 3(c)]. To our knowledge, this is the first evidence for the existence of a length scale ℓ* ∼ 1/Δz in local response measurements㉕. Essentially the same characteristic length was shown by Wyart et al. to govern the vibrational density of states [9,10]. This scale is identified as the maximum linear size ℓ* of a domain that can deform freely by pushing on the bonds at its surface. Equating the number of surface bonds (∼ℓ^d−1) with the number of excess bonds in the bulk (∼Δz ℓ^d) yields ℓ* ∼ 1/Δz [9,10].

中译为提取这一尺度，作者考察中心颗粒无穷小膨胀所激发的响应，以避免图 1 和图 3(a) 中局域定向力带来的强各向异性。通过将径向应力拟合到弹性响应来归一化力；对接触力变化的径向分量 df_r 作同心环平均得到 ⟨df_r(r)⟩，并定义均方根涨落 h(r) = √⟨[df_r(r) − ⟨df_r(r)⟩]²⟩。逼近堵塞转变时，涨落影响范围增大 [图 3(b)]；以 rΔz 为横轴后，h 的数据发生塌缩 [图 3(c)]。据作者所知，这是局域响应测量中支持 ℓ* ∼ 1/Δz 的首个证据㉕。Wyart 等人此前已表明，本质相同的尺度控制振动态密度 [9,10]。ℓ* 可理解为能通过推动边界接触而自由形变的区域的最大线尺度。令表面接触数（∼ℓ^d−1）与体内过剩接触数（∼Δzℓ^d）相等，得到 ℓ* ∼ 1/Δz。

ENBoth the average response ⟨df_r(r)⟩ and the fluctuation h(r) decay as 1/r²; the relative fluctuations do not decay far from the perturbed grain. The response is not self-averaging, and there is no finite correlation length of fluctuations㉖. The asymptotic relative fluctuation h(r)/⟨df_r(r)⟩ ∼ h(r)r² grows as 1/Δz² [Fig. 3(d)]; one must coarse-grain over O((Δz)⁻⁴) grains before the response begins to converge toward an average continuum-like stress field.

中译平均响应 ⟨df_r(r)⟩ 与涨落 h(r) 都按 1/r² 衰减，因此远离扰动颗粒后，相对涨落并不衰减。该响应不具自平均性，涨落也不存在通常意义下的有限关联长度㉖。相对涨落的渐近值 h(r)/⟨df_r(r)⟩ ∼ h(r)r² 按 1/Δz² 增长 [图 3(d)]；必须对 O((Δz)⁻⁴) 个颗粒进行粗粒化，响应才开始收敛到平均的类连续介质应力场。

ENOutlook.—Our analysis of the (local) linear response substantiates and extends the concept that the jammed phase of weakly compressed frictionless particles is dominated by the proximity of floppy modes [8–11]. We identified the increasingly nonaffine response as giving rise to the scaling Δz ∼ √δ and presented direct evidence for the previously introduced scale ℓ* ∼ 1/Δz [9,10]. The emerging scenario favors a microscopic, geometric interpretation of these scalings and has several implications that deserve further study: (i) What is the finite-size scaling form of u_⊥? (ii) What happens for non-power-law contact interactions such as f_ij ∼ exp[−δ_ij^1−β] = exp[−1/δ_ij^β−1] with β > 1? Our analysis suggests Δz ∼ δ^β/2㉘. (iii) What happens to the square-root divergence in the near-contact radial distribution g(r) [3,25,26] when the packing algorithm does not allow floppy-mode-like rearrangements during annealing, as may occur in local-rearrangement algorithms or packings of truly hard spheres?

中译展望。——对局域线性响应的分析支持并扩展了一个观点：弱压缩无摩擦颗粒的堵塞态受松弛模邻近效应支配 [8–11]。作者把越来越非仿射的响应与 Δz ∼ √δ 联系起来，并为此前提出的尺度 ℓ* ∼ 1/Δz [9,10] 给出直接证据。由此产生的微观几何图像提出三个待检验问题：(i) u_⊥ 的有限尺寸标度函数是什么？(ii) 对非幂律接触作用，例如 f_ij ∼ exp[−δ_ij^1−β] = exp[−1/δ_ij^β−1]（β > 1），响应如何变化？该分析预言 Δz ∼ δ^β/2㉘。(iii) 若退火算法不允许类松弛模重排，例如局域重排算法或真正硬球的堆积，近接触径向分布 g(r) 的平方根发散将如何改变？

ENAcknowledgments. We thank K. Shundyak, J. H. Snoeijer, and M. Depken for discussions and M. Wyart for critical correspondence which led us to realize the divergence (3) and (4). W. G. E. acknowledges support from physics foundation FOM, and M. v. H. acknowledges support from NWO/VIDI.

中译致谢。感谢 K. Shundyak、J. H. Snoeijer 与 M. Depken 的讨论，感谢 M. Wyart 的关键通信——正是它使我们认识到 (3)、(4) 的发散。W. G. E. 感谢荷兰物理基金会 FOM 的资助，M. v. H. 感谢 NWO/VIDI 的资助。