David Aristoff·Charles Radin
Abstract
We introduce and simulate a two-dimensional Edwards-style model of granular matter at vanishing pressure. The model incorporates some of the effects of gravity and friction, and exhibits a random loose packing density whose standard deviation vanishes with increasing system size, a phenomenon that should be verifiable for real granular matter.
我们引入并模拟了压力消失时的二维 Edwards 式颗粒物质模型。该模型包含了重力和摩擦力的部分影响，并表现出随机松散堆积密度，其标准偏差随系统规模增大而消失，这一现象在真实的颗粒物质中应该是可以验证的。
An Edwards-Style Model
We introduce and analyze a crude model for the random loose packings of granular matter. These packings, as well as random close packings, were carefully prepared by Scott et al. in the 1960s, in samples of steel ball bearings.
Gently pouring samples of $20,000$ to $80,000$ spheres into a container, the lowest possible volume fraction obtainable—the socalled random loose packing density—was determined to be $0.608 \pm 0.006$.
我们介绍并分析了颗粒物质随机松散堆积的粗略模型。Scott 等人于 20 世纪 60 年代在钢球轴承样品中仔细制备了这些堆积以及随机紧密堆积。
将 $20000$ 至 $80000$ 的球体样品轻轻倒入容器中，可获得的最低体积分数–即所谓的随机松散堆积密度–被确定为 $0.608 \pm 0.006$.
The above refers to monodisperse steel spheres immersed in air; they also worked with spheres of other materials immersed in other fluids; variations in the coefficient of friction and in the effective gravitational force lead to somewhat different values for the random loose packing density.
以上提到的是浸入空气中的单分散钢球；他们也曾处理过浸入其他流体中的其他材料的球体；摩擦系数和有效重力的变化导致随机松散堆积密度的数值略有不同。
Matter is generally described as “granular” if it is composed of a large number of noncohesive subunits each of which is sufficiently massive that its gravitational energy is much larger than its thermal energy. A common example is a sand pile.
如果物质是由大量非内聚的亚单位组成，而每个亚单位的质量又足够大，以至于其引力势能远大于热能，那么这种物质通常被描述为 “颗粒状”。沙堆就是一个常见的例子。
There are several classic phenomena characteristic of static granular matter, in particular dilatancy, random close packing, and random loose packing, none of which can yet be considered well-understood; see [3] for a good review. A basic question about these phenomena is whether they are sharply defined or inherently vague.
Dilatancy has recently been associated with a phase transition measured by the response of the material to shear, which answers the question for this phenomenon. The case of random close packing is controversial and awaits further experiment; see [16, 21]. Our main goal here is to analyze this question with respect to random loose packing, to determine whether or not traditional theoretical approaches to granular matter predict a sharply defined random loose packing density.
It is clear that any experimental determination of a random loose packing density will vary with physical conditions such as coefficient of friction, and we will take this variation into account in our analysis below.
静态颗粒物质有几种典型的特征现象，特别是扩张性、随机紧密堆积和随机松散堆积，其中没有一种现象可以被认为是很好理解的；详见[3]。关于这些现象的一个基本问题是: 它们可以被明确定义, 还是本质上就模糊不清?
最近，根据材料对剪切力的响应测量，扩散性(Dilatancy)与相变有关，这回答了这一现象的问题。随机紧密堆积的情况尚存争议，有待进一步实验；见 [16, 21]。我们在此的主要目标是分析随机松散堆积的这一问题，以确定颗粒物质的传统理论方法是否预测了明确定义的随机松散堆积密度。
显然，任何 RLP 密度的实验测定值都会随着摩擦系数等物理条件的变化而变化，我们将在下面的分析中考虑这些变量。
We begin by contrasting two common approaches to modelling static granular matter. One, the more common, is the “protocol-dependent simulation”, in which one studies properties of dense packings by exploring a variety of methods of preparation of the packings; see [5, 21]or[23] for examples.
Another approach goes under the name of Edwards theory [4], in which, basically, one adds the effects of friction and a strong gravitational force to the hard sphere model of equilibrium statistical mechanics. We note that appropriate specification of the added forces fully determines an Edwards model; there are no adjustable parameters beyond those familiar from statistical mechanics, such as density and pressure.
(Of course one can always introduce further approximations or features, for instance mean field theory, soft core, attraction, etc.) In particular, in an Edwards model all Markov chain Monte Carlo simulations will, if done correctly, give the same result; there is no freedom in preparing the packings the way there is in protocol-dependent simulation.
我们首先对比两种常见的静态颗粒物质建模方法。其中一种更为常见，即 “Protocol-Dependent Simulation”，即通过探索各种制备堆积的方法来研究致密堆积的特性；相关例子见[5, 21]或[23]。
如果要详细了解该方法, 就要回到布朗运动以及发展出来的 Langevin Dynamics, 这种随机梯度动力学要详细讲就是另一个页面了.
另一种方法称为 Edwards 理论[4]，主要是在平衡统计力学的硬球模型中加入摩擦力和强重力的影响。我们注意到，对引入力的适当说明完全决定了 Edwards 模型；除了统计力学中熟悉的参数（如密度和压力）外，没有其他可调参数。
当然，我们还可以引入更多的近似或特征值，如平均场理论、软核、吸引力等）特别是，在爱德华兹模型中，如果操作正确，所有的马尔科夫链蒙特卡罗模拟都会得到相同的结果；不存在像 Protocol-Dependent Simulation 那样需要预先定义堆积方式的自由度。
These two approaches—the protocol-dependent simulations and the Edwards approachhave different strengths. There have been serious claims that the former approach has serious difficulty making sense of some granular phenomena, in particular random close packing. We have previously shown how Edwards theory allows a clean definition of random close packing, and in this paper we show, by a very different mechanism, how it allows for an understanding of random loose packing.
In that sense our choice of using an Edwards-type model is central to our argument. (We do not claim that the Edwards approach has been proven the most accurate theory of static granular matter, but only that it is a serious contender.)
这两种方法–协议依赖型模拟和 Edwards 方法–各有所长。有人严肃地指出，前一种方法很难理解某些颗粒现象，特别是随机紧密堆积现象。我们之前已经展示了 Edwards 理论是如何清晰定义随机紧密堆积的，而在本文中，我们通过一种非常不同的机制展示了 Edwards 理论是如何理解随机松散堆积的。
从这个意义上说，我们选择使用 Edwards 模型是我们论证的核心。(我们并没有说 Edwards 方法已被证明是最准确的静态颗粒物质理论，而只是说它是一个有力竞争者)。
We briefly summarize our Edwards-style model as follows: We consider arrangements of hard-core parallel squares in a fixed rectangular box, where each square has to rest on either two squares below it or on the box’s floor, and we put a uniform probability distribution on the set of all such arrangements. Then we run Markov chain Monte Carlo simulations and measure the packing fraction of the Markov chain configurations.
我们将 Edwards 模型简要概括如下：我们考虑在一个固定的矩形盒子中排列刚性平行方格，每个方格必须靠在它下面的两个方格或盒子的地板上。然后，我们运行马尔科夫链蒙特卡罗模拟，并测量马尔科夫链构型的堆积率。
We begin more ambitiously by discussing a more realistic model. As is standard in Edwards theory we take as a starting point a variant of the hard sphere model of equilibrium statistical mechanics. Consider a model consisting of large collections of impenetrable, unit mass, unit diameter spheres in a large container, acted on by gravity and with infinite coefficient of friction between themselves and with the container.
Put a probability density on the set of all mechanically stable packings of the spheres in their container, with the probability density of a packing $c$ proportional to $\text{exp}[−E(c)]$, where $E(c)$ is the sum of the heights, from the floor of the container, of the centers of the spheres in the packing $c$. We expect, but cannot show, that such an ensemble will exhibit a gradient in the volume fraction (with volume fraction decreasing with height) and that there is a well-defined random loose packing density as one approaches the top of the packing (where the analogue of hydrostatic pressure goes to zero).
By a “well-defined random loose packing density” we mean that as one takes an infinite volume limit, the probability distribution for the volume fraction of the top layer of the packing becomes concentrated at a single nonzero value. We emphasize that we are focusing on a bulk property near the top of the configuration, not a surface phenomenon.
我们将更大胆地开始讨论一个更现实的模型。按照 Edwards 理论的标准，我们从平衡统计力学中的硬球模型的变式出发。考虑这样一个模型：在一个大容器中，有大量刚性、单位质量、单位直径的球体，这些球体受重力作用，并且球体之间以及球体与容器之间的摩擦系数为无穷大。
在容器中所有机械稳定的球体堆积集合上赋予一个概率密度，堆积方式 $c$ 的概率密度与 $\text{exp}[-E(c)]$ 成正比，其中 $E(c)$ 是堆积 $c$ 中球体中心距容器底面的高度之和。我们预计（但无法证明），这样一个集合体的体积分数会呈现梯度（体积分数随高度递减），而且当接近集合体顶部时（静水压力在此为零），会出现良好定义的随机松散堆积密度。
我们所说的 “良好定义的随机松散堆积密度” 是指在取体积无限的极限下，堆积顶层体积分数的概率分布会集中在一个非零值上。我们要强调的是，我们关注的是构型顶部附近的体积特性，而不是表面现象。
The above determines a well defined zero pressure probability distribution for packings $c$. One could imagine simulating the distribution with Monte Carlo or molecular dynamics, but this is not practical at the high densities which are necessary in a granular model.
(We emphasize that any such simulation should reproduce the above probability distribution; in this Edwards-style model the equilibrium probability distribution is completely determined, so there is no freedom available in deciding how packings are simulated.)
以上确定了堆积 $c$ 的良好定义的的零压概率分布。我们可以想象用蒙特卡洛或分子动力学来模拟这种分布，但这在颗粒模型所需的高密度下并不现实。
(我们强调，任何此类模拟都应再现上述概率分布；在这种 Edwards 模型中，平衡概率分布是完全确定的，因此其结果与如何确定模拟的堆积方式无关)。
To make Monte Carlo simulations feasible, we make several simplifications in the way gravity and friction are incorporated in the above model. First we switch to an ensemble consisting of packings which are limits, as the gravitational constant goes to zero, of mechanically stable packings; we effect this by setting $E(c) = 0$ in the relative density $\text{exp}[−E(c)]$. With this simplification configurations are now, in their entirety, representative of the top layer in the original model. Next we consider the two dimensional version of the above: congruent frictional unit disks in mechanically stable configurations under vanishingly small gravity.
Note that each such disk must be in contact with either a pair of supporting disks below it or part of the container. (Here and elsewhere in this paper we neglect events of probability zero, such as one sphere perfectly balanced on another.) We simplify the role of gravity and friction in the model one last time by replacing the disks by congruent squares, with edges aligned with the sides of the (rectangular) container, each square in contact with either a pair of supporting squares below it or the floor of the container. This is now a granular version of the old model of “(equilibrium) hard squares” [8], which is a simplification of “hard disks” and “hard spheres” (see [1] for a review), in which gravity and friction is neglected but kinetic energy plays a significant role.
We emphasize that in our granular model there is no longer any need to concentrate on the “top layer”; in fact we will eventually be concerned with an infinite volume limit which, as usual, focuses on the middle of the collection of squares and lets the boundaries grow to infinity. (We note that the model is capable of handling higher densities by constraining the squares to lie in a tightly containing box. We also note recent work by Song et al. [20, 22] which takes a different path, employing a mean field approximation instead of a simplified short range model which can be fully simulated, as we have done.)
为了使蒙特卡罗模拟可行，我们对上述模型中的重力和摩擦力进行了一些简化。首先，我们切换到一个由堆积组成的系综，当引力常数归零时，这些填料是机械稳定填料的极限；我们通过在相对密度 $\text{exp}[-E(c)]$ 中设置 $E(c) = 0$ 来实现这一点。经过这样的简化，现在的构型完全可以代表原始模型中的顶层。接下来, 我们考虑上述模型的二维版本：在重力极小的情况下，机械稳定构型中的全等摩擦单元盘。
请注意，每个这样的圆盘都必须与它下面的一对支撑圆盘或容器的一部分接触。(本文在这里和其他地方都忽略了概率为零的事件，例如一个球体完全平衡地放在另一个球体上)。我们最后一次简化模型中重力和摩擦力的作用，把圆盘换成边缘与（矩形）容器边对齐的全等正方形，每个正方形要么与下面的一对支撑正方形接触，要么与容器的地板接触。这是旧的"（平衡）硬方块 “模型[8]的颗粒化版本，是 “硬磁盘 “和 “硬球体 “的简化版(综述见[1])，其中重力和摩擦力被忽略，但动能起着重要作用。
我们要强调的是，在我们的粒状模型中，不再需要关注 “顶层”；事实上，我们最终要关注的是无限体积极限，即像往常一样，关注方块集合的中间部分，并让边界增长到无穷大。(我们注意到，通过限制方块位于一个紧密包含的盒子中，该模型能够处理更高的密度)。我们还注意到 Song 等人最近的研究，他们采用了不同的方法，用平均场近似代替了简化的短程模型，而短程模型可以像我们所做到的一样完全模拟）。
We have run Markov chain Monte Carlo simulations on this model with the following results.
We initialize the squares in an allowed configuration of some well-defined volume fraction anywhere between $0.5$ and $1$. If the initial volume fraction $\phi$ is not approximately $0.76$, the simulation gradually expands or contracts the packings until the packing fraction reaches the range $0.76 \pm 0.01$; see Figs. 1 and 2. Furthermore, as the size of the packings increases, the standard deviation of the volume fraction tends towards zero.
我们对该模型进行了马尔科夫链蒙特卡罗模拟，结果如下.
我们以某个良好定义的, 体积分数在 $0.5$ 到 $1$ 之间的允许构型来初始化这些方块。如果初始体积分数 $\phi$ 不是大约 $0.76$，模拟会逐渐膨胀或紧缩堆积，直到堆积率达到 $0.76 \pm 0.01$；见图 1 和图 2。此外，随着填料尺寸的增大，体积分数的标准偏差趋向于零。
The process is insensitive to the dimensions of the containing box except for extremes. We choose the height of the box to be large enough so that the configurations of squares cannot reach the ceiling (so the box height becomes irrelevant). We must choose the box width more carefully, since if the side walls of the containing box abut a closely-packed initial configuration, the simulation cannot significantly change the volume fraction; alternatively, if the width of the box is much larger than that of the initial configuration, the simulation will produce a monolayer on the floor.
除开一些极端情况，该过程对容器盒的尺寸并不敏感。我们选择的盒子高度要足够大，这样方块结构就无法触及天花板（因此盒子高度变得无关紧要）。我们必须更谨慎地选择盒子的宽度，因为如果内装盒子的侧壁与紧密堆积的初始构型相邻，模拟就无法显著改变体积分数；反之，如果盒子的宽度远大于初始配置的宽度，模拟就会在地板上产生单层。
We ignore both extremes, however, and find that the equilibrium volume fraction is otherwise insensitive to the width of the box. More precisely, we found that the equilibrium volume fraction should be accurate if the box width is between $2\sqrt{N}$ and $8\sqrt{N}$, where $N\geq 100$ is the number of squares.
To understand these limits, first note that since we will be conjecturing the behavior of the model in the infinite volume limit, the equilibrium configuration should be a single bulk pile, so the box width should be on the order of $\sqrt{N}$.
然而，我们忽略了这两个极端，并且发现平衡态的体积分数对盒子的宽度并不敏感。更准确地说，我们发现如果盒子的宽度介于 $2\sqrt{N}$ 和 $8\sqrt{N}$（其中 $N\geq 100$ 是方格数）之间，平衡体积分数应该是准确的。
Regarding the lower bound, note that at any volume fraction a configuration occupies the least amount of floor space when the squares are arranged in a single full triangle. The bottom level of such a triangle has just under $\sqrt{2N}$ squares.
Assume the containing box fits tightly around the triangle; if the triangle has volume fraction greater than $0.754$ then the configuration will not be able to decrease to this equilibrium volume fraction. We avoid this by ensuring that the box width is at least $2\sqrt{N} > (0.754)^{-1}\sqrt{2N}$.
To arrive at the upper bound we performed simulations on fixed particle number and let the box width vary. We found that the equilibrium volume fraction was reliable so long as the box width was less than about $8\sqrt{N}$, at least for $N\geq 100$.
关于下限，请注意，在任何体积分数下，当方格排列成一个完整的三角形时，该构型所占的地面空间最小。这样一个三角形的底层只有不到 $\sqrt{2N}$ 的方格。
假设包含的盒子紧贴三角形; 如果三角形的体积分数大于 $0.754$，那么构型将无法减小到这一平衡体积分数。我们通过确保方盒宽度至少为 $2\sqrt{N} > (0.754)^{-1}\sqrt{2N}$ 来避免这种情况。
为了得出上限，我们对固定的粒子数进行了模拟，并让盒子的宽度变化。我们发现，只要箱宽小于大约 $8\sqrt{N}$，至少在 $N\geq 100$ 的情况下，平衡体积分数是可靠的。
We conclude that, for box widths in the aforementioned acceptable range, the equilibrium volume fraction depends only on the number of squares in the system. The main goal of our work is an analysis of the distribution of volume fraction—both the mean and standard deviation—as the number of particles increases. We conclude that the limiting standard deviation as particle number goes to infinity is zero, so the model exhibits a sharp value for the random loose packing density, which we estimate to be approximately $0.754$.
我们的结论是，对于上述可接受范围内的箱宽，平衡体积分数只取决于系统中的方块数。我们工作的主要目标是分析颗粒数量增加时体积分数的分布–对平均值和标准偏差的考察也包含在内。我们的结论是，当粒子数达到无穷大时，极限标准偏差为零，因此该模型显示出随机松散堆积密度的一个尖锐值，我们估计该值约为 $0.754$。
The heart of our argument is the degree to which we can demonstrate that in this model there is a sharp value, approximately $0.754$, for the equilibrium volume fraction of large systems, and we postpone analysis of error bars to later sections. But to understand the value $0.754$, consider the following crude estimate of the volume in phase space of all allowable packings at fixed volume fraction $\phi$. First notice that the conditions defining the model prevent the possibility of any “holes” in a configuration.
Furthermore, if we consider any rectangle in the interior of a configuration, each horizontal row in the rectangle contains the same number of squares. (One consequence is that in the infinite volume limit each individual configuration must have a sharply defined volume fraction; of course this says nothing about the width of the distribution of volume fraction over all configurations.)
我们论证的核心是我们能在多大程度上证明在这个模型中大系统的平衡体积分数有一个尖锐的值，大约是 $0.754$，我们把误差条的分析推迟到后面的章节。但为了理解 $0.754$ 这个值，请考虑以下对固定的体积分数 $\phi$ 时所有允许的堆积在相空间中的体积的粗略估计。首先要注意的是，定义模型的条件阻止了在构型中出现任何 “洞 “的可能性。
此外，如果我们考虑配置内部的任何矩形，矩形中的每一横行都包含相同数量的正方形。(结果之一是，在取体积无穷大极限时，每个单独的构型都必须有一个明确定义的体积分数；当然，这并没有说明所有构型的体积分数分布宽度。)
Now consider a very symmetrical configuration of squares at any desired volume fraction $\phi$, with the squares in each horizontal row equally spaced, and gaps between squares each of size $(1-\phi)/\phi$ centered over squares in the next lower horizontal row; see Fig. 3. Consider these squares to represent average positions, fix all but one square in such a position, and consider the (horizontal) degree of motion allowed to the remaining square.
现在考虑任意一个要求体积分数为 $\phi$ 的极其对称的方块构型，每个横行的方块间距相等，每个正方形之间的间隙大小为 $(1-\phi)/\phi$，位于下一个较低水平行正方形的中心；详见图 3。将这些方格现在的位置视为代表平均位置，将除一个方格外的所有方格固定在这样的位置上，并考虑剩下的方格允许的(水平)运动(自由)度。
There are two constraints on its movement: the gap size separating it from its two neighbors in its horizontal row, and the length to which its top edge and bottom edge intersects the squares in the horizontal rows above and below it. These two constraints are to opposite effect: increasing the gap size decreases the necessary support in the rows above and below. A simple calculation shows that the square has optimum allowed motion when the gap size is $1/3$, corresponding to a volume fraction of $0.75$, roughly as found in the simulations.
In other words, this argument suggests that the volume in phase space (which for $N$ squares we estimate to be $L^{N}$, where $L$ is the allowed degree of motion of one square considered above) is maximized among allowed packings of fixed volume fraction by the packings of volume fraction about $0.75$. Note that this is only a free volume-type estimate, so it is by no means a proof that a sharp entropy-maximizing volume fraction exists or is equal to or near $0.75$.
它的移动有两个限制：一是它与横行中两个相邻方块之间的间隙大小，二是它的顶边和底边与上下两横行方块相交的长度。这两个限制条件的作用正好相反：间隙增大，上下两行所需的支撑就会减小。一个简单的计算表明，当间隙大小为 $1/3$，相当于 $0.75$ 的体积分数时，正方形具有最佳允许运动，这与模拟中的结果大致相同。
换句话说，这一论证表明，相空间的体积（对于 $N$ 个方块，我们估计为 $L^{N}$，其中 $L$ 是上面考虑的一个方块允许的运动度）在体积分数约为 $0.75$ 时, 在允许堆积方式中取最大值。需要注意的是，这只是一个自由体积类型的估计，因此绝不能证明存在一个尖锐的熵最大化体积分数，或其等于或接近 $0.75$。
To obtain accurate physical measurements a fluidization/sedimentation method has been developed to prepare samples of millions of grains in a controlled manner; see $[9, 13]$ and references therein for the current state of the experimental data. In these experiments a fluidized bed of monodisperse grains sediment in a fluid.
The sediment is of uniform volume fraction, at or above $0.55$ depending on various experimental parameters. Recall that the old experiments of Scott et al. $[17, 18]$ reported a value of $0.608$ for ball bearings in air; to achieve the low value $(0.55 \pm 0.001)$ the grains need to have a high friction coefficient and the fluid needs to have mass density only slightly lower than the grains to minimize the destabilizing effect of gravity. (In the absence of gravity one could still produce a granular bed by pressure; we do not know of experiments reporting a random loose packing value for such an environment.)
为了获得精确的物理测量结果，我们开发了一种流化/沉积方法，用于以受控方式制备数百万颗粒的样品；有关当前的实验数据，请参阅 [9, 13] 及其参考文献。在这些实验中，流化床中的单分散颗粒在流体中沉积。
沉积物的体积分数是均匀的，达到还是超过 $0.55$ 取决于各种实验参数。回顾 Scott 等人的旧实验 $[17, 18]$，空气中球轴承的密度数值为 $0.608$；要达到较低的数值 $(0.55 \pm 0.001)$，颗粒需要具有较高的摩擦系数，流体的质量密度只需略低于颗粒，以尽量减少重力的破坏效应。(在没有重力的情况下，人们仍然可以通过压力产生颗粒床；我们并不清楚是否有在这种环境下会产生 RLP 值的实验报告)。
Given the dependence of the lowest achievable density on the characteristics of the experiment, we need to clarify the goal of this paper. From the physical perspective it is interesting that, for any fixed coefficient of friction and fixed relative density between the grains and background fluid, there seems to be a sharply defined lowest volume fraction achievable by bulk manipulation.
It is possible furthermore that by suitably varying the coefficient of friction and relative density there is a single lowest possible volume fraction (currently believed to be about $0.55$ [9]); we expect that this is the case, and that this has a simple geometrical interpretation in terms of ensembles of frictional hard spheres under gravity, as discussed above.
This was the motivation of this work, and it is supported by simulations of our model. Our results suggest that whatever the initial local volume fraction of the fluidized granular bed, on sedimentation (in low effective gravity) most samples would have a well-defined volume fraction, the random loose packing density, with no intrinsic lower bound on the standard deviation of the distribution of volume fraction.
鉴于可达到的最低密度取决于实验的特性，我们需要明确本文的目标。从物理角度来看，有趣的是，对于任何固定的摩擦系数和固定的颗粒与背景流体之间的相对密度，似乎都有一个明确定义的可通过批量操作实现的最低体积分数。
此外，通过适当改变摩擦系数和相对密度，有可能存在一个单一的最低体积分数（目前认为约为 $0.55$ [9]）；我们希望情况确实如此，而且如上所述，这在重力作用下的摩擦硬球集合体方面有一个简单的几何解释。
这正是这项工作的动机所在，我们的模型模拟也证明了这一点。我们的研究结果表明，无论流化颗粒床的初始局部体积分数是多少，在沉积过程中（在低有效重力条件下），大多数样品都会有一个明确的体积分数，即 RLP 密度，且体积分数分布的标准偏差 没有内在下限。
There have been previous probabilistic interpretations of the random loose packing density, for instance, as well as the recent mean field model of Song et al. [20, 22]. A distinguishing feature of our results is our analysis of the degree of sharpness of the basic notion, which, as we shall see below, requires unusual care in the treatment of error analysis. In summary, we have performed Markov chain Monte Carlo simulations on a two dimensional model of low pressure granular matter of the general Edwards probabilistic type.
Our main result, superficially summarized in Fig. 8, is that in this model the standard deviation of the volume fraction decays to zero as the particle number increases, which indicates a well-defined random loose packing density for the model. This suggests that real granular matter exhibits sharply defined random loose packing; this could be verified by repeating sedimentation experiments at a range of physical dimensions.
Our argument is only convincing to the extent that the confidence intervals in Fig. 8 are small and justified, which required a statistical treatment of the data unusual in the physics literature. We hope that our detailed error analysis may be useful in other contexts.
例如，以前曾有过对随机松散堆积密度的概率解释，以及 Song 等人最近提出的平均场模型。我们结果的一个显著特点是我们对基本概念尖锐程度的分析，正如我们将在下文中看到的，这要求我们在处理误差分析时异常谨慎。总之，我们对一般 Edwards 概率型低压颗粒物质二维模型进行了马尔科夫链蒙特卡罗模拟。
图 8 从表面上概括了我们的主要结果，即在该模型中，随着粒子数的增加，体积分数的标准偏差衰减为零，这表明该模型具有定义明确的 RLP 密度。这表明真实的颗粒物质表现出清晰的随机松散堆积；这可以通过在一系列物理尺度下重复沉降实验来验证。
我们的论证只有在图 8 中的置信区间较小且合理的情况下才有说服力，这需要对数据进行物理学文献中少见的统计处理。我们希望我们的详细误差分析在其他情况下也能派上用场。
Analysis of Simulations
We performed Markov chain Monte Carlo simulations on our granular model, which we now describe more precisely. We begin with a fixed number of unit edge squares contained in a large rectangular box $B$. A collection of squares is “allowed” if they do not overlap with positive area, their edges are parallel to those of the box $B$, and the lower edge of each square intersects either the floor of the box $B$ or the upper edge of each of two other squares; see Fig. 3. Note that although the squares have continuous translational degrees of freedom in the horizontal direction, this is not in evidence in the vertical direction because of the stability condition: the squares inevitably appear at discrete horizontal “levels”.
我们对颗粒模型进行了马尔科夫链蒙特卡罗模拟，现在对其进行更精确的描述。首先，我们在一个长方形大盒子 $B$ 中设置了固定数量的单位方块。如果一组方块没有正面积重叠，它们的边与方框 $B$ 的边平行，并且每个正方形的下边与方框 $B$ 的底面或其他两个正方形的上边相交，那么这组正方形就是 “允许的”；见图 3。请注意，虽然方块在水平方向上有连续的平移自由度，但由于稳定性条件的限制，在垂直方向上却没有这种自由度：方块不可避免地出现在离散的 “水平面 “上。
Markov chain simulations were performed as follows. In the rectangular container $B$ a fixed number of squares are introduced in a simple “crystalline” configuration: squares are arranged equally spaced in horizontal rows, the spacing determined by a preassigned volume fraction $\phi$ , and with squares centered above the centers of the gaps in the row below it; see Fig. 4. The basic step in the simulation is the following.
A square is chosen at random from the current configuration and all possible positions are determined to which it may be relocated and produce an allowed configuration. Note that if the chosen square supports a square above it then it can only be allowed a relatively small horizontal motion; otherwise it may be placed atop some pair of squares, or the floor.
马尔可夫链模拟如下。在矩形容器 $B$ 中，以简单的 “结晶” 构型引入固定数量的方块：方块等间距排列成水平行，间距由预先分配的体积分数 $\phi$ 决定，方块的中心位于其下一行间隙中心的上方；见图 4。模拟的基本步骤如下。
从当前构型中随机选择一个方格，然后确定它可以移动到的所有可能位置，并生成允许的构型。需要注意的是，如果被选中的方格支撑着上方的一个方格，那么只能允许它做相对较小的水平移动；否则，它可能会被放置在一对方格或地板上。
So the boundary of the configuration plays a crucial role in the ability of the chain to change the volume fraction. In any case the positions to which the chosen square may be moved constitute a union of intervals. A random point is selected from this union of intervals and the square is moved. The random movement of a random square is the basic element of the Markov chain.
It is easy to see that this protocol is transitive and satisfies detailed balance, so the chain has the desired uniform probability distribution as its asymptotic state. See Fig. 5 for a configuration of $399$ squares after $10^{6}$ moves. Our interest is in random loose packing, which occurs in the top (bulk) layer of a granular pile, and we assume that the entirety of each of our configurations represents this top layer.
We emphasize that our protocol is not particularly appropriate for studying other questions such as the statistical shape of the boundary of a granular pile, or properties associated with high volume fraction, such as random close packing.
因此，构型的边界对链条改变体积分数的能力起着至关重要的作用。在任何情况下，所选方块可移动的位置都是一个区间的集合。从这个区间的集合中随机选择一个点，然后移动方块。随机方块的随机移动是马尔科夫链的基本要素。
不难看出，该协议是传递性的，并且满足细致平衡，因此该链的渐近状态就是我们需要的均匀概率分布。经过 $10^{6}$ 次移动后，$399$ 个方格的构型见图 $5$。我们感兴趣的是随机松散堆积，它发生在颗粒堆的顶层（散装），而我们假定每个构型的整体都代表这个顶层。
Protocol 可以在上面的阐述中充分展示其含义.
Detailed Balance(细致平衡)
即一个系统处于平衡态 $i$ 的概率为 $\pi(i)$, 系统从平衡态 $i$ 转变为 $j$ 的概率为 $P_{i\rightarrow j}$, 那么就有细致平衡条件:
$$ \pi(i)P_{i\rightarrow j} = \pi(j)P_{j\rightarrow i} $$
After a prescribed number of moves, a volume fraction is computed for the collection of squares as follows. Within horizontal level $L_{j}$, where $j=0$ corresponds to the squares resting on the floor, the distances between the centers of neighboring squares is computed. (Such a distance is $1+g$ where $g$ is the gap between the squares.) Suppose that $n_{j}$ of these neighboring distances are each less than $2$, and that the sum of these distances in the level is $s_{j}$.
At this point our procedure will be complicated by the desire to obtain information during the simulation about inhomogeneities in the collection, for later use in analyzing the approach to equilibrium. For this purpose we introduce a new parameter, $p$.
经过规定次数的移动后，方块集合的体积分数计算如下。在水平水平面 $L_{j}$（其中 $j=0$ 对应于静止在地面上的方格）内，计算相邻方格中心之间的距离。(假设这些相邻距离中的 $n_{j}$ 都小于 $2$，则水平面上这些距离的总和为 $s_{j}$。
For fixed $0 < p < 1$ we consider those levels, beginning from $j = 0$, for which $n_{j}$ is at least $0.75p$ times the length of the box’s floor. Suppose $L_{J(p)}$ is the highest level such that it, and all levels below it, satisfy the condition. We then assign the volume fraction
$$ \phi(p) = \frac{\sum_{j=0}^{J(p)}{n_{j}}}{\sum_{j=0}^{J(p)}{s_{j}}} $$
to the assembly of squares. (The factor $0.75$ represents the volume fraction we expect the box’s floor to reach in equilibrium. Note that any two such calculations of volume fraction of the same configuration may only differ by a term proportional to the length of the boundary of the configuration, so any inhomogeneity is limited to this size.)
对于固定的 $0 < p < 1$，我们考虑从 $j = 0$ 开始的那些层，其中 $n_{j}$ 至少是盒子底板长度的 $0.75p$ 倍。假设 $L_{J(p)}$ 是最高的层，它和它下面的所有层都满足条件。然后我们分配体积分数
$$ \phi(p) = \frac{\sum_{j=0}^{J(p)}{n_{j}}}{\sum_{j=0}^{J(p)}{s_{j}}} $$
来组装正方形。(系数 $0.75$ 代表我们期望盒子底板在平衡状态下达到的体积分数。需要注意的是，对同一构型的体积分数进行的任何两次计算，其差异可能仅与构型边界长度成正比，因此任何不均匀性都仅限于此大小)。
Such a calculation of volume fraction was performed regularly, after approximately $10^{6}$ moves, producing a time series of volume fractions $\phi_{t}$ for the given number of squares. (We suppress reference to the variable $p$ for ease of reading. As will be seen later all our results correspond to the choice $p = 0.4$, so one can, without much loss, ignore other possible values.) Variables $\phi_{t}$ and $\phi_{t+1}$ are highly dependent, but we can be guaranteed that if the series is long enough then the sample mean:
$$ \frac{1}{N}\sum_{t=1}^{N}\phi_{t} $$
will be a good approximation to the true mean of the target (uniform) probability distribution for the given number of squares.
这样的体积分数计算是定期进行的，大约每走 $10^{6}$ 步，就会产生给定格数的体积分数时间序列 $\phi_{t}$。(为了阅读方便，我们不再提及变量 $p$。稍后我们将看到，我们的所有结果都与选择 $p = 0.4$ 相对应，因此我们可以忽略其他可能的值，而不会有太大损失)。变量 $\phi_{t}$ 和 $\phi_{t+1}$ 是高度依赖的，但我们可以保证，如果序列足够长，那么样本平均数：
$$ \frac{1}{N}\sum_{t=1}^{N}\phi_{t} $$
将是给定方格数下目标（均匀）概率分布真实平均值的良好近似值。
