Abstract

Formulating a statistical mechanics for granular matter remains a significant challenge, in part due to the difficulty associated with a complete characterization of the systems under study.

We present a fully characterized model of a granular material consisting of $N$ two-dimensional, frictionless hard disks, confined between hard walls, including a complete enumeration of all possible jammed structures. We show that the properties of the jammed packings are independent of the distribution of defects within the system and that all the packings are isostatic.

This suggests that the assumption of equal probability for states of equal volume, which provides one possible way of constructing the equivalent of a microcanonical ensemble, is likely to be valid for our model.

An application of the second law of thermodynamics involving two subsystems in contact shows that the expected spontaneous equilibration of defects between the two is accompanied by an increase in entropy and that the equilibrium, obtained by entropy maximization, is characterized by the equality of compactivities.

Finally, we explore the properties of the equivalent to the canonical ensemble for this system.

制定颗粒物质的统计力学仍然是一项重大挑战，部分原因是很难对所研究的系统进行完整的表征。

我们提出了一个由 $N$ 个二维无摩擦硬盘组成的颗粒材料的全特性模型，这些硬盘被限制在硬壁之间，其中包括对所有可能的堵塞结构的完整枚举。我们的研究表明，堵塞堆积的特性与系统内的缺陷分布无关，而且所有堆积都是等静压的。

这表明，体积相等的状态概率相等的假设(这是构建等效于微观规范集合的一种可能方法)很可能对我们的模型有效。

涉及两个接触子系统的热力学第二定律的应用表明，两者之间缺陷的预期自发平衡伴随着熵的增加，而通过熵最大化获得的平衡具有紧凑性相等的特点。

最后，我们探讨了该系统等效于经典团簇的特性。

Introduction

Granular materials such as sand are too massive to be influenced by the thermal fluctuations that affect the motion of particles on the atomic scale. If left undisturbed, a sand pile remains in a rigid, solidlike structure, but when poured, sand flows just like a fluid.

Foams, made up of a collection of macroscopic bubbles, can support a yield stress like a solid but they flow like a fluid when sheared above a certain threshold rate. In both cases, the particle-particle structure and particle dynamics are similar to that observed in atomic and molecular liquids and glasses, suggesting that many athermal systems may exhibit a wide variety of physical phenomena that were originally thought to occur only in thermal systems.

While there have been a number of efforts to capture the connection between the different systems, including the formulation of a phase diagram for jammed and glassy materials, we do not have an underlying and general theory to describe granular matter.

The machinery of classical thermodynamics and thermal statistical mechanics, which provides us with the tools for the study of complex behavior in molecular systems, can not be used to explain the behavior of a granular system because the energy required to move a macroscopic particle far exceeds that available from thermal excitation.

Consequently, there is an extensive effort to develop a form of statistical mechanics that can account for the properties of athermal systems.

沙子等颗粒状材料的质量太大，不会受到热波动的影响，而热波动会影响原子尺度上的粒子运动。如果不加扰动，沙堆会保持坚硬的固态结构，但如果将其倾倒，沙子就会像流体一样流动。

由宏观气泡集合而成的泡沫可以像固体一样承受屈服应力，但当剪切速率超过一定临界速率时，泡沫会像流体一样流动。在这两种情况下，粒子与粒子之间的结构和粒子动力学与在原子液体、分子液体和玻璃中观察到的相似，这表明许多非热(平衡)系统可能表现出各种各样的, 原本被认为只发生在热(平衡)系统中的物理现象.

虽然我们已经做了很多努力来捕捉不同系统之间的联系，包括绘制堵塞材料和玻璃态材料的相图，但我们还没有一个基本的通用理论来描述颗粒物质。

经典热力学和热统计力学机制为我们提供了研究分子系统复杂行为的工具，但却不能用来解释粒状系统的行为，因为移动一个宏观粒子所需的能量远远超过热激发所能提供的能量。

因此，人们正在努力开发一种能够解释热系统特性的统计力学形式。

An appropriate starting point for a statistical mechanics of granular systems is to develop a probability distribution function that is equivalent to the microcanonical ensemble where all the states of the same energy are assumed to have an equal probability of being sampled.

One possibility, suggested by Edwards, is to assume that all jammed states of equal volume have equal probabilities so that the probability of finding a given configuration $i$ is

$$ P_{i} = e^{-S/\lambda}\delta(V - W_{i})\Theta, $$

where $V$ is the volume, $W_{i}$ is the Hamiltonian-type volume function for the configuration, $S$ is the entropy, and $\lambda$ is the analog of the Bolzmann constant. The normalization in Eq. (1) is

$$ e^{S/\lambda} = \int \delta(V - W_{i})\Theta \mathrm{\textbf{d}}(\text{All degrees of freedom}) $$

and

$$ \Theta = \begin{cases} 1 & \text{if collectively jammed}\\ 0 & \text{otherwise} \end{cases} $$

颗粒系统统计力学的一个适当起点是建立一个概率分布函数，该函数等同于微正则系综，其中假定具有相同能量的所有状态被采样的概率相等。

Edwards 提出的一种可能性是，假设所有 体积 相等的堵塞状态具有相等的概率，因此找到给定构型 $i$ 的概率为

$$ P_{i} = e^{-S/\lambda}\delta(V - W_{i})\Theta, $$

其中，$V$ 是体积，$W_{i}$ 是构型的汉密尔顿式体积函数，$S$ 是熵，$\lambda$ 是类似的博尔兹曼常数。公式 (1) 中的归一化为

$$ \Theta = \begin{cases} 1 & \text{集体堵塞}\\ 0 & \text{otherwise} \end{cases} $$

Evaluating the integral in Eq. (2) and taking the natural log of both sides defines the entropy as

$$ S/\lambda = \ln\Omega(V), $$

where $\Omega(V)$ is the number of collectively jammed configurations with a given $V$.

$\Omega(V)$ grows exponentially with the number of particles $N$, so entropy is extensive and the compactivity, which is equivalent to the temperature in a thermal system, can be defined as

$$ X = \frac{\partial V}{\partial S}. $$

对公式 (2) 中的积分求值并取两边的自然对数，熵的定义为

$$ S/\lambda = \ln\Omega(V), $$

其中，$\Omega(V)$ 是给定 $V$ 的集体堵塞构型的数量。

$Omega(V)$ 随着粒子数 $N$ 的增加而呈指数增长，因此熵是广延量. 紧致度（相当于热系统中的温度）可定义为

$$ X = \frac{\partial V}{\partial S}. $$

这个量在其他论文里也被 $\chi$ 或者其它字母进行标记. 理解其原始定义.

In general, $\Omega(V)$ is not known. One difficulty is that little is known about how particles share volume in jammed materials, and while a number of different tessellation methods have been suggested, the construction of W remains a significant challenge.

Furthermore, the global nature of the collective jamming condition does not allow us to determine which configurations are jammed on the basis of the local geometric properties of the individual particles. The effects of friction and stress in jammed configurations add further complications, leading to the proposal of alternative distribution functions.

一般来说，$\Omega(V)$ 是未知的。其中一个困难是，人们对粒子如何在堵塞材料中分配体积知之甚少，虽然已经提出了许多不同的细分方法，但 $W$ 的构造仍然是一个巨大的挑战。

$W_{i}$ 是构型的汉密尔顿式体积函数.

此外，由于集体堵塞条件的全局性，我们无法根据单个粒子的局部几何特性来确定哪些构型被堵塞。干扰配置中的摩擦和应力效应进一步增加了复杂性，因此我们提出了其他分布函数。

A numerical simulation of a slowly, sheared granular material suggests that a thermodynamic temperature, related to compactivity, can be extracted from particle mobility measurements, but simulations of frictionless hard disks appear to cast doubt on the assumption that packings of equal volume are equiprobable.

Recent experiments on externally agitated granular systems found that, while compactivity did not equilibrate between two subsystems, both subsystems shared a number of reproducible properties, strongly supporting the notion that a thermodynamic approach to granular materials is a real possibility, even though we do not have a complete picture of how this should be developed.

对一种缓慢剪切的颗粒材料进行的数值模拟表明，可以从颗粒流动性测量结果中提取出与致密性有关的热力学温度，但对无摩擦硬盘的模拟似乎使人对等体积堆积可等效的假设产生怀疑。

最近在外力搅拌颗粒系统上进行的实验发现，虽然两个子系统之间的致密性并不平衡，但两个子系统共享一些可重现的特性，这有力地支持了一种观点，即(建立)颗粒材料的热力学方法具有真正的可能，尽管我们还不完全了解应该如何开发这种方法。

However, for all the systems studied to date, the distribution function and a detailed knowledge of the particle packings is lacking. The goal of this paper is to examine the statistical mechanics of a simple model granular system that has the potential to be examined experimentally and for which the full density of states can be obtained exactly.

We analyze the distribution of jammed packings for a system of highly confined hard disks in Sec. II, and show that the assumption of equal probability for states of equal volume is likely to be valid for this model. In Sec. III, we examine the predictions of the second law of thermodynamics by considering the equilibrium state obtained by bringing together two isolated systems into the granular equivalent of thermal contact. The canonical ensemble is studied in Sec. IV, while Sec. V contains our discussion and conclusions.

然而，迄今为止研究的所有系统都缺乏分布函数和颗粒堆积的详细知识。本文的目的是研究一个简单模型颗粒系统的统计力学，该系统有可能通过实验进行研究，并且可以精确获得其全部状态密度。

在 Sec. II 中，我们分析了一个高度密闭硬盘系统的堵塞堆积分布，并证明等体积状态的等概率假设很可能对该模型有效。在 Sec. III 中，我们通过考虑将两个孤立系统结合到热接触的颗粒等效物中所得到的平衡状态，来研究热力学第二定律的预测。Sec. IV 研究了典型系综，Sec. V 包含我们的讨论和结论。

PACKINGS OF HIGHLY CONFINED HARD DISKS

Our model granular material consists of $N$ two-dimensional (2D) frictionless hard disks, with diameter $\sigma$ , trapped between two hard walls separated by a distance $H/\sigma < 1 + \sqrt{3/4}$, and we will ignore the effects of gravity. The density of states for this system was previously described in Ref. [16] and is included here for the sake of completeness and to highlight the fact that the distribution of packings for this system can be obtained using simple combinatorial arguments.

In addition, we also show here that all the packings of the system are isostatic and that packings of the same volume are all structurally equivalent.

我们的模型颗粒材料由 $N$ 个直径为 $\sigma$ 的二维（2D）无摩擦硬盘组成，它们被困在两面硬壁之间，两面硬壁之间的距离为 $H/\sigma < 1 + \sqrt{3/4}$，我们将忽略重力的影响。该系统的状态密度先前在参考文献[16]中描述过。[16]中已有描述，这里将其包括在内是为了完整起见，并强调这一系统的堆积分布可以通过简单的组合论证得到。

此外，我们还在此表明，该系统的所有堆积都是等(静)压的，相同体积的堆积在结构上都是等价的。

In 2D, a particle is locally jammed if it has at least three rigid contacts that are not all in the same semicircle. However, while local jamming of all particles is a necessary requirement for a collectively jammed state, it is not sufficient because the concerted motion of a group of particles can cause a packing to collapse.

By confining the disks between walls separated by $H/\sigma < 1 + \sqrt{3/4}$, particles are only able to contact their nearest neighbor on each side of the wall, so there are only two local packing environments.

In addition, the confinement prevents collective motions that would allow the locally jammed structures to collapse, and the complete distribution of collectively jammed states can be obtained from local geometric considerations alone. Increasing the channel diameter $H/\sigma > 1 + \sqrt{3/4}$ allows additional second neighbor contacts leading to a more complicated density of jammed states.

在二维空间中，如果一个粒子至少有三个不在同一半圆内的刚性接触点，则该粒子局部堵塞。然而，虽然所有粒子的局部受阻是集体堵塞状态的必要条件，但这还不够，因为一组粒子的协同运动会导致堆积坍塌。

通过将磁盘限制在间隔为 $H/\sigma < 1 + \sqrt{3/4}$ 的墙壁之间，粒子只能与墙壁两侧的近邻接触，因此只有两种局部堆积环境。

此外，束缚阻止了集体运动，而集体运动会使局部堵塞结构坍缩，因此仅从局部几何因素就可以得到集体干扰态的完整分布。增大通道直径 $H/\sigma > 1 + \sqrt{3/4}$ 可以增加第二阶相邻接触，从而导出更复杂的堵塞态密度。

Figure 1(a) shows a jammed configuration containing the two possible local disk arrangements. The most dense state involves two particles contacting across the channel, while the defect state consists of two neighboring particles contacting along the channel, which results in a less dense arrangement.

Both the most dense local arrangement and the defect have well-defined volumes $v_{0} = H\sqrt{(2\sigma-H)H}$ and $v_{1} = H\sigma$, respectively, that are additive, so the volume function $W$ can be written as

$$ \begin{aligned} V &= W = (N - M)v_{0} + Mv_{1}\\ &= NH[\sqrt{(2-H)H}(1-\theta) + \theta] \end{aligned} $$

where $M$ is the number of defects in the system, $\theta = M/N$ is the fraction of defects, and $\sigma$ is the unit of length. The occupied volume fraction for a packing is $\phi_{J} = \pi/\{4H[\sqrt{(2-H)H}(1-\theta) + \theta]\}$.

图 1(a) 显示了包含两种可能的局部圆盘排列的堵塞构型。最密集的状态是两个粒子穿过通道接触，而缺陷状态则是两个相邻的粒子沿着通道接触，从而形成密度较低的排列。

最密集的局部排列和缺陷都有定义明确的体积，分别是 $v_{0} = H\sqrt{(2\sigma-H)H}$ 和 $v_{1} = H\sigma$ ，它们是相加的，因此体积函数 $W$ 可以写成

$$ \begin{aligned} V &= W = (N - M)v_{0} + Mv_{1}\\ &= NH[\sqrt{(2-H)H}(1-\theta) + \theta] \end{aligned} $$

其中 $M$ 是系统中缺陷的数量，$\theta = M/N$ 是缺陷分数，$\sigma$ 是长度单位。堆积的占位体积分数为 $\phi_{J} = \pi/{4H[\sqrt{(2-H)H}(1-\theta) + \theta]}$。

To count the number of packings with a given $V$, we develop a lattice gas, or Ising-model-like, description of each configuration of disks by drawing a bond between the centers of neighboring disks and labeling the defect bonds as “$1$” and the most dense bonds as “$0$” (see Fig. 1).

This means that the total number of bonds equals $N$. Following Hill, if $M$ is the number of defect bonds, we can divide a configuration into blocks consisting of consecutive “ones” in the chain and blocks consisting of consecutive “zeros” in the chain.

为了计算具有给定 $V$ 的堆积数，我们在相邻圆盘的中心之间绘制了一个键，并将缺陷键标记为"$1$"，将最密集的键标记为"$0$"，从而对每个圆盘构型进行了类似于晶格气体或伊辛模型的描述（见图 1）。

这意味着键的总数等于 $N$。按照 Hill 的说法，如果 $M$ 是缺陷键的数量，我们可以把一个构型分为由链中连续的 “1 “组成的块和由链中连续的 “0 “组成的块。

A block of ones is necessarily separated from a block of zeros by a boundary consisting of a $1–0$ or $0–1$ bond. The total number of configurations with $M$ defects and $M_{01}$ boundaries is obtained by considering the number of different ways of arranging $M$ ones among the $(M_{01} + 1)/2$ possible block of ones, such that there is at least one “$1$” in a block, and $(N − M)$ zeros among the $(M_{01} + 1)/2$ possible block of zeros. This gives

$$ \Omega_{J}(M,M_{01}) = \frac{M!(N-M)!}{[M-\frac{M_{01}}{2}]![N-M-\frac{M_{01}}{2}]![\frac{M_{01}}{2}!]^{2}} $$

in the limit of large numbers.

“$1$ 块” 必然由 $1-0$ 或 $0-1$ 键组成的边界与 “$0$ 块” 分开。具有 $M$ 个缺陷和 $M_{01}$ 个边界的构型总数是通过考虑在 $(M_{01} + 1)/2$ 个可能的 $1$ 块中排列 $M$ 个 $1$ 的不同方式，使得至少有一个 “1” 在一个块中，以及在 $(M_{01} + 1)/2$ 个可能的 $0$ 块中排列 $(N-M)$ 个 $0$ 而得到的。这表明在大数极限中,

$$ \Omega_{J}(M,M_{01}) = \frac{M!(N-M)!}{[M-\frac{M_{01}}{2}]![N-M-\frac{M_{01}}{2}]![\frac{M_{01}}{2}!]^{2}} $$

这是一个高中生都能解决的排列组合问题.

The volume of a packing of $N$ disks only depends on $\theta = M/N$ and is independent of $M_{01}$. If the defects could be distributed totally randomly throughout the system, then Eq. (7) would reduce to the usual Ising model expression $\Omega_{J}(V) = N!/M!(N-M)!$.

However, not all defect arrangements result in truly jammed states. When two defects appear next to each other in a “$1-1$” arrangement, the central particle does not satisfy the local jamming condition [Fig. 1(b)]. This arrangement represents a saddle point in configuration space that will collapse if the central particle is perturbed in a direction normal to the wall.

These configurations can be eliminated from the distribution by setting $M_{01} = 2M$, which ensures that every defect is isolated from the other defects and gives

$$ \Omega_{J}(V) = \frac{(N-M)!}{M!(N-M)!}. $$

由 $N$ 磁盘组成的堆积体的体积只取决于 $\theta = M/N$，而与 $M_{01}$ 无关。如果缺陷可以完全随机地分布在整个系统中，那么公式 (7) 将简化为通常的伊辛模型表达式 $\Omega_{J}(V) = N!/M!(N-M)!$.

然而，并非所有的缺陷排列都会导致真正的堵塞态。当两个缺陷以 “$1-1$“的排列方式相邻出现时，中心粒子并不满足局部堵塞条件[图 1(b)]。这种排列代表了构型空间中的一个鞍点，如果中心粒子沿壁的法线方向受到扰动，它就会坍塌。

通过设定 $M_{01} = 2M$，可以从分布中消除这些(不稳定的)构型，从而确保每个缺陷都与其他缺陷隔离，并得出

$$ \Omega_{J}(V) = \frac{(N-M)!}{M!(N-M)!}. $$

The distribution of packings is binomial with a single most dense structure, containing no defects, a single least dense structure with $\theta = 0.5$, and a a maximum number of packings when $\theta = \frac{1}{2} - \frac{\sqrt{5}}{10}$.

堆积的分布是二项式的，其中包含了一个最密集结构(不含缺陷)，还有一个在 $\theta = 0.5$ 时的最不密集结构，而当 $\theta = \frac{1}{2} - \frac{\sqrt{5}}{10}$ 时堆积的数量最多。

注意这样一个事实: 堆积得密和堆积得多是两个概念!

Mechanical equilibrium is an important property of granular packings that can be understood in terms of a balance between the total degrees of freedom in a system and the number of force equations that constrain or counteract them.

In our system, each disk has two translational degrees of freedom, resulting in a total of $2N$ degrees of freedom. Each disk in a jammed packing contacts two other disks and the wall, which results in a total of $N$ disk–disk force equations, noting each contact involves two disks, and $N$ disk–wall force equations, giving a total of $2N$ equations.

The number of degrees of freedom is exactly balanced by the number of force equations, which shows that all the jammed packings for this model are isostatic, independent of the number of defects. Furthermore, since each particle only contacts one of its neighbors on either side, it does not feel any interaction from particles beyond its nearest neighbors, so there is no interaction between defect states.

As a result, the properties of the collectively jammed packings, with the same volume, should be independent of how the defects are distributed, i.e., there is no distinction between a packing with an ordered arrangement of defects compared to one with a more random arrangement. These properties strongly suggest that the assumption of equal probability for states of equal volume is valid for the current model.

机械平衡是颗粒堆积的一个重要特性，可以从系统的总自由度与约束或抵消这些自由度的力方程数量之间的平衡来理解。

在我们的系统中，每个圆盘有两个平移自由度，因此共有 $2N$ 个自由度。在堵塞堆积中，每个圆盘都会接触到其他两个圆盘和盘壁，这就产生了 $N$ 的圆盘-圆盘力方程（注意每个接触都涉及两个圆盘）和 $N$ 的圆盘-盘壁力方程，总共产生了 $2N$ 的方程。

自由度的数量正好与力方程的数量相平衡，这表明该模型的所有堵塞堆积都是等静态的，与缺陷的数量无关。此外，由于每个粒子只与其两侧的一个相邻粒子接触，它不会感受到来自其近邻粒子之外的任何相互作用，因此缺陷态之间不存在相互作用。

注意区别 isostatic(等静压) 和 quasi-static(准静态), 两者所指的概念完全不同.

因此，具有相同体积的集体堵塞堆积的性质应该与缺陷的分布方式无关，也就是说，缺陷有序排列的堆积与随机排列的堆积之间没有区别。这些特性有力地表明，等体积状态的等概率假设在当前模型中是有效的。

THE SECOND LAW OF THERMODYNAMICS

The second law in classical thermodynamics postulates the existence of entropy as a state function in order to understand the driving force behind spontaneous processes. Furthermore, it states that $\mathrm{d}S\geq 0$ for any spontaneous process in an isolated system so that the entropy is maximized when the system is in equilibrium.

The most widely used example of the second law involves bringing two metal bars at different initial temperatures, say $T_{1} < T_{2}$, into thermal contact. If the composite system is isolated, then experience (experiment) tells us that heat will flow from the hot bar into the cold bar until it comes to equilibrium and the temperature in each bar is the same.

为了理解自发过程背后的驱动力，经典热力学第二定律假定了熵作为一种态函数的存在。此外，该定律还指出，对于一个孤立系统中的任何自发过程都有 $\mathrm{d}S\geq 0$, 使得系统处于平衡状态时的熵达到最大。

第二定律最广泛应用的例子是将两根初始温度不同的金属棒（例如 $T_{1} < T_{2}$ ）进行热接触。如果这个复合系统是孤立的，那么经验（实验）告诉我们，热量将从热棒流入冷棒，直到达到平衡，每根金属棒的温度相同。

We can construct an analogous experiment with our granular system as follows: First, we note that, on the basis of Eqs. (4) and (8), the entropy of the system with a given volume is

$$ \frac{S}{N\lambda} = (1 - \theta)\ln{(1-\theta)} - (1-2\theta)\ln{(1-2\theta)} - \theta\ln{\theta}, $$

and the compactivity is

$$ \begin{aligned} X &= \frac{\partial V}{\partial} = \frac{\partial V}{\partial \theta}\frac{\partial \theta}{\partial S}\\ &=\frac{H[\sqrt{(2-H)H} - 1]}{\ln{(1-\theta)} + \ln{\theta} - 2\ln{(1-2\theta)}}, \end{aligned} $$

where we have made use of the derivatives of Eqs. (6) and (9) with respect to $\theta$. In the lower limit, $X\rightarrow 0$ as $\theta\rightarrow 0$, i.e., when the system is in its most dense state. In the upper limit, $X\rightarrow \infty$ as $\theta\rightarrow 1/2-\sqrt{5}/10$ which occurs at the maximum in the distribution of jammed packings. Packings with higher concentrations of defects are not sampled in equilibrium because these would give rise to negative compactivities.

我们可以用颗粒系统构建一个类似的实验，具体如下：首先，根据公式 (4) 和 (8)，给定体积系统的熵为

$$ \frac{S}{N\lambda} = (1 - \theta)\ln{(1-\theta)} - (1-2\theta)\ln{(1-2\theta)} - \theta\ln{\theta}, $$

紧致度为

$$ \begin{aligned} X &= \frac{\partial V}{\partial} = \frac{\partial V}{\partial \theta}\frac{\partial \theta}{\partial S}\\ &=\frac{H[\sqrt{(2-H)H} - 1]}{\ln{(1-\theta)} + \ln{\theta} - 2\ln{(1-2\theta)}}, \end{aligned} $$

其中我们使用了公式 $(6)$ 和 $(9)$ 对于 $\theta$ 的导数。

在下限，当系统处于最密集状态时，$X\rightarrow 0$为$\theta\rightarrow 0$;

在上限，$X\rightarrow \infty$ 为 $\theta\rightarrow 1/2-\sqrt{5}/10$，这发生在堵塞堆积分布的最大值处。

在平衡状态下不会对缺陷浓度较高的堆积进行采样，因为这些堆积会产生负的紧致度。

To prepare two subsystems of $N_{1}$ and $N_{2}$ disks at compactivities $X_{1}$ and $X_{2}$, respectively, we must, in principle, place each subsystem in contact with a “thermal” reservoir with which it can exchange volume until equilibrium is reached.

However, since both the volume and compactivity are functions of the fraction of defects, it is sufficient to select starting configurations with the initial defect concentrations $\theta_{1}’$ and $\theta_{2}’$, which correspond to some initial conditions with $X_{1} \neq X_{2}$.

We can now bring the two subsystems into contact so that they can exchange volume. This can be achieved by placing the last particle of system one and the first particle of system two in contact so that the configuration remains jammed.

In the context of this experiment, an isolated system is one that can not exchange volume with an external reservoir, so the composite system has a fixed total volume $V_{T} = V_{1} + V_{2}$, where we use large enough system sizes that the small volumes associated with the end effects can be ignored.

要制备两个分别由紧致度为 $X_{1}$ 和 $X_{2}$ 的 $N_{1}$ 和 $N_{2}$ 磁盘组成的子系统，原则上我们必须让每个子系统都与一个 “热 “库接触，并与之交换体积，直到达到平衡。

然而，由于体积和密实度都是都是缺陷分数($\theta$)的函数，因此只需选择初始缺陷浓度为 $\theta_{1}’$ 和 $\theta_{2}’$ 的起始配置即可，它们对应于一些初始条件，即 $X_{1}\neq X_{2}$。

现在，我们可以让两个子系统接触，让它们交换体积。这可以通过将【系统一】的最后一个粒子和【系统二】的第一个粒子接触来实现，从而使构型保持堵塞状态。

在本实验中，孤立系统是指不能与外部库交换体积的系统，因此复合系统有一个固定的总体积 $V_{T} = V_{1} + V_{2}$; 我们使用的系统体积足够大，因此可以忽略与末端效应相关的小体积。

The challenge of dealing with a realistic set of dynamics, such as shaking, will be addressed in more detail in our discussion but, in the meantime, we need to use an idealized dynamics that will allow the system to jump from jammed state to jammed state under the conditions of fixed volume.

As the system is perturbed, we expect it to move between jammed states by a series of random local particle rearrangements that will require the movement of defects.

If all the particles are made from the same material, our physical intuition tells us that the defects will spontaneously move from a region of high concentration to a region of low concentration, until eventually the system comes to equilibrium and the defects are, on average, equally distributed throughout.

Since both subsystems are made from the same material and have the same size, a uniform distribution of defects also corresponds to a uniform compactivity.

我们将在讨论中更详细地探讨如何处理一组现实的动力学问题，例如晃动，但与此同时，我们需要使用一种理想化的动力学，使系统能够在体积固定的条件下从某一堵塞态跃变为另一堵塞态。

当系统受到扰动时，我们预计它将通过一系列随机的局部粒子重排，在堵塞态之间切换，而这需要缺陷的移动。

如果所有粒子都由相同的材料制成，我们的物理直觉告诉我们，缺陷会自发地从高浓度区域向低浓度区域移动，直到最终系统达到平衡，缺陷平均地分布在整个系统中。

这里的物理直觉是基于我们对于热平衡的朴素理解移植到颗粒物理中, 但是有什么严格化的方法吗?

由于两个子系统由相同的材料制成，具有相同的尺寸，因此均匀的缺陷分布也对应于均匀的紧致度。

Application of the second law of thermodynamics implies that the equilibrium distribution of defects can be obtained by maximizing the total entropy of the composite system $S_{T} = S_{1}(\theta_{1}) + S_{2}(\theta_{2})$. The conservation of volume gives

$$ \theta_{2} = \frac{N_{1}(\theta_{1}’-\theta_{1}) + N_{2}\theta_{2}’}{N_{2}} $$

which can be used in Eq. (9) to yield an expression for $S_{T}$ solely in terms of $\theta_{1}$ and the initial defect concentrations consistent with the starting compactivities for each subsystem.

热力学第二定律的应用意味着缺陷的平衡分布可以通过最大化复合系统的总熵 $S_{T} = S_{1}(\theta_{1}) + S_{2}(\theta_{2})$ 来获得。体积守恒得出

$$ \theta_{2} = \frac{N_{1}(\theta_{1}’-\theta_{1}) + N_{2}\theta_{2}’}{N_{2}} $$

可用于式 (9) 以得出 $S_{T}$ 的表达式，该表达式仅以 $\theta_{1}$ 和与每个子系统的初始紧致度一致的初始缺陷浓度来表示。

The equilibrium value of $\theta_{1}$ in the composite system, obtained from $\mathrm{d}S_{T}/\mathrm{d}\theta_{1} = 0$, is then

$$ \theta_{1} = \frac{N_{1}\theta_{1}’ + N_{2}\theta_{2}’}{N_{1} + N_{2}} $$

This is just the average number of defects, or uniform distribution, which is consistent with our physical expectation. Furthermore, the compactivity in both subsystems is the same at equilibrium.

根据 $\mathrm{d}S_{T}/\mathrm{d}\theta_{1} = 0$，复合系统中 $\theta_{1}$ 的平衡值为

$$ \theta_{1} = \frac{N_{1}\theta_{1}’ + N_{2}\theta_{2}’}{N_{1}+ N_{2}} $$

这正好是缺陷的平均数量，或者说是均匀分布，与我们的物理预期一致。此外，在平衡状态下，两个子系统的紧致度是相同的。

This model can also be used to study the effects of bringing different types of granular materials into contact. The simplest example to study is the case where a subsystem of $N_{1}$ disks with diameter $\sigma_{1}$ is brought into contact with a second subsystem of $N_{2}$ disks with diameter $\sigma_{2}$, under the conditions $\sigma_{1} < \sigma_{2}$ and $H/\sigma_{1} < 1 + \sqrt{3/4}$, which ensures that the interface between the two subsystems can still be made through a single nearest-neighbor contact. $V_{1}$ is given Eq. (6) with $N = N_{1}$ and the volume of subvolume $2$ is

$$ V_{2} = N_{2}H[\sqrt{(2\sigma_{2}- H)H}(1-\theta) + \sigma_{2}\theta]. $$

At a fixed $V_{\text{T}}$, we now have

$$ \theta_{2} = \theta_{2}’ - \frac{N_{1}(\theta_{1}-\theta_{1}’)[\sqrt{(2-H)H} - 1]}{N_{2}[\sqrt{2(\sigma_{2} - H)H} - \sigma_{2}]}, $$

which reduces to Eq. (11) as $\sigma_{2}\rightarrow \sigma_{1} = 1$ and can be used in Eq. (9) to provide an expression for $S_{2}$ in terms of $\theta_{1}$.

该模型还可用于研究不同类型颗粒材料接触的效果。最简单的例子是：在直径为 $\sigma_{1}$ 的条件下，一个由直径为 $\sigma_{1}$ 的 $N_{1}$ 圆盘组成的子系统与第二个由直径为 $\sigma_{2}$ 的 $N_{2}$ 圆盘组成的子系统接触, 并且条件是 $\sigma_{1} < \sigma_{2}$ 和 $H/\sigma_{1}< 1 + \sqrt{3/4}$，这确保了两个子系统之间的界面仍然可以通过单个最近邻接触实现。$V_{1}$ 由公式 (6) 得出，其中 $N = N_{1}$，子体积 $2$ 的体积为

$$ \theta_{2} = \theta_{2}’ - \frac{N_{1}(\theta_{1}-\theta_{1}’)[\sqrt{(2-H)H} - 1]}{N_{2}[\sqrt{2(\sigma_{2} - H)H} - \sigma_{2}]}, $$

在固定的 $V_{\text{T}}$，我们现在有

$$ \theta_{2} = \theta_{2}’ - \frac{N_{1}(\theta_{1}-\theta_{1}’)[\sqrt{(2-H)H} - 1]}{N_{2}[\sqrt{2(\sigma_{2} - H)H} - \sigma_{2}]}, $$

由于 $\sigma_{2}\rightarrow \sigma_{1} = 1$，可将其还原为式 (11)，并可用于式 (9) 以提供 $\theta_{1}$ 的 $S_{2}$ 表达式。

Figure 2 shows the volume and the entropy per particle for the two subsystems and the composite system with the initial conditions chosen so that $\theta_{1}’ = 0$ and $\theta_{2}’=0.27$, which corresponds to $X_{1} = 0$ and $X_{2} = \infty$, respectively.

At equilibrium, where $S_{\text{T}}$ is at a maximum, $X_{1} = X_{2} = 0.318$, but neither the volumes nor entropies of the two subsystems are equal because the change in volume associated with a defect is different in each subsystem as a result of the different particle sizes.

图 2 显示了在初始条件为 $\theta_{1}’ = 0$ 和 $\theta_{2}’=0.27$，分别对应 $X_{1} = 0$ 和 $X_{2} = \infty$ 的情况下，两个子系统以及复合系统每个粒子的体积和熵。

平衡时，$S_{\text{T}}$ 为最大值，$X_{1} = X_{2} = 0.318$，但两个子系统的体积和熵都不相等，因为粒子大小不同，每个子系统中与缺陷相关的体积变化也不同。

To show that the compactivities in the two subsystems are equal when $S_{T}$ is maximized at fixed $V_{T}$, we begin by assuming $X_{1}\neq X_{2}$ and show that this leads to a contradiction to the conservation of the total volume. From Eq. (10), we have

$$ X_{1} = \left(\frac{\partial V_{1}}{\partial \theta_{1}}\right)\left(\frac{\partial \theta_{1}}{\partial S_{1}}\right)\neq \left(\frac{\partial V_{2}}{\partial \theta_{2}}\right)\left(\frac{\partial \theta_{2}}{\partial \theta_{1}}\right)\left(\frac{\partial \theta_{1}}{\partial S_{2}}\right) = X_{2}, $$

where we have used the fact that $\theta_{2}$ is a function of $\theta_{1}$ at fixed $V_{\text{T}}$. Maximizing the total entropy yields

$$ \left(\frac{\partial S_{1}}{\partial \theta_{1}}\right) = -\left(\frac{\partial S_{2}}{\partial \theta_{1}}\right) $$

which, in combination with Eq. (15), gives

$$ \left(\frac{\partial V_{1}}{\partial \theta_{1}}\right)\neq -\left(\frac{\partial V_{2}}{\partial \theta_{2}}\right)\left(\frac{\partial \theta_{2}}{\partial \theta_{1}}\right) = - \left(\frac{\partial V_{2}}{\partial \theta_{1}}\right) $$

This completes the proof as this last inequality suggests that the volume lost from one system is not equal to the volume gained by the other as the number of defects in system 1 is varied, which violates the conservation of volume. Therefore, $X_{1} = X_{2}$ at equilibrium.

为了证明当 $S_{T}$ 在固定的 $V_{T}$ 下达到最大时，两个子系统的紧凑性是相等的，我们首先假设 $X_{1}\neq X_{2}$，并证明这会导致与总体积守恒的矛盾。根据公式 (10)，我们可以得出

$$ X_{1} = \left(\frac{\partial V_{1}}{\partial \theta_{1}}\right)\left(\frac{\partial \theta_{1}}{\partial S_{1}}\right)\neq \left(\frac{\partial V_{2}}{\partial \theta_{2}}\right)\left(\frac{\partial \theta_{2}}{\partial \theta_{1}}\right)\left(\frac{\partial \theta_{1}}{\partial S_{2}}\right) = X_{2}, $$

其中我们使用了 在 $V_{\text{T}}$ 为定值时, $\theta_{2}$ 是 $\theta_{1}$ 的函数这一事实。将总熵最大化可以得到

$$ \left(\frac{\partial S_{1}}{\partial \theta_{1}}\right) = -\left(\frac{\partial S_{2}}{\partial \theta_{1}}\right) $$

结合公式 (15) 可以得出

$$ \left(\frac{\partial V_{1}}{\partial \theta_{1}}\right)\neq -\left(\frac{\partial V_{2}}{\partial \theta_{2}}\right)\left(\frac{\partial \theta_{2}}{\partial \theta_{1}}\right) = - \left(\frac{\partial V_{2}}{\partial \theta_{1}}\right) $$

这就完成了证明，因为最后一个不等式表明，随着系统 $1$ 中缺陷数量的变化，一个系统损失的体积不等于另一个系统获得的体积，这违反了体积守恒定律。因此，在平衡状态下，$X_{1} = X_{2}$。

CANONICAL EQUILIBRIUM

The equivalent to the canonical ensemble in a thermal system is obtained by integrating over all possible degrees of freedom and providing a Boltzmann-type weight to each jammed state on the basis of its volume, which (for the present system) gives

$$ \begin{aligned} e^{-\frac{Y}{\lambda X}} &= \int \Theta e^{-W/\lambda X}\mathrm{d}(\text{all degrees of freedom})\\ &= \sum_{M=0}^{M=N/2}\Omega(M,N;V)e^{-W/\lambda X}\\ &= {e^{\frac{-v_{0}N}{\lambda X}}}_{2}F_{1}\left[\frac{1}{2}-\frac{N}{2}, -\frac{N}{2};-N;-4e^{\frac{v_{0}}{\lambda X}-\frac{H}{\lambda X}}\right] \end{aligned} $$

where $Y$ is the analog to the free energy, $_2F_{1}$ is a hypergenometric function, and the integral has been reduced to a sum over the discrete set of jammed states.

The free energy can now be used in conjunction with the equivalent thermodynamic machinery to obtain other thermodynamic variables. For example, in analogy with the Helmholtz free energy, given as

$$ Y = V - XS, $$

the entropy of the system can be obtained from

$$ S = -\left(\frac{\partial Y}{\partial X}\right)_{V}. $$

通过对所有可能的自由度进行积分，并根据每个堵塞态的体积为其提供一个波尔兹曼型权重，就可以得到 热(平衡)系统中的 类比于经典系综的 结果（对于本系统）

$$ \begin{aligned} e^{-\frac{Y}{\lambda X}} &= \int \Theta e^{-W/\lambda X}\mathrm{d}(\text{all degrees of freedom})\\ &= \sum_{M=0}^{M=N/2}\Omega(M,N;V)e^{-W/\lambda X}\\ &= {e^{\frac{-v_{0}N}{\lambda X}}}_{2}F_{1}\left[\frac{1}{2}-\frac{N}{2}, -\frac{N}{2};-N;-4e^{\frac{v_{0}}{\lambda X}-\frac{H}{\lambda X}}\right] \end{aligned} $$

其中，$Y$ 是自由能的类似值，$_2F_{1}$ 是超几何函数，积分已退化为对离散堵塞状态系综的求和。现在，自由能可以与等效热力学机制结合使用，以获得其他热力学变量。例如，与亥姆霍兹自由能类比，其值为

$$ Y = V - XS, $$

系统的熵值可由以下公式求得

$$ S = -\left(\frac{\partial Y}{\partial X}\right)_{V}. $$

Figure $3$ shows the entropy and volume relative to the most dense packing for a granular system of disks confined within channels of diameter $H/\sigma = 1.86$ and $1.50$ as a function of the compactivity. The underlying distributions of jammed states are the same for both systems and the difference in behavior between the two arises from the larger change in volume associated with the gain or loss of a defect state in the wider channel.

As a result, the relative change in volume is greater, and occurs gradually as a function of $X$ for the wider channel. The properties of the narrow channel are relatively insensitive to changes in $X$ at the higher compactivities, but we see a rapid decrease in entropy and volume as $X$ approaches zero.

图 $3$ 显示了在直径分别为 $H/\sigma = 1.86$ 和 $1.50$ 的通道中，相对于最致密堆积的盘粒系统的熵和体积与致密性的函数关系。这两个系统的基本堵塞态分布是相同的，两者的行为差异是由于在较宽的通道中，缺陷态的增减会引起较大的体积变化。

因此，对于较宽的通道来说，体积的相对变化更大，而且是作为 $X$ 的函数逐渐发生的。在密度较高的情况下，窄通道的特性对 $X$ 的变化相对不敏感，但当 $X$ 接近零时，我们会看到熵和体积迅速减少。

DISCUSSION

The goal of this paper was to explore the statistical mechanics of a simple model granular system. We are able to carry out a complete analysis of the distribution of jammed states for a system of highly confined hard disks and find that all the packings of a fixed volume are structurally the same.

This suggests that the Edwards’ assumption of equiprobability for jammed states of equal volume is well justified in the present model, even if it is not valid for more complex granular systems. The model also represents a genuine granular system that can be realized experimentally, and it is rare to find cases where the exact results for real experiments are available.

本文的目标是探索一个简单模型颗粒系统的统计力学。我们能够对一个高度密闭的硬盘系统的堵塞状态分布进行完整的分析，并发现固定体积的所有堆积在结构上都是相同的。

这表明，Edwards 关于体积相等的堵塞状态的等概率假设在本模型中是完全合理的，即使它对更复杂的粒状系统无效。该模型还代表了一个可以通过实验实现的真正的粒状系统，而实际实验的确切结果是很少见的。

这篇文章只介绍了在 2D 下的情况, 我们是如何将其有效地推广至 3D 情况并且做出有意义的实验的?

However, there are aspects of granular systems not covered in this model. First, we point out that our analysis required an idealistic dynamics that allowed the system to move directly between jammed states at fixed volume. Shaken granular materials need to generate a momentary increase in free volume to allow the particles to unjam, move, and then return to a jammed state.

The effects on the probability distribution of jammed states of mechanically injecting energy into the system and allowing particles the free volume needed to rearrange have not been included here. The presence of friction between particles has also been ignored and we would expect this to alter the number and type of packings included in the distribution, as well as possibly influencing the assumption of equiprobable volume states.

For example, it is immediately obvious that the neighboring defect state [Fig. 1(b)] would become stable in the presence of friction, but other “bridge-type” particle contacts may also need to be included. Nevertheless, the simplicity of the present system and its accessibility to experiment suggest that it may be an invaluable tool in understanding some of the elementary aspects in the statistical mechanics of granular systems.

不过，这个模型还没有涵盖颗粒系统的某些方面。首先，我们要指出的是，我们的分析需要一种理想主义的动力学，即允许系统在固定体积的堵塞状态之间直接移动。摇动的颗粒材料需要产生自由体积的瞬间增加，以便让颗粒松开、移动，然后返回堵塞状态。

这里没有包括机械地向系统注入能量和允许粒子有重新排列所需的自由体积对堵塞状态概率分布的影响。粒子之间的摩擦也被忽略了，我们预计这会改变分布中包含的堆积的数量和类型，并可能影响等容积状态的假设。

例如，显而易见的是，邻近的缺陷状态[图 1(b)]会在摩擦力的作用下变得稳定，但其他 “桥型” 粒子接触可能也需要包括在内。尽管如此，本系统的简单性和实验的可及性表明，它可能是理解颗粒系统统计力学某些基本方面的宝贵工具。