Blumenfeld Raphael, Edwards Sam F
Biological and Systems, Cavendish Laboratory, Cambridge, UK.
J Phys Chem B. 2009 Mar 26;113(12):3981-7. doi: 10.1021/jp809768y.
We discuss the microstates of compressed granular matter in terms of two independent ensembles: one of volumes and another of boundary force moments. The former has been described in the literature and gives rise to the concept of compactivity: a scalar quantity that is the analogue of temperature in thermal systems. The latter ensemble gives rise to another analogue of the temperature: an angoricity tensor. We discuss averages under either of the ensembles and their relevance to experimental measurements. We also chart the transition from the microcanoncial to a canonical description for granular materials and show that one consequence of the traditional treatment is that the well-known exponential distribution of forces in granular systems subject to external forces is an immediate consequence of the canonical distribution, just as in the microcanonical description E = H leads to exp (-H/kT). We also put this conclusion in the context of observations of nonexponential forms of decay. We then present a Boltzmann-equation and Fokker-Planck approaches to the problem of diffusion in dense granular systems. Our approach allows us to derive, under simplifying assumptions, an explicit relation between the diffusion constant and the value of the hitherto elusive compactivity. We follow with a discussion of several unresolved issues. One of these issues is that the lack of ergodicity prevents convenient translation between time and ensemble averages, and the problem is illustrated in the context of diffusion. Another issue is that it is unclear how to make use in the statistical formalism the emerging ability to exactly predict stress fields for given structures of granular systems.
一个是体积系综，另一个是边界力矩系综。前者在文献中已有描述，并引出了压缩性的概念：一个标量，类似于热系统中的温度。后者系综引出了另一个温度类似物：角能张量。我们讨论了任一系综下的平均值及其与实验测量的相关性。我们还绘制了颗粒材料从微正则描述到正则描述的转变，并表明传统处理的一个结果是，颗粒系统中在外力作用下众所周知的力的指数分布是正则分布的直接结果，就像在微正则描述中E = H导致exp(-H/kT)一样。我们还将这一结论置于非指数衰减形式观测的背景下。然后，我们提出了玻尔兹曼方程和福克 - 普朗克方法来解决密集颗粒系统中的扩散问题。我们的方法使我们能够在简化假设下，推导出扩散常数与迄今难以捉摸的压缩性值之间的明确关系。接着，我们讨论了几个未解决的问题。其中一个问题是，缺乏遍历性使得时间平均和系综平均之间难以方便地转换，并且在扩散的背景下说明了这个问题。另一个问题是，不清楚如何在统计形式体系中利用新兴的能力，即对于给定颗粒系统结构精确预测应力场的能力。
