Adrian Baule

    • School of Mathematical Sciences, Queen Mary University of London, London, United Kingdom

&Bullet; physics 14, 93

The results of new experiments confirm the idea that there is a statistical mechanics framework for granular materials.


Illustration 1: New experiments measuring the volume statistics of a container filled with plastic beads validate a statistical mechanical framework for granular materials.

Beach sand, sugar and coffee powder are just three examples of ubiquitous granular materials in life, made up of small particles or grains. Despite this ubiquity, a basic understanding of the properties of these materials has so far remained unclear. Now, Yujie Wang of Shanghai Jiao Tong University, China, and coworkers are taking a step towards developing this understanding by conducting experiments that validate a statistical mechanics framework for granular systems [1] . Their results could help researchers determine new methods for predicting the properties of granular materials.

Granular materials behave similarly to other multiparticulate systems such as gases, liquids, and solids. For example, granules can flow like liquids when poured, and like solids they can absorb stress when jammed. The macroscopic observables of a granular system, such as its packing density, can be reproduced by controlling a few parameters of the system, which makes it clear that it is governed by statistical principles. Gases, liquids and solids in thermal equilibrium are also subject to statistical principles and can be successfully described by the theory of statistical equilibrium mechanics. In contrast, the establishment of a similarly comprehensive statistical theory of granular systems has been a long-standing problem.

In statistical equilibrium mechanics, the central observable is the total energy of the system. This energy relates both to the entropy of the system (the number of microstates accessible at a given energy) and to the likelihood that the system will sit in a configuration with a given energy. In the case of granular systems, however, the total energy is either not retained (as in systems made up of frictional, dissipative particles) or is irrelevant (as in systems made up of frictionless, hard particles) and thus energy is unsuitable as a state variable.

A promising state variable for granular materials is the volume of the system. More than three decades ago, the late Sam Edwards of the University of Cambridge and colleagues hypothesized that volume could play the role of energy in granular systems. They showed that, using the volume of the system, they can formulate a statistical mechanics-based theory that includes concepts such as granular entropy, “compact activity” (a temperature-like variable), and a canonical volume ensemble that expresses the probability of observing a. determines configuration with a certain volume [2, 3] . This framework has received a lot of attention. But testing the theory’s predictions directly through experiments or simulations was a challenge [4] .

The challenge arises because the properties of a granular system typically depend on the manufacturing protocol. For example, it matters whether grains form a jammed aggregate under gravity after being poured into a container, or whether they are compacted from a loose configuration by moving the container walls. The final packaging structure may contain remnants of the initial state and the manufacturing details that are not taken into account in theory. In addition, it is difficult to faithfully reproduce the underlying conditions of the theory such as the canonical ensemble condition, which requires an exchange of volume between the system and a surrounding “heat bath”. Wang and colleagues have now solved these problems in their new experiments.

The work of the team follows in the footsteps of previous experiments that mimicked the effect of random heat fluctuation by repeatedly “stimulating” the system [5, 6] . Under such stimuli, the system can reach one of several stationary states, each of which has a certain packing density and which are independent of how the system was prepared. Researchers have interpreted these stationary states as equilibrium-like states, which in principle should be describable by a formalism of statistical mechanics.

In the new experiments, Wang and colleagues examined a granular material made up of 3D-printed plastic beads several mm in diameter held in a container. For different iterations of the experiment, the team used beads with different roughness to investigate the influence of friction on the packing statistics. They excited the material by periodically tapping the container and monitored the bead configurations using X-ray tomography.

The Edwards framework considers the volume of a given system that fluctuates under a volume exchange with its environment. Instead, Wang and colleagues consider “subsystems” within the container, where a subsystem is defined as a spherical area of ​​fixed diameter around a particle, the volume of which can be precisely determined. These subsystems are large enough to remain uncorrelated and therefore can be viewed as independent implementations of the same system for which ensemble statistics can be compiled.

With this subsystem idea, Wang and colleagues found that the probability distribution of the associated volumes follows the functional form of Edwards’ canonical volume ensemble. They were also able to calculate the compactness of the system and its granular entropy using two different methods. Surprisingly, they found that the relationship between compactivity and knock intensity is independent of friction, suggesting that different but identically excited granular systems adopt the same compactivity. In the context of statistical equilibrium mechanics, this behavior corresponds to what is observed when two systems in contact with the same heat bath reach the same temperature – a manifestation of the zeroth law of thermodynamics.

Wang and colleagues’ data support the validity of a granular zeroth law of thermodynamics within experimental uncertainty, which is in contrast to an earlier finding that the zeroth law does not hold true for a 2D compression experiment with friction disks [7] . This discrepancy could arise from the different packing protocols used in the two sets of experiments or from subtle differences in the way compactivity was measured. The new data can also be summarized in a phase diagram, which characterizes the packings with regard to the packing density, the coefficient of friction and the average number of contacts per particle and thus extends the previous results of the mean field theory [8] . Different coefficients of friction represent different curves in this phase diagram, but with suitable re-scaling the curves collapse into a single curve, which indicates a universal behavior of the friction packings.

The new results complement a growing body of work that supports the validity of statistical mechanics approaches for granular materials [9, 10] . However, it remains a great challenge to use these approaches for specific computations [4] . Much more theoretical and experimental work is required to fully exploit the explanatory power of granular statistical mechanics for real applications.


  1. Y. Yuan et al., “Experimental test of the Edwards volume ensemble for tapped granule packs”, Phys. Rev. Lett.127, 018002 (2021).
  2. SF Edwards and RBS Oakeshott, “Theory of Powders”, Physica A157, 1080 (1989).
  3. A. Mehta and SF Edwards, “Statistical Mechanics of Powder Mixtures”, Physica A157, 1091 (1989).
  4. A. Baule et al., “Edwards Statistical Mechanics for Trapped Granular Matter”, Rev. Mod. Phys.90, 015006 (2018).
  5. E. Nowak et al., “Density fluctuations in vibrated granular materials”, Phys. Rev. E57, 1971 (1998).
  6. M. Schröter et al., “Volume fluctuations in the steady state in a granular medium”, Phys. Rev. E71, 030301 (2005).
  7. JG Puckett and KE Daniels, “Equilibrating temperature-like variables in jammed granular subsystems”, Phys. Rev. Lett.110, 058001 (2013).
  8. C. Song et al., “A phase diagram for trapped matter”, nature453, 629 (2008).
  9. S. Martiniani et al., “Numerical test of the Edwards conjecture shows that all packages are equally likely to jam” Nat. Phys.13, 848 (2017).
  10. IT cheap et al., “Protocol dependency and state variables in the force-torque ensemble”, Phys. Rev. Lett.122, 038001 (2019).

About the author

Image by Adrian Baule

Adrian Baule is a lecturer in applied mathematics at Queen Mary University of London (QMUL). After studying at the University of Tübingen and the University of Münster, both in Germany, he received his doctorate as Dr. at the University of Leeds, UK, in 2008. He was then postdoc at Rockefeller University, New York, and the City College of New York. He joined QMUL in 2011. His research deals with various aspects of statistical non-equilibrium mechanics and their application to granular, soft and active matter.

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