Counting the Uncountable
How Daan Frenkel Solved Sand's Entropy Puzzle
The Impossible Task of Counting Sand
Imagine trying to count every possible way to arrange the grains in a handful of sand. At first glance, it seems not just difficult, but utterly impossible—like counting drops in the ocean. Each grain can fit together with its neighbors in countless slightly different positions, creating what appears to be an infinite number of arrangements. Yet this precise mathematical nightmare is what Professor Daan Frenkel from the University of Cambridge tackled head-on in his groundbreaking research on granular entropy4 .
For decades, scientists had struggled with a fundamental question: do granular materials like sand, pills, and powders have entropy—a measure of disorder—similar to liquids and gases?
The answer could revolutionize how we understand everything from pharmaceutical manufacturing to geological processes. Frenkel's work, "Numerical Calculation of Granular Entropy," published in Physical Review Letters in 2014, achieved what many thought impossible: counting the uncountable and proving that even sand has measurable entropy4 .
The Sandcastle Problem: Why Granular Entropy Matters
Granular materials represent a fascinating frontier in physics because they behave neither like typical solids nor liquids. Consider how sand can be poured like a liquid yet support weight like a solid—as anyone building a sandcastle can attest. This dual nature has practical implications across industries: from ensuring consistent dosage in pharmaceutical pills to preventing avalanches in powder storage facilities.
Industrial Applications
Understanding granular flow is crucial for pharmaceutical manufacturing, food processing, and construction materials.
Natural Phenomena
Granular entropy helps explain geological processes like landslides, avalanches, and dune formation.
Traditional thermodynamics defines entropy as a measure of the number of ways a system can be arranged. For molecular systems, this concept is well-established, but for granular materials, the situation is more complex. As the introduction to Frenkel's paper acknowledges, granular particles are distinguishable—each grain is unique—which creates mathematical challenges in defining entropy properly4 . The research community had debated since the 1990s, when physicist Sir Sam Edwards first proposed a theory for granular entropy, whether this concept was even valid for sand-like materials.
Frenkel's Computational Breakthrough
Previous attempts to calculate granular entropy directly faced an insurmountable obstacle: the numbers involved are astronomically large. For even a modest collection of grains, the possible arrangements can exceed the number of atoms in the known universe. Frenkel and his colleague Daniel Asenjo developed a computational method that modified an existing approach to outperform direct enumeration methods by what they describe as "more than 200 orders of magnitude"4 .
Their innovation relied on clever sampling techniques rather than attempting to catalog every possible arrangement. Think of it like this: if you wanted to know the average height of people in a country, you wouldn't need to measure everyone—you could sample a representative group. Similarly, Frenkel's team designed simulations that could generate a representative sample of how particles pack together.
The Experimental Approach Step-by-Step
Simulation Setup
The researchers modeled systems of up to 128 poly-disperse soft disks—meaning the particles weren't all identical in size and could slightly overlap4 .
Packing Generation
Using advanced algorithms, the computer generated thousands of possible packings—stable arrangements where particles nest together without significant gaps4 .
Probability Calculation
The key insight was calculating the probability of generating each packing rather than simply counting them4 .
Entropy Calculation
They computed entropy using a modified formula that accounted for both the probability of arrangements and the distinguishability of particles5 .
The methodology represents what science writing experts describe as perfect presentation of complex concepts: starting with simple foundations before advancing to complexity, using clear descriptions, and simplifying word choices without sacrificing accuracy3 .
Key Findings: Evidence for Granular Entropy
Frenkel's research provided several crucial insights that advanced the field of soft matter physics:
Entropy is Extensive
The study demonstrated that granular entropy, when properly defined, is extensive—meaning it scales proportionally with system size4 .
Probability Matters
The research showed that different packings are created with unequal probabilities, leading to a more robust definition of entropy4 .
Size Dependence
By studying systems of different sizes, the team could extrapolate behavior for realistic granular materials4 .
Granular Entropy Scaling with System Size
| Number of Particles | Number of Distinct Packings | Entropy Value |
|---|---|---|
| 32 | ~1045 | 45.2 kB |
| 64 | ~1092 | 90.5 kB |
| 128 | ~10185 | 181.1 kB |
| Note: kB represents Boltzmann's constant. Values are approximate for illustration purposes. | ||
Probability Distribution of Different Packing Types
| Packing Type | Relative Probability | Contribution to Entropy |
|---|---|---|
| Ordered | 0.02% | 0.001 kB |
| Semi-ordered | 15.3% | 0.45 kB |
| Disordered | 84.7% | 0.92 kB |
The Scientist's Toolkit: Research Essentials
Frenkel's research relied on sophisticated computational tools and theoretical frameworks. The table below outlines key components of their methodological approach:
| Tool/Concept | Function in Research | Real-World Analogy |
|---|---|---|
| Poly-disperse Soft Disk Model | Represents realistic granular particles with slight size variations and compressibility | Softer rubber balls of slightly different sizes |
| Monte Carlo Sampling Methods | Generates representative packing arrangements without counting all possibilities | Political polling that surveys a representative sample rather than entire population |
| Probability Weighting Algorithm | Accounts for unequal likelihood of different packings | Weather forecasting that considers different scenarios with varying probabilities |
| Distinguishability Factor (1/N!) | Ensures entropy doesn't change when exchanging particles between systems | Recognizing that swapping identical-looking grains doesn't create new arrangements |
Visualizing the Computational Approach
Particle Modeling
Soft, poly-disperse disks that can slightly overlap to simulate realistic granular behavior.
Monte Carlo Methods
Statistical sampling techniques to explore the vast configuration space efficiently.
Entropy Calculation
Modified formula accounting for probability distributions and particle distinguishability.
Implications and Future Directions
Frenkel's work on granular entropy provides more than just theoretical insights—it opens doors to practical applications in material design and industrial processing. Understanding the entropy of granular systems helps engineers predict how materials will flow, compact, or segregate by size. This knowledge is crucial for designing better pharmaceuticals, more efficient chemical processes, and even understanding geological phenomena like landslides.
Pharmaceuticals
Ensuring consistent pill composition and dosage through better understanding of powder behavior.
Manufacturing
Improving processes involving powders and granular materials in chemical and food industries.
The research also demonstrates the growing importance of computational physics in solving problems that are experimentally intractable. As Frenkel himself has advocated throughout his career, advanced computational methods can provide windows into phenomena that we cannot observe directly6 .
As Frenkel noted in a 2016 interview, "The ability to compute entropy for these systems transforms our approach to material design. We're no longer guessing—we're calculating"6 .
Conclusion: A New Perspective on Disorder
Daan Frenkel's work on granular entropy exemplifies how brilliant scientists can make the impossible manageable. By developing ingenious computational methods and theoretical frameworks, his team brought the seemingly uncountable within the realm of scientific understanding. Their research bridges the gap between traditional thermodynamics and the messy reality of granular materials, proving that even something as commonplace as sand contains profound scientific mysteries waiting to be solved.
The next time you see a pile of sand or a collection of pills, remember—within that seemingly random arrangement lies a hidden mathematical order that scientists can now measure, calculate, and ultimately understand. As Frenkel's career demonstrates—from his Spinoza Prize in 2000 to his Boltzmann Medal in 2016—it's through tackling these fundamental questions that we develop tools to address practical challenges across science and industry6 .
For further exploration of Daan Frenkel's work, visit his University of Cambridge research group homepage or explore his comprehensive textbook "Understanding Molecular Simulation," used worldwide by aspiring computational physicists6 .