̽��ֱ�� of Cambridge - Stefano Martiniani

Study reveals mysterious equality with which grains pack it in

tdk25 — Mon, 26 Jun 2017 15:00:36 +0000

At the moment they come together, the individual grains in materials like sand and snow appear to have exactly the same probability of combining into any one of their many billions of possible arrangements, researchers have shown.

̽��ֱ��finding, by an international team of academics at the ̽��ֱ�� of Cambridge, UK, and Brandeis ̽��ֱ�� in the US, appears to confirm a decades-old mathematical theory which has never been proven, but provides the basis for better understanding granular materials – one of the most industrially significant classes of material on the planet.

A granular material is anything that comprises solid particles that can be seen individually with the naked eye. Examples include sand, gravel, snow, coal, coffee, and rice.

If correct, the theory demonstrated in the new study points to a fact of remarkable – and rather mysterious – mathematical symmetry. It means, for example, that every single possible arrangement of the grains of sand within a sand dune is exactly as probable as any other.

̽��ֱ��study was led by Stefano Martiniani, who is based at New York ̽��ֱ�� but undertook the research while completing his PhD at St John’s College, ̽��ֱ�� of Cambridge.

“Granular materials are so widely-used that understanding their physics is very important,” Martiniani said. “This theory gives us a very simple and elegant way to describe their behaviour. Clearly, something very special is happening in their physics at the moment when grains pack together in this way.”

̽��ֱ��conjecture that Martiniani tested was first proposed in 1989 by the Cambridge physicist Sir Sam F. Edwards, in an effort to better understand the physical properties of granular materials.

Globally, these are the second-most processed type of material in industry (after water) and staples of sectors such as energy, food and pharmaceuticals. In the natural world, vast granular assemblies, such as sand dunes, interact directly with wind, water and vegetation. Yet the physical laws that determine how they behave in different conditions are still poorly understood. Sand, for example, behaves like a solid when jammed together, but flows like a liquid when loose.

Understanding more about the mechanics of granular materials is of huge practical importance. When they jam during industrial processing, for example, it can cause significant disruption and damage. Equally, the potential for granular materials to “unjam” can be disastrous, such as when soil or snow suddenly loosens, causing a landslide or avalanche.

At the heart of Edwards’ proposal was a simple hypothesis: If one does not explicitly add a bias when preparing a jammed packing of granular materials – for example by pouring sand into a container – then any possible arrangement of the grains within a certain volume will occur with the same probability.

This is the analogue of the assumption that is at the heart of equilibrium statistical mechanics – that all states with the same energy occur with equal probability. As a result the Edwards hypothesis offered a way for researchers to develop a statistical mechanics framework for granular materials, which has been an area of intense activity in the last couple of decades.

But the hypothesis was impossible to test – not least because above a handful of grains, the number of possible arrangements becomes unfathomably huge. Edwards himself died in 2015, with his theory still the subject of heated scientific debate.

Now, Martiniani and colleagues have been able to put his conjecture to a direct test, and to their surprise they found that it broadly holds true. Provided that the grains are at the point where they have just jammed together (or are just about to separate), all possible configurations are indeed equally likely.

Helpfully, this critical point – known as the jamming transition – is also the point of practical significance for many of the granular materials used in industry. Although Martiniani modelled a system comprising soft spheres, a bit like sponge tennis balls, many granular materials are hard grains that cannot be compressed further once in a packed state.

“Apart from being a very beautiful theory, this study gives us the confidence that Edwards’ framework was correct,” Martiniani said. “That means that we can use it as a lens through which to look at a whole range of related problems.”

Aside from informing existing processes that involve granular materials, there is a wider significance to better understanding their mechanics. In physics, a “system” is anything that involves discrete particles operating as part of a wider network. Although bigger in scale, the way in which icebergs function as part of an ice floe, or the way that individual vehicles move within a flow of traffic (and indeed sometimes jam), can be studied using a similar theoretical basis.

Martiniani’s study was undertaken during his PhD, while he was a Gates Scholar, under the supervision of Professor Daan Frenkel from the Department of Chemistry. It built on earlier research in which he developed new methods for calculating the probability of granular systems packing into different configurations, despite the vast numbers involved. In work published last year, for example, he and colleagues used computer modelling to work out how many ways a system containing 128 tennis balls could potentially be arranged. ̽��ֱ��answer turned out to be ten unquadragintilliard – a number so huge that it vastly exceeds the total number of particles in the universe.

In the new study, the researchers employed a sampling technique which attempts to compute the probability of different arrangements of grains without actually looking at the frequency with which these arrangements occur. Rather than taking an average from random samples, the method involves calculating the limits of the possibility of specific arrangements, and then calculates the overall probability from this.

̽��ֱ��team applied this to a computer model of 64 soft spheres - an imaginary system which could therefore be “over-compressed” after reaching the jamming transition point. In an over-compressed state, the different arrangements were found to have different probabilities of occurrence. But as the system decompressed to the point of the jamming transition, at which the grains were effectively just touching, the researchers found that all probabilities became equal – exactly as Edwards predicted.

“In 1989, we didn’t really have the means of studying whether Edwards was right or not,” Martiniani added. “Now that we do, we can understand more about how granular materials work; how they flow, why they get stuck, and how we can use and manage them better in a whole range of different situations.”

̽��ֱ��study, Numerical test of the Edwards conjecture shows that all packings become equally probable at jamming is published in the journal Nature Physics. DOI: 10.1038/nphys4168.

For the first time, researchers have been able to test a theory explaining the physics of how substances like sand and gravel pack together, helping them to understand more about some of the most industrially-processed materials on the planet.

Granular materials are so widely-used that understanding their physics is very important. Clearly, something very special is happening at the moment when grains pack together in this way.

Stefano Martiniani

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A huge range of materials are classified as granular – including sand, gravel, snow, nuts, coal, rice, barley, coffee and cereals. Globally, they are the second-most processed type of material in industry, after water.

̽��ֱ��text in this work is licensed under a Creative Commons Attribution 4.0 International License. For image use please see separate credits above.

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Lost in high-dimensional space: Study improves the cure for the “Curse Of Dimensionality”

tdk25 — Thu, 13 Oct 2016 08:05:16 +0000

Researchers have developed a new technique for making calculations in “high-dimensional space” – mathematical problems so wide-ranging in their scope, that they seem at first to be beyond the limits of human calculation.

In what sounds like the title of a rejected script for an Indiana Jones movie, the method improves on existing approaches to beat a well-known problem known as “ ̽��ֱ��Curse Of Dimensionality”. It was devised by a team of researchers at the ̽��ֱ�� of Cambridge.

In rough terms, the “Curse”, refers to the apparent impossibility of making calculations in situations where the number of variables, attributes, and possible outcomes is so large that it seems futile even to try to comprehend the problem in the first place.

A simple example is this: Imagine that you have a cup containing 100 grains of rice. You pick it up, shake it, and put it down again. ̽��ֱ��arrangement within the cup changes, but what are the chances of that arrangement occurring, relative to all other possibilities?

While most people would reasonably consider that problem not just impossible, but largely pointless, it illustrates the type of maths needed to make predictions about much bigger – and more meaningful – issues.

Those include, for example, trying to model the likely shape and impact of a decaying ecosystem, such as a developing area of deforestation, or the potential effect of different levels of demand on a power grid. More fundamentally, the same class of calculation would theoretically enable us to get to grips with the statistical probability of our own existence on Earth, or the chances that life might happen again, elsewhere in the Universe.

̽��ֱ��new study was led by Stefano Martiniani, a Gates Scholar at St John’s College, Cambridge, who carried out the work with colleagues in the Department of Chemistry and at Microsoft Research in Cambridge.

“There is a very large class of problems that can be solved through the sort of approach that we have devised,” Martiniani said. “It opens up a whole world of possibilities in the study of things like dynamical systems, chemical structure prediction, or artificial neural networks.”

Most people understand “dimensions” to mean height, width, depth and time, but in Mathematics the term is also used flexibly to describe the number of parameters needed to specify a “state” for any given problem. ̽��ֱ��more complicated the problem in question, the greater the space you need to express the parameters. They therefore become “high-dimensional spaces”.

Similarly, working out the likelihood of a particular outcome in a situation where all sorts of different variables apply – such as the grains of rice in a cup arranging themselves in a particular way – is a high-dimensional problem. Expressing and plotting the combined impact of the many parameters that might affect the outcome involves imagining a graph with multiple axes, as if working in numerous dimensions at once.

̽��ֱ��method devised by Martiniani and colleagues, like other approaches, begins by characterising such challenges as an “energy landscape”. ̽��ֱ��range of possible states in which a system such as the cup of rice may exist is envisaged as a landscape of mountains and valleys, in which the base of each valley is a stable state.

̽��ֱ��set of initial conditions leading to this stable state is called a “basin of attraction”. ̽��ֱ��fundamental theory is that, if the volume of each basin of attraction can be calculated, then this begins to provide some sort of indication of the probability of a given state’s occurrence.

To do that, researchers build computer software which models high dimensional systems, using the landscape analogy, and makes calculations within it.

̽��ֱ��simplest model is a brute force approach, which essentially takes a reading, shakes the system up, takes another reading, and repeats the process – many millions of times – in an attempt to establish the probability of certain outcomes. A more sophisticated strategy recurrently starts in the same place and measures the average distance within the energy landscape in which the system finds the same basin of attraction, through which the user gradually develops an appreciation of its volume.

“In most cases you are like a blindfolded person, walking around drunk in the energy landscape,” Martiniani said. “At any given moment, you only really know where you are and where you have just come from.”

In the new study, however, the team applied a different approach to the same kind of problem. Borrowing a technique widely used in biomolecular simulations, called the Multistate Bennett Acceptance Ratio, they developed a method which systematically tests the limits of one particular basin of attraction. Rather than gauging its volume by just taking an average from random samples, it looks for the furthest and least likely limits.

̽��ֱ��net result is a much more efficient sampling technique, which enables a much broader range of calculations in high-dimensional space.

To test this, the team modelled an imaginary 93-dimensional “system” made up of 32 soft spheres that could be packed together in multiple ways. They found that they were able to sample and quantify outcomes within that system that would only be found randomly one in every 10100 times. In other words, the chances of stumbling across those outcomes by chance would be one in ten duotrigintillion.

“In basic terms it goes where brute force sampling never will, because if you started to try, you would never finish,” Martiniani added. “Technically, the limits of the problems we can solve are now not those of the approach, but the computing power we need to simulate the underlying energy landscape. When addressing these kinds of problems in high-dimensional space, this should now be the technique of choice.”

̽��ֱ��study, Structural analysis of high-dimensional basins of attraction, is published in the journal Physical Review E.

Researchers have developed a new method for making effective calculations in “high-dimensional space” – and proved its worth by using it to solve a 93-dimensional problem.

In most cases you are like a blindfolded person, walking around drunk in the energy landscape.

Stefano Martiniani

Tom Ruen via Wikimedia Commons.

Projection of a 9-dimensional cube. High-dimensional spaces pose considerable problems when trying to make calculations and predictions – something that the new method devised by researchers aims to address.

̽��ֱ��text in this work is licensed under a Creative Commons Attribution 4.0 International License. For image use please see separate credits above.

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How many ways can you arrange 128 tennis balls? Researchers solve an apparently impossible problem

tdk25 — Wed, 27 Jan 2016 16:16:05 +0000

In research carried out at the ̽��ֱ�� of Cambridge, a team developed a computer program that can answer this mind-bending puzzle: Imagine that you have 128 soft spheres, a bit like tennis balls. You can pack them together in any number of ways. How many different arrangements are possible?

̽��ֱ��answer, it turns out, is something like 10²⁵⁰ (1 followed by 250 zeros). ̽��ֱ��number, also referred to as ten unquadragintilliard, is so huge that it vastly exceeds the total number of particles in the universe.

Far more important than the solution, however, is the fact that the researchers were able to answer the question at all. ̽��ֱ��method that they came up with can help scientists to calculate something called configurational entropy – a term used to describe how structurally disordered the particles in a physical system are.

Being able to calculate configurational entropy would, in theory, eventually enable us to answer a host of seemingly impossible problems – such as predicting the movement of avalanches, or anticipating how the shifting sand dunes in a desert will reshape themselves over time.

These questions belong to a field called granular physics, which deals with the behaviour of materials such as snow, soil or sand. Different versions of the same problem, however, exist in numerous other fields, such as string theory, cosmology, machine learning, and various branches of mathematics. ̽��ֱ��research shows how questions across all of those disciplines might one day be addressed.

Stefano Martiniani, a Gates Scholar at St John’s College, ̽��ֱ�� of Cambridge, who carried out the study with colleagues in the Department of Chemistry, explained: “ ̽��ֱ��problem is completely general. Granular materials themselves are the second most processed kind of material in the world after water and even the shape of the surface of the Earth is defined by how they behave.”

“Obviously being able to predict how avalanches move or deserts may change is a long, long way off, but one day we would like to be able to solve such problems. This research performs the sort of calculation we would need in order to be able to do that.”

At the heart of these problems is the idea of entropy – a term which describes how disordered the particles in a system are. In physics, a “system” refers to any collection of particles that we want to study, so for example it could mean all the water in a lake, or all the water molecules in a single ice cube.

When a system changes, for example because of a shift in temperature, the arrangement of these particles also changes. For example, if an ice cube is heated until it becomes a pool of water, its molecules become more disordered. Therefore, the ice cube, which has a tighter structure, is said to have lower entropy than the more disordered pool of water.

At a molecular level, where everything is constantly vibrating, it is often possible to observe and measure this quite clearly. In fact, many molecular processes involve a spontaneous increase in entropy until they reach a steady equilibrium.

In granular physics, however, which tends to involve materials large enough to be seen with the naked eye, change does not happen in the same way. A sand dune in the desert will not spontaneously change the arrangement of its particles (the grains of sand). It needs an external factor, like the wind, for this to happen.

This means that while we can predict what will happen in many molecular processes, we cannot easily make equivalent predictions about how systems will behave in granular physics. Doing so would require us to be able to measure changes in the structural disorder of all of the particles in a system - its configurational entropy.

To do that, however, scientists need to know how many different ways a system can be structured in the first place. ̽��ֱ��calculations involved in this are so complicated that they have been dismissed as hopeless for any system involving more than about 20 particles. Yet the Cambridge study defied this by carrying out exactly this type of calculation for a system, modelled on a computer, in which the particles were 128 soft spheres, like tennis balls.

“ ̽��ֱ��brute force way of doing this would be to keep changing the system and recording the configurations,” Martiniani said. “Unfortunately, it would take many lifetimes before you could record it all. Also, you couldn’t store the configurations, because there isn’t enough matter in the universe with which to do it.”

Instead, the researchers created a solution which involved taking a small sample of all possible configurations and working out the probability of them occurring, or the number of arrangements that would lead to those particular configurations appearing.

Based on these samples, it was possible to extrapolate not only in how many ways the entire system could therefore be arranged, but also how ordered one state was compared with the next – in other words, its overall configurational entropy.

Martiniani added that the team’s problem-solving technique could be used to address all sorts of problems in physics and maths. He himself is, for example, currently carrying out research into machine learning, where one of the problems is knowing how many different ways a system can be wired to process information efficiently.

“Because our indirect approach relies on the observation of a small sample of all possible configurations, the answers it finds are only ever approximate, but the estimate is a very good one,” he said. “By answering the problem we are opening up uncharted territory. This methodology could be used anywhere that people are trying to work out how many possible solutions to a problem you can find.”

̽��ֱ��paper, Turning intractable counting into sampling: computing the configurational entropy of three-dimensional jammed packings, is published in the journal, Physical Review E.

A bewildering physics problem has apparently been solved by researchers, in a study which provides a mathematical basis for understanding issues ranging from predicting the formation of deserts, to making artificial intelligence more efficient.

̽��ֱ��brute force way of doing this would be to keep changing the system and recording the configurations. Unfortunately, it would take many lifetimes before you could record it all. Also, you couldn’t store them, because there isn’t enough matter in the universe.

Stefano Martiniani

Atomic Taco via Flickr

Tennis balls

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