Imagine that you have 128 tennis balls, and can arrange them in any way you like. How many arrangements are possible? According to a new paper, the answer is about 10^250, also known as ten unquadragintilliard: that's a number so big that it exceeds the total number of particles in the universe.
Such “configurational entropy” - a term used to describe how structurally disordered the particles in a physical system are - could lead to a model for the sort of maths that would be needed to solve bigger problems still, ranging from predicting avalanches, or artificial intelligence systems.
Being able to calculate configurational entropy would eventually enable us to answer a host of seemingly impossible problems in granular physics, which deals with the behavior of materials such as snow, soil or sand, not to mention resurrecting string theory and artificial intelligence hopes.
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 under study, so for example it could mean all the water in a lake, or all the water molecules in a single ice cube.
Credit: Tennis balls Atomic Taco Flickr
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. The 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, modeled on a computer, in which the particles were 128 soft spheres, like tennis balls.
The brute force way of doing this would be to keep changing the system and recording the configurations - that would take many lifetimes and 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.
Citation: Turning intractable counting into sampling: Computing the configurational entropy of three-dimensional jammed packings. Published in Physical Review E: DOI:http://dx.doi.org/10.1103/PhysRevE.93.012906
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