Some events are typically uncontrollable and occur at a seemingly-unpredictable moment. For example, traffic jams sometimes happen with no apparent reason in a crowded but still fluid highway. Or a stock-market index dips suddenly, only to recover a few hours later, with nobody to blame for it. Sometimes a song from an obscure band becomes a hit overnight.

These sort of events have another common characteristic besides appearing capricious. When they happen, they happen very fast. A good example of this is the following video, in which you will see a crowd go wild.

It is clearly difficult to guess a priori when (and if) the crowd will go dancing. Naively, one can look for a sudden change in some variable. For instance, the speed at which people join the dancing group: during the first minute and 12 seconds, up to three persons dance, but it takes only another minute to get a few hundreds to join. Thus, we can say that minute 1:12 is the onset of the dancing madness. But, because from the onset time on, the number of people standing up and joining the dance grows very fast, watching for this sudden change becomes pretty useless. Once we realize that the stock market is crashing or that we are involved in a traffic jam, it is already too late.

It turns out though that this rather infrequent but quickly-happening events are not totally unpredictable after all. Physicists have been interested in this so-called rare events for some decades, not only because they would like to avoid traffic jams when driving to the lab., but also because rare events are not that rare in nature: the formation of rain droplets, the crystallization of water into ice, or the formation of air bubbles when boiling pasta have a lot in common. The basic understanding of this phenomena can be very-well described by the theory of homogeneous nucleation. The theory can be understood in very simple terms, and I next attempt to do this by referring to the dancing event in the video above.

Let’s assume that the sitting crowd is in the mood for dancing. This means that happiness increases when people move from ‘laying down and chilling out’ to ‘up and dancing’, and this increase is proportional to the volume of people dancing. However, there is a cost — a decrease in happiness — associated to the group in the ‘up and dancing’ state. This cost is related to the embarrassment of dancing while being observed by a crowd — we all have experienced that feeling. The ‘embarrassment cost’ also grows with the size of the group, but in this case it is not proportional to the volume of people dancing as a group, but only to the number of people dancing at the boundary of it. This is because only those at the boundary feel being observed by the sitting crowd. In the schematic graph below I have drawn the ‘embarrassment cost’ (labeled interfacial cost) and the volumetric ‘happiness cost’ (which is, in cost terms, negative). Clearly, the interfacial cost grows slower with size than the volumetric cost. The sum of the two is the total cost, and it displays a maximum.

The shape of the total-cost curve explains nucleation phenomena qualitatively. For sizes smaller than the critical size S*, increasing the size of the dancing group comes at an interfacial cost. Beyond S* all is downhill, and the slope heading down is steep. In the video above, S* is 3, which is the critical nucleus of people needed to jump start rapid growth toward group happiness. From minute 1:12 on, the size of the group grows as fast as people can stand up and join the dancers. Before the critical nucleus is reached, only very few persons seem to have enough energy (or alcohol) to overcome the barrier to nucleation.

Similarly, when the temperature of water is below that of at which it crystallizes (zero degrees Celsius at atmospheric pressure), ice forms via a nucleation process. Molecules of water jiggle in the fluid, and fluctuations in their energy can drive a few molecules to order likewise they do in ice (the image at the top of this page shows this for hard spheres). The size of this small nucleus of ice-like molecules will fluctuate — shrink and grow back again — if there is not enough energy available to overcome the interfacial cost of forming a critical nucleus, which is the energy needed to form the ice-water interface. This is why water does not freeze spontaneously at its freezing temperature, but some degrees below it. Decreasing the temperature below the freezing point increases the negative volumetric cost and diminishes the size of the nucleation barrier, making it easier that spontaneously-forming nuclei of ice reach the critical size.

For technical information about nucleation theory, follow this link to a scientific talk by professor Daan Frenkel, who works at the University of Cambridge, UK, and was my scientific advisor some time ago. (You will need a basic knowledge in thermodynamics to understand it.)

Now, what happens when we boil water to cook pasta and add salt right before the water starts to boil? Yes, nucleation at work again, but this time helped by the small grains of salt.