We tend to have a static conception of randomness, something like a coin toss, or a dice roll, or even a lottery. Simulations encompass a dynamic conception of randomness. So they necessarily deal with changes in randomness over time and across space. The technical literature is rather terse for precisely defining and working with simulations in a wide array of fields. Definitely not the sort of thing that I enjoy blogging about. But delving into the intuitive side is stimulating stuff.

Incidentally, the title of this post comes from a poem by Emily Dickinson. It eloquently hints at the intuitions that are very useful for appreciating simulations. She managed to express a living, morphing, breathing image of time and space, in just a dozen succinct lines. That’s the metaphoric equivalent of anything which is dynamic and random. It is born. It breathes. It lives. And it even dies! Throughout all that, it’s made of many different instances, or “Nows”.

Dickinson clearly grasped incremental progress toward an asymptote, which is the essence of simulations as I conceptualise them. My favourite examples of simulations are actually far outside my own field. I’ve been curious about weather forecasting and solar system evolution for a long time. So in the rest of this post I’ll try to illuminate my Dickinsonian intuitions about simulations using those two fields.

Gerd Gigerenzer’s book Risk Savvy (2014) makes a clever point about the chance of rain tomorrow, e.g., what does a 30% chance of rain tomorrow actually mean? Without any further clarification, you might interpret it along the following lines.

  1. Around 30% of the suburbs will experience rain at some point tomorrow, but there’s no indication as to which ones.
  2. Around 30% of the meteorologists in the committee agree that it will rain at some point tomorrow, but there’s no consensus as to when.

Even though it’s a simplified example, it nonetheless illustrates an important difference between static and dynamic views of randomness. On the one hand, a static view of randomness means that the two interpretations above are mutually exclusive: so it’s either the first or the second, but not both. On the other hand, a dynamic view means that both are valid along with the added specification of where and when it’s likely to rain.

The dynamic view is indeed what modern weather forecasting uses. Giant supercomputers run regular simulations of the fluid-dynamical models corresponding to your local atmosphere and terrain. Thus tomorrow’s forecast informs you about which suburbs are likely to experience rain, as well as roughly what time of day that’s expected, among other variables such as temperature, humidity, wind, etc. And of course the forecast is regularly updated to keep pace with the changing atmosphere and latest measurements.

Much has been popularised about the chaotic nature of fluid dynamics (e.g., read James Gleick), along with the challenges of accurate weather forecasting. However, 21st century weather forecasting has become ridiculously good! Nate Silver spent a whole chapter on this topic in his book The Signal and the Noise (2012). He said everything that I ever thought of saying about the accuracy of weather forecasting. But as far as the intuitive side is concerned, it’s worth stating the common knowledge that although tomorrow’s forecast is trustworthy, the ten day forecast is to be taken with a pinch of salt, because such distant horizons have large uncertainties. And forecasts covering vast territories are also inconveniently inaccurate, as rural folks know all too well.

The main explanation behind that common knowledge is chaos theory, whereby the errors from numerical rounding of initial measurements compound to produce vastly different (even contradictory) predictions beyond a small window of time and space. Even though chaos theory is deterministic and it’s the numerical errors that compound, the uncertainty in its predictions cannot be eliminated due to the intense sensitivity to starting conditions and our finite computers. So to make progress with weather forecasting, meteorologists run lots of simulations and then average the results. This step is akin to composing forever from nows, as in Dickinson’s poem. It’s a way of reaching ever closer to an asymptote that’s forever unreachable.

Marcus du Sautoy’s book What We Cannot Know (2016) reviews the history of chaos theory in a thoroughly enlightening manner. He points out that computing the extremely long term evolution of the solar system suffers from the same error compounding effects as weather forecasting. The main difference is that the window of validity is a few million years for the solar system due to its very slow changes compared to the weather anywhere on earth. He cites a paper published by the journal Nature in 2009 that has the apocalyptic title ‘Existence of collisional trajectories of Mercury, Mars and Venus with the Earth’. We can all rest assured that this potential collision is well over three billion years from now.

I’m rather fond of this example because it is endlessly fascinating to ponder the possibility that the orbits of the terrestrial planets could become so unstable in the very distant future that they end up smashing into each other (though not all at the same time). The authors of the paper make it clear that they estimate a probability of only 1% for this possibility to manifest after three billion years from now, based on their simulations of the solar system evolution under general relativity. That’s just about as unlikely an event as you can imagine. However, it’s the amazing range and sequence of possibilities uncovered by their simulations that are worth reading in the paper. In meteorological simulations the possibilities are not as striking as this (the butterfly effect notwithstanding).

Whether it’s forecasting planetary trajectories or tomorrow’s weather, the role of simulations in helping us make sense of the chaos is worth appreciating more. We happen to be a bit lucky with deterministic theories of fluid dynamics and general relativity to the extent that they can be simulated with mainly the numerical errors being compounded, and that reality happens to match our theories exceedingly well in these cases. Simulations naturally lead us into a cumulative trajectory, where we compose a grainy picture of reality after numerous iterations. It’s necessarily a slow grind, similar to what Dickinson said, “Let Months dissolve in further Months - / And Years - exhale in Years -“.