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Sunday 27 October 2013

Laplace's Demon

This time last year I did a post about Maxwell’s Demon, which featured in a TV documentary by the physicist and science popularizer Jim Al-Khalili. The same subject is discussed in his excellent book Paradox, together with another distantly related demon attributed to Pierre-Simon Laplace (the photo shows a statue of Laplace that I saw at the Palace of Versailles).

The name of Laplace isn’t as well-known to non-physicists as that of Maxwell, although his role in the history of science was just as important. In much the same way that Maxwell consolidated everything that was known about electromagnetism in his famous set of equations, Laplace consolidated Newtonian dynamics in an elegant new formulation that is much easier to work with than Newton’s own rather clunky equations.

If people have heard of Laplace at all, it’s because he supposedly told Napoleon “I had no need of that hypothesis” when asked a question about God. As with many well-known scientific soundbites (cf. Spooky Action at a Distance and Not Even Wrong), this doesn’t mean what it appears to mean. Laplace wasn’t saying he had scientific proof of the nonexistence of God, simply that it wasn’t necessary to invoke divine intervention, as Newton had done, in order to explain the long-term stability of the Solar System.

This brings us to the subject of Laplace’s Demon. Laplace believed the entire workings of the universe were determined by mathematical equations, and that neither random chance nor divine intervention had any role to play. Hence “an intelligence that, at a given instant, knows all the forces by which nature is animated and the respective situation of the entities that compose it, if moreover it was sufficiently vast to submit these data to analysis... nothing would be uncertain for it, and the future, like the past, would be present to its eyes.”

This is an awesome concept if you think about it. The term “demon” was applied retrospectively, by analogy with Maxwell’s demon, but Laplace himself used the more neutral term “intelligence”. In modern terms, he might have been thinking about a supercomputer. But was he right? Is it possible to compute the entire course of future events simply by making a complete set of measurements at a single instant in time?

Two centuries of scientific progress haven’t changed Laplace’s basic premise – the physical universe still seems to conform to immutable laws that can be expressed in the form of mathematical equations. We don’t know all the equations, but we know more of them than Laplace did. Saying that a mathematical equation is rigorously obeyed, however, isn’t the same as saying you can predict its observable consequences. In fields like statistical mechanics and quantum physics, all the important equations are stochastic – they deal in probabilities. The equations allow you to calculate the probability of an event to whatever precision you want, but they don’t tell you if the event will actually occur or not – that’s down to random chance.

In fact, as Jim Al-Khalili points out in his book, you don’t even have to invoke post-Laplacian physics to see the fallacy of Laplace’s Demon – only post-Laplacian number-crunching. The equations Laplace worked with are nonlinear differential equations, which are notoriously difficult to solve. In the days before computers, people had to stick to a small subset of easy solutions, and that’s what Laplace did. But we now know there are always some regions of phase space where the solutions are effectively stochastic – the tiniest change in initial conditions leads to a completely different phase trajectory. This result isn’t limited to complex systems – it’s true even for the classic three-body problem (e.g. Earth – Moon – Sun) that Newton and Laplace worked on.

For my own attempt to calculate the evolution of the Galaxy (using Laplace’s equation) see A Virtual Spaceship.

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