New Hard Steps Results

If planets like ours are common but intelligent life like ours is rare, then it should be rare that life on a planet evolves to our level of development before life is no longer possible on that planet.  If Earth was “lucky” in this way, and if life had to go through a series of stages of varying difficulty to reach our level, how long should each stage have taken?

Now these stages could be of quite different difficulties, taking quite different unconditional expected times to complete.  But back in ’98 I noticed (and posted) an interesting non-intuitive result: if each stage is “exponential,” with a constant per time chance c to jump to the next level, then all “hard step” durations are similarly distributed, no matter what their relative difficulty.  (Joint step times are drawn from a uniform distribution.)  So we should see a history of roughly equally spaced hard step transition events in Earth’s history.

Prof. David J. Aldous, of U.C. Berkeley Dept. of Statistics, has just posted some generalizations of this result. While my result generalizes trivially to any per time success chance function C(t) that is nearly a constant C(t) = c near t=0, Aldous also generalized my similarly-distributed result to any function that is nearly linear C(t) = c*t near t=0.  He also generalized my result to any arbitrary tree of possible paths.  Each link in the tree can have arbitrarily varying difficulty, at each node in the tree many processes compete to be the first to succeed, and the one that wins this contest determines the system’s direction in the tree.

While Aldous warns us against over-reliance on simple models, this does I think gives a bit more reason to expect our history to consist of a sequence of roughly equally spaced hard step transitions.

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  • Jess Riedel

    Does this relate to your perspective on human economic growth rates: periods of constant growth punctuated by sudden increases in growth rate, where the periods are of decreasing length?

    • Jess Riedel

      More specifically, is there some way you could argue that the relevant time scale against which to measure the period between evolutionary hurdles is roughly constant, but the time scale to measure economic hurdles is related to the current economic growth rate (and is therefore decreasing)?

  • Khoth

    I’m not following the logic from “each interval has about the same expected length” to “you’d expect the intervals to be roughly equally spaced”.

    If you roll a die many times, the expected length of the gap between each six is the same, but there will amost certainly be a great deal of clumpiness in the distribution you actually get.

    • Khoth

      (for example, in your paper, for your monte carlo simulations, the deviation for each step is generally not much less than the mean, which suggests that although things are on average uniform, you’ll have to sample rather a lot of worlds before you see that uniformity)

    • Jess Riedel

      I think that’s what he means by “roughly”. The intervals might reasonably vary by a factor of 5, or so, but that’s nowhere near the variance you’d might naively expect since the “hardness” of each jump might differ by many orders of magnitude.

    • Jess is right here.

  • Can’t explain why, but I read this and keep thinking of Mandelbrot.

  • These papers both seem on the impenetrable side to me 🙁

    • Khoth

      I think the basic idea behind them is that if an event is so unlikely that the chance of it occurring in a planet’s lifetime is small, but the event is vital for our existence, then we can ignore the possibility that it took more than a few billion years to happen (because if it had, we wouldn’t be here).

      If you look at just the very first part of an exponential distribution, before it’s had a chance to decay much, it looks pretty flat, ie a uniform distribution from zero to a few billion years. The thing is, that’s the same whether the event was 1/100-chance-per-planet rare or 1/1000000 chance. By conditioning on the event actually having happened in only a few billion years, you’ve sliced off the massive long tail from the distributions, leaving only an initial part that looks much the same for each.

    • Khoth is right here.

  • Page 3 of the Hard Step paper from 1998 seems problematical – and its conclusions seem pretty unbelievable and far out – essentially because it apparently derives a lot from an armchair with little reference to evidence. Readers will probably have limited attention spans for this kind of material. I think it is in need of some examples or graphs – or something. It would of course be nice to make this kind of statement about evolution but things like irregular large asteroid impacts make the project challenging.