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Author Topic: How are fractals affected/changed by entropy?  (Read 2626 times)

Offline Alinta

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How are fractals affected/changed by entropy?
« on: 18/12/2013 05:37:00 »
Hi all,

Can someone give me a simple explanation for how entropy affects the fractal dimensions of things in the natural world? For example, I've just been reading about how the coastline of Britain has an approximate fractal dimension of 1.26 (1 being not fractal, and heading towards increasing fractal complexity, shown mathematically as 2). Due to the book I am reading, I am only assuming that this number was found by Felix Hausdorff in the early 1900s. According to the same book, a typical cloud has a fractal dimension of 1.35, so a bit more fractal.

So, taking the coastline example (but it could also apply to mountains and trees), my question is, if we can give fractal dimensions, as a number (1.26), to natural edges like this, how do we posit this against changes in the landscape due to things like ocean levels and wind erosion? Does this mean that something like a coastline would become less fractal with time? Or will these things (and entropy in general) actually make no difference to the fractal nature of the world? In other words, will things change, but remain as fractal as they ever were, even when the phenomena that is changing it is something other/outside of itself? (eg the sea or the wind). - Sorry, that last sentence sounded a bit silly, I guess I am having trouble articulating this when I don't yet know very much about it, which is why I am here. :)

I'm sure there are a number of articles that may explain this connection between fractals and entropy, however I have no formal training in any of this, so I would really super duper appreciate a more accessible and practical explanation. :)

Cheers,
Alinta

PS. Not sure what subheading to put this under (physics or the environment) so I'm just going to put it here...


 

Offline evan_au

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Re: How are fractals affected/changed by entropy?
« Reply #1 on: 18/12/2013 10:46:01 »
I can't give a general answer, but here are a few specific examples, on different scales:
  • A single rough stone in a stream eventually tends towards becoming a smoother pebble. Friction tends to rub off the rough edges, over time. Having rubbed off, there are no counteracting forces in the stream trying to stick on extra material in random places to make it rougher again.
  • The edge of a pebble-bottomed stream has a fractal dimension. As individual pebbles get worn down to nothing, over time, new pebbles are eroded from the bedrock to take their place.
  • On a cold, solid planet, weathering and gravity will tend to move material from the high points to the low points, tending towards a spheroid.
  • However, on a planet with active internal convection in a liquid core, the surface will tend to move up and down as driven by the internal fluid flows. If plate tectonics is operating, this will oppose the tendency towards a perfect sphere
So considerations of the energy are important, as expenditure of energy can counteract entropy. 
 

Offline dlorde

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Re: How are fractals affected/changed by entropy?
« Reply #2 on: 18/12/2013 15:23:32 »
Coastlines are complicated and dynamic with regard to fractals, erosion will tend to smooth off rough surfaces reducing their individual fractal dimensions, but also breaks off chunks of land and breaks down larger pieces to smaller pieces, increasing the fractal dimension on a larger scale, but I suspect that overall you get a dynamic equilibrium and the fractal dimension remains relatively constant.
 

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Re: How are fractals affected/changed by entropy?
« Reply #2 on: 18/12/2013 15:23:32 »

 

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