On Thu, Jan 9, 2014 at 12:24 PM, John Clark <johnkcl...@gmail.com> wrote:
> On Wed, Jan 8, 2014 at 2:41 PM, Jesse Mazer <laserma...@gmail.com> wrote > > > For example, in Life one could define macrostates in terms of the ratio >> of white to black cells [...] >> > > In the Game of Life the number of black cells is always infinite, so I > don't see how you can do any ratios. > > John K Clark > > Maybe that would be true for some ideal Platonic version of the Game of Life on an infinite board, but any real-world implementation of a cellular automaton involves a finite number of squares--usually this is done with a periodic boundary condition, so squares on the left edge of the finite grid are defined to be neighbors of squares on the right edge, and squares on the top edge of the grid are defined to be neighbors of squares on the bottom edge. Another alternative would be to imagine you do have an infinite grid, but with a starting state where there are only a finite pattern of black squares surrounded by an infinite number of white squares, then you can expand the size of the simulated grid if the region of black squares approaches its border, so that the grid always remains larger than the region of black squares (you don't have to simulate regions beyond that because any region that's all-white on a given time-step, and doesn't have any black squares on its immediate border, will stay all-white on the next time-step). In either of these cases (though it's easier to analyze the periodic example since the grid size remains constant), the ratio of black squares to white squares on the simulated grid region at any given time is well-defined, so one can use this ratio to define the macrostate. And since the rules of the Game of Life aren't reversible, and many different initial states end up either in an all-white end state or an end-state with mostly white and a few blinking black shapes, I'm pretty sure this would be a case where an "entropy" defined in terms of these macrostates would tend to decrease from a randomly-chosen initial finite pattern of black squares. Do you disagree? (even if you're not as confident as I am that this would be true for the Game of Life, one could easily define less "interesting" transition rules where this is obviously the case, like a transition rule that says that only if a black square has a single black neighbor will it remain white, in every other case the square will turn white--hopefully you'd at least agree that in this case, entropy would tend to decrease from a random initial state on a periodic grid). Jesse -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
