On 1/14/2014 8:33 AM, John Clark wrote:

     > but rather as the number of possible microstates the system might be in 
at this
    moment given that we only know the macrostate

We don't even know for a fact that some macroscopic objects, like Black Holes for example, even contain microstates; in fact the present thinking (a minority disagrees) is that probably they don't and a Black Hole can be completely described by just 3 numbers, its mass, spin, and electric charge. A Black Hole contains enormous entropy because there are a gargantuan number of ways it could have been formed, but if you know those 3 numbers then you know all there is to know about a particular Black Hole. And in the real world only 2 numbers are important because the electric charge is always zero.

    > For example, suppose we consider a very small 2x2 board with only 4 cells 

What are the laws of physics in this new game? A 2x2 board is MUCH too small for the traditional rules of the Game of Life to be applicable.

    > And if the macrostate is "0 black:4 white" there's only one possible 
    (same for "4 black:0 white"), so this is the lowest possible entropy

I don't know about this new game of yours because I don't know what the rules are but in the Game of Life a solid block of nothing but active cells would be in the lowest possible entropy state because the fewest previous states could have produced it. Actually I should have said the lowest impossible entropy state because NO previous state could have produced it, zero.

A solid block of nothing but dead cells would have the highest entropy because more previous states than any other could have produced it, and entropy is the logarithm of the number of those states.

You seem to have a non-standard view of entropy in statistical mechanics. It is NOT the log of the number of ways a macro-state could form. That would be ambiguous in any case (do different order of events count as different ways?...different paths to the events?). The entropy is the log of the number of micro-states consistent with the macro-state. A black hole has entropy because it has a temperature, which implies that it has micro-states. And the relationship between energy and temperature implies that the number of micro-states is the event horizon area measured in Planck units. To say it only has mass, charge, and angular momentum is just to give a classical macro-state description, like saying gas in a vessel just has pressure, temperature, and volume.


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