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
microstate
(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.
Brent
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