Posted by
Russell Standish on
Jul 24, 2006; 12:36pm
URL: http://friam.383.s1.nabble.com/Definition-of-Complexity-tp522229p522246.html
On Mon, Jul 24, 2006 at 06:38:49AM -0600, Robert Holmes wrote:
> Interesting paper Russell but I don't think I get it yet. Could you clarify
> why entropy is emergent under your definition (" An emergent phenomenon is
> simply one that is described by atomic concepts available in the
> macrolanguage, but cannot be so described in the microlanguage")?
Entropy is give by the Boltzmann-Gibbs formula once the thermodynamic
state variables are fixed (total energy, pressure, temperature and so
on). Nothing in the microscopic description of matter says these are
the relevant state variables.
>
> As far as I can make out this definition could be satisfied by pretty much
> anything that appears in both statistical mechanics and thermodynamics (both
> fields, not just one).
Almost. The classic borderline case is temperature, which is
essentially the average energy per molecule per degree of freedom. Are
averages (or sums) emergent? This is obviously a degenerate case, and
most people get around this by calling this sort of emergence
"resultant". Sort of like saying it is emergent, but not very
interesting. Entropy, by the way is not resultant.
> For example, I can calculate a specific heat capacity
> for an ideal gas in two ways - either the thermo route or the stat mech
> route - and both descriptions work (i.e. give the right answer). Does this
> mean specific heat capacity is emergent? If not, why not? And if it does
> mean that specific heat capacity is emergent does it mean that we need to
> challenge this definiton of emergence because to be honest, I'm not sure
> that Cp would generally be considered an emergent property?
As for heat capacity, this is proportionality between heat put in, and
the resulting rise in temperature. This is given by the number of
degrees of freedom of the molecule (and also typically renormalised by
molecular mass), so you would have to say the specific heat is
actually microscopic term (just with a different name) as well, so is
not emergent.
Where it might be emergent is in a heterogenous
substance, where the molecular degrees of freedom differs from
molecule to molecule. Then the specific heat is an average over all
molecules, and is of the same resultant type as temperature.
> Robert
>
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