Posted by
Russell Standish on
Jul 24, 2006; 5:50pm
URL: http://friam.383.s1.nabble.com/Definition-of-Complexity-tp522229p522252.html
On Mon, Jul 24, 2006 at 09:38:50PM -0600, Robert Holmes wrote:
> On 7/24/06, Russell Standish <r.standish at unsw.edu.au> wrote:
> >
> >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.
>
>
> Still don't get it. I suppose it depends on what you mean by "nothing in
> the microscopic description says these are the relevant state variables". If
> this means that Boltzmann's postulate S = k.ln(omega) doesn't explicitly
> contain U, F, etc. then you are right. But actually I don't need many more
> equations to derive precise equations for U and F. Specifically, all I need
> is dU = dQ + dW and dQ = TdS and after a page or two of math I've got
> equations for U, S and F in terms of the partition function (see for example
> Glazer & Wark, "Statistical Mechanics: A Survival Guide"). So I don't quite
> see how you can say that the microscopic description doesn't tell us about
> the macro description. As G & W put it: "if we know the partition function
> for a particular system we then know all of the thermodynamic functions. It
> is difficult to overstress the importance of this."
>
> What am I missing?
>
> Robert
One can certainly start from the partition function. But the partition
function is something that is additional to the microscopic
description, hence emergent. Indeed, the partition function is
different depending on whether you are using microcanonical, canonical
or grand canonical ensembles, each of which is a thermodynamic, not
microscopic concept.
Alternatively, your thermodynamic equations dQ=TdS is something
additional also (there is no S for a start, and the equation can only
be applied adiabatically (which is a thermodynamic, not microscopic
concept)). Even the dU=dQ+dW equation is not microscopic - as this
refers to a partitioning of the universe into a system and its
environment.
Cheers
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