Definition of Complexity

Posted by Robert Holmes on Jul 26, 2006; 4:30am
URL: http://friam.383.s1.nabble.com/Definition-of-Complexity-tp522229p522265.html

OK, I think I'm getting it. It really is down to whether the microlanguage
is the same as the macrolanguage (no emergence) or different (emergence).

But doesn't that lead to an extraordinarily broad definition of emergence?
For example, my macrolanguage for describing gravity involves mass and G and
inverse square laws. But my microlanguage either involves gravitons (if I'm
a particle physicist) or curved spacetime (if I'm a general relativist). The
fact that either of these microlanguages give the same results as the
macrolanguage in the classical limit in no way implies that the micro-and
macro-languages are the same (exactly as with the micro- and macro-language
descriptions of entropy). So gravity is emergent.

So if entropy is emergent and gravity is emergent and any other force
mediated by a subatomic particle is emergent, just how useful is it to label
something 'emergent' in this way? If the definition of emergence is so
broad, how can we usefully use it?

Robert



On 7/24/06, Russell Standish <r.standish at unsw.edu.au> wrote:

>
> On Tue, Jul 25, 2006 at 06:46:12PM -0600, Robert Holmes wrote:
> > >
> > >
> > >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.
> >
> >
> > I'm surprised that you consider the partition function as being "in
> > addition" to the microscopic description. Is this the common view in
> > statistical mechanics? Just to be specific, if I've got a system of
> > distinguishable particles and the energy levels aren't degenerate, the
> > single particle partition function Zsp is given by:
> >
> > Zsp = sum( exp( -ei/k.T ) )
> > where ei is the energy of the energy level i, the sum is over all i (i.e
> .
> > over all energy levels), k is the Boltzmann constant and T is the
> > temperature.
> >
> > Now that seems about as microscopic description of a system as you can
> get.
> > Could you explain why it's not please?
> >
> > Thanks for your patience!
> >
> > Robert
>
> You have just written the canonical partition function. This assumes
> that the universe is divided into two parts, the system, and its
> environment, and that these are in thermal contact with each other.
>
> If you further assume that particles can move between the system and
> environment, then you get the grand canonical partition function:
>
> Z=\sum_{N=0}^{\infty}\sum_{{n_i}}\prod_i exp(-n_i(E_i-\mu)/kT)
>
> These assumptions are not microscopic in nature, but how we choose
> to divide up physical reality. (The choice is needn't be arbitrary - in
> most stat phys situations, there is a clear "best choice", and choosing
> any other way of looking at the system is crazy, but you must
> recognise that it is still a choice independent of microscopic dynamics).
>
> Cheers
>
> --
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