Definition of Complexity

[Joseph L. Breeden]

I'm usually very quiet on this group. I almost always follow the
discussions and often look up the references, but I must say that you've
hit on a topic that has been bothering me for a decade. I did my thesis
work applying chaos theory to astrophysical systems (about 15 years
ago). It was always critically important that we could define what a
chaotic system was, we had statistical tools for showing that a system
was probably chaotic according to the scientific definition, and there
was a rapidly growing body of mathematical literature (not all of which
I could follow) providing a theoretical basis.

Complexity theory troubles me because it is treated like pornography. "I
know it when I see it." I remember a brief discussion around the launch
of the Journal of Complexity (I think it was that one), where someone
asked, "Don't we need a definition of complexity to have a journal of
complexity?" They were rebuffed by the editors with the comment that
"the submitting authors will create the definition".

I am sympathetic to the difficulty in defining complexity, but I have
always felt that the lack of a clear definition is the primary thing
holding back complexity theory. With chaos theory, if someone publishes
a book on "chaos theory in literary review of the renaissance" (don't
laugh), we have tools to point out that they are abusing a
mathematically grounded scientific term (even if the choice of the word
"chaos" is partly responsible for the abuses). In complexity, I lack the
tools to go to the author of a book on "complexity theory in business
management" and discuss whether it is being used properly or the author
is just stealing a term for purposes of marketing.

So, this is where I am out of date. At this point, do you all consider
chaos theory to be a subset of complexity? (I have my doubts, since
three bodies in orbit are chaotic, but are they "complex"?) Owen listed
some useful statistics to compute to identify chaos theory, but are any
of these or the Reynolds number really viewed as a definition of
complexity? (Robert is pursuing this question and I'm glad to read it.)
Do you believe that a definition (verbal or mathematical) of complexity
now exists which would allow a practitioner to confirm that a system is
"complex"? Again, I'm showing how long ago I worked in this area, but
complexity always seemed to be defined in terms of "emergence", which
also had a troubling definition -- along the lines of "something we
didn't expect". Again very bad.

I've asked too many questions for this kind of forum, but if a seminal
paper has come along in the last decade which resolves all this, I would
greatly appreciate a reference.

Thanks much, and I'm sorry if I've stepped on any toes. I tend to go
stomping about without my glasses rather often.

Joe Breeden

[Russell Standish]

Have you read my paper "On Complexity and Emergence"? I wrote it not
to propose new definitions of complexity or emergence, but rather to
explain that some rather old ideas on the subject actually do work,
and show how the many varied and disparate attempts are related to
each other, and also why people get so confused on the topic. In fact
I tried to explain that emergence had a well-defined definition on a
radio chat show back in 2002, but got gazumped by the "we can't define
emergence, but we know it when we see it" from one of the other
panelists. It was a bit of an eye-opener for me as to how radio science shows
work :)

Since I never considered the ideas in the paper to be original, I have
been enormously surprised at the citation popularity of the paper.

Its not a big paper, so its worth a read. As always, I appreciate
comments...

Cheers

On Sun, Jul 23, 2006 at 09:56:11PM -0600, Joseph L. Breeden wrote:

>
> I'm usually very quiet on this group. I almost always follow the
> discussions and often look up the references, but I must say that you've
> hit on a topic that has been bothering me for a decade. I did my thesis
> work applying chaos theory to astrophysical systems (about 15 years
> ago). It was always critically important that we could define what a
> chaotic system was, we had statistical tools for showing that a system
> was probably chaotic according to the scientific definition, and there
> was a rapidly growing body of mathematical literature (not all of which
> I could follow) providing a theoretical basis.
>
> Complexity theory troubles me because it is treated like pornography. "I
> know it when I see it." I remember a brief discussion around the launch
> of the Journal of Complexity (I think it was that one), where someone
> asked, "Don't we need a definition of complexity to have a journal of
> complexity?" They were rebuffed by the editors with the comment that
> "the submitting authors will create the definition".
>
> I am sympathetic to the difficulty in defining complexity, but I have
> always felt that the lack of a clear definition is the primary thing
> holding back complexity theory. With chaos theory, if someone publishes
> a book on "chaos theory in literary review of the renaissance" (don't
> laugh), we have tools to point out that they are abusing a
> mathematically grounded scientific term (even if the choice of the word
> "chaos" is partly responsible for the abuses). In complexity, I lack the
> tools to go to the author of a book on "complexity theory in business
> management" and discuss whether it is being used properly or the author
> is just stealing a term for purposes of marketing.
>
> So, this is where I am out of date. At this point, do you all consider
> chaos theory to be a subset of complexity? (I have my doubts, since
> three bodies in orbit are chaotic, but are they "complex"?) Owen listed
> some useful statistics to compute to identify chaos theory, but are any
> of these or the Reynolds number really viewed as a definition of
> complexity? (Robert is pursuing this question and I'm glad to read it.)
> Do you believe that a definition (verbal or mathematical) of complexity
> now exists which would allow a practitioner to confirm that a system is
> "complex"? Again, I'm showing how long ago I worked in this area, but
> complexity always seemed to be defined in terms of "emergence", which
> also had a troubling definition -- along the lines of "something we
> didn't expect". Again very bad.
>
> I've asked too many questions for this kind of forum, but if a seminal
> paper has come along in the last decade which resolves all this, I would
> greatly appreciate a reference.
>
> Thanks much, and I'm sorry if I've stepped on any toes. I tend to go
> stomping about without my glasses rather often.
>
> Joe Breeden
>
> ============================================================
> FRIAM Applied Complexity Group listserv
> Meets Fridays 9a-11:30 at cafe at St. John's College
> lectures, archives, unsubscribe, maps at http://www.friam.org

--
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is of type "application/pgp-signature". Don't worry, it is not a
virus. It is an electronic signature, that may be used to verify this
email came from me if you have PGP or GPG installed. Otherwise, you
may safely ignore this attachment.

----------------------------------------------------------------------------
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Mathematics                               0425 253119 (")
UNSW SYDNEY 2052                 R.Standish at unsw.edu.au            
Australia                                http://parallel.hpc.unsw.edu.au/rks
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[Robert Holmes]

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")?

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). 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?
Robert

On 7/21/06, Russell Standish <r.standish at unsw.edu.au> wrote:

>
> Have you read my paper "On Complexity and Emergence"? I wrote it not
> to propose new definitions of complexity or emergence, but rather to
> explain that some rather old ideas on the subject actually do work,
> and show how the many varied and disparate attempts are related to
> each other, and also why people get so confused on the topic. In fact
> I tried to explain that emergence had a well-defined definition on a
> radio chat show back in 2002, but got gazumped by the "we can't define
> emergence, but we know it when we see it" from one of the other
> panelists. It was a bit of an eye-opener for me as to how radio science
> shows
> work :)
>
> Since I never considered the ideas in the paper to be original, I have
> been enormously surprised at the citation popularity of the paper.
>
> Its not a big paper, so its worth a read. As always, I appreciate
> comments...
>
> Cheers
>
> On Sun, Jul 23, 2006 at 09:56:11PM -0600, Joseph L. Breeden wrote:
> >
> > I'm usually very quiet on this group. I almost always follow the
> > discussions and often look up the references, but I must say that you've
> > hit on a topic that has been bothering me for a decade. I did my thesis
> > work applying chaos theory to astrophysical systems (about 15 years
> > ago). It was always critically important that we could define what a
> > chaotic system was, we had statistical tools for showing that a system
> > was probably chaotic according to the scientific definition, and there
> > was a rapidly growing body of mathematical literature (not all of which
> > I could follow) providing a theoretical basis.
> >
> > Complexity theory troubles me because it is treated like pornography. "I
> > know it when I see it." I remember a brief discussion around the launch
> > of the Journal of Complexity (I think it was that one), where someone
> > asked, "Don't we need a definition of complexity to have a journal of
> > complexity?" They were rebuffed by the editors with the comment that
> > "the submitting authors will create the definition".
> >
> > I am sympathetic to the difficulty in defining complexity, but I have
> > always felt that the lack of a clear definition is the primary thing
> > holding back complexity theory. With chaos theory, if someone publishes
> > a book on "chaos theory in literary review of the renaissance" (don't
> > laugh), we have tools to point out that they are abusing a
> > mathematically grounded scientific term (even if the choice of the word
> > "chaos" is partly responsible for the abuses). In complexity, I lack the
> > tools to go to the author of a book on "complexity theory in business
> > management" and discuss whether it is being used properly or the author
> > is just stealing a term for purposes of marketing.
> >
> > So, this is where I am out of date. At this point, do you all consider
> > chaos theory to be a subset of complexity? (I have my doubts, since
> > three bodies in orbit are chaotic, but are they "complex"?) Owen listed
> > some useful statistics to compute to identify chaos theory, but are any
> > of these or the Reynolds number really viewed as a definition of
> > complexity? (Robert is pursuing this question and I'm glad to read it.)
> > Do you believe that a definition (verbal or mathematical) of complexity
> > now exists which would allow a practitioner to confirm that a system is
> > "complex"? Again, I'm showing how long ago I worked in this area, but
> > complexity always seemed to be defined in terms of "emergence", which
> > also had a troubling definition -- along the lines of "something we
> > didn't expect". Again very bad.
> >
> > I've asked too many questions for this kind of forum, but if a seminal
> > paper has come along in the last decade which resolves all this, I would
> > greatly appreciate a reference.
> >
> > Thanks much, and I'm sorry if I've stepped on any toes. I tend to go
> > stomping about without my glasses rather often.
> >
> > Joe Breeden
> >
> > ============================================================
> > FRIAM Applied Complexity Group listserv
> > Meets Fridays 9a-11:30 at cafe at St. John's College
> > lectures, archives, unsubscribe, maps at http://www.friam.org
>
> --
> *PS: A number of people ask me about the attachment to my email, which
> is of type "application/pgp-signature". Don't worry, it is not a
> virus. It is an electronic signature, that may be used to verify this
> email came from me if you have PGP or GPG installed. Otherwise, you
> may safely ignore this attachment.
>
>
> ----------------------------------------------------------------------------
> A/Prof Russell Standish                  Phone [phone] (mobile)
> Mathematics                                    0425 253119 (")
> UNSW SYDNEY 2052                         R.Standish at unsw.edu.au
> Australia
> http://parallel.hpc.unsw.edu.au/rks
>             International prefix  +612, Interstate prefix 02
>
> ----------------------------------------------------------------------------
>
>
> ============================================================
> FRIAM Applied Complexity Group listserv
> Meets Fridays 9a-11:30 at cafe at St. John's College
> lectures, archives, unsubscribe, maps at http://www.friam.org
>

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[Russell Standish]

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
>

--
*PS: A number of people ask me about the attachment to my email, which
is of type "application/pgp-signature". Don't worry, it is not a
virus. It is an electronic signature, that may be used to verify this
email came from me if you have PGP or GPG installed. Otherwise, you
may safely ignore this attachment.

----------------------------------------------------------------------------
A/Prof Russell Standish                  Phone [phone] (mobile)
Mathematics                               0425 253119 (")
UNSW SYDNEY 2052                 R.Standish at unsw.edu.au            
Australia                                http://parallel.hpc.unsw.edu.au/rks
            International prefix  +612, Interstate prefix 02
----------------------------------------------------------------------------

[Phil Henshaw-2]

I think you're right on point.  There is a lot of theory without
referent in this general field (as there is or was in various other
complex systems theory movements).  Maybe because this one came more out
of physics the fuzzy language has a physics twang.  There's that magical
notion of "the edge of chaos" for example.   I think it has some
mathematical meaning in a specific chaotic system, but then I saw it
defined as an environmental variable in a virtual complex systems
computer model, and then saw it widely used in the popular press as the
statistical middle ground between predictability and unpredictability.
The idea conveyed in discussion with both uses is that there's something
like "the edge of play" in a game of daring that in inherently creative
in nature, but nobody can point to what they mean by it, or who the
player is.   I have some examples of things sort of like that, and it's
tempting to play on the popularity of the term, but it may not mean
anything.


Phil Henshaw                       ????.?? ? `?.????
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
680 Ft. Washington Ave
NY NY 10040                      
tel: [phone]                
e-mail: pfh at synapse9.com          
explorations: www.synapse9.com    

> -----Original Message-----
> From: friam-bounces at redfish.com
> [mailto:friam-bounces at redfish.com] On Behalf Of Joseph L. Breeden
> Sent: Sunday, July 23, 2006 11:56 PM
> To: friam at redfish.com
> Subject: [FRIAM] Definition of Complexity
>
>
>
> I'm usually very quiet on this group. I almost always follow
> the discussions and often look up the references, but I must
> say that you've hit on a topic that has been bothering me for
> a decade. I did my thesis work applying chaos theory to
> astrophysical systems (about 15 years ago). It was always
> critically important that we could define what a chaotic
> system was, we had statistical tools for showing that a
> system was probably chaotic according to the scientific
> definition, and there was a rapidly growing body of
> mathematical literature (not all of which I could follow)
> providing a theoretical basis.
>
> Complexity theory troubles me because it is treated like
> pornography. "I know it when I see it." I remember a brief
> discussion around the launch of the Journal of Complexity (I
> think it was that one), where someone asked, "Don't we need a
> definition of complexity to have a journal of complexity?"
> They were rebuffed by the editors with the comment that "the
> submitting authors will create the definition".
>
> I am sympathetic to the difficulty in defining complexity,
> but I have always felt that the lack of a clear definition is
> the primary thing holding back complexity theory. With chaos
> theory, if someone publishes a book on "chaos theory in
> literary review of the renaissance" (don't laugh), we have
> tools to point out that they are abusing a mathematically
> grounded scientific term (even if the choice of the word
> "chaos" is partly responsible for the abuses). In complexity,
> I lack the tools to go to the author of a book on "complexity
> theory in business management" and discuss whether it is
> being used properly or the author is just stealing a term for
> purposes of marketing.
>
> So, this is where I am out of date. At this point, do you all
> consider chaos theory to be a subset of complexity? (I have
> my doubts, since three bodies in orbit are chaotic, but are
> they "complex"?) Owen listed some useful statistics to
> compute to identify chaos theory, but are any of these or the
> Reynolds number really viewed as a definition of complexity?
> (Robert is pursuing this question and I'm glad to read it.)
> Do you believe that a definition (verbal or mathematical) of
> complexity now exists which would allow a practitioner to
> confirm that a system is "complex"? Again, I'm showing how
> long ago I worked in this area, but complexity always seemed
> to be defined in terms of "emergence", which also had a
> troubling definition -- along the lines of "something we
> didn't expect". Again very bad.
>
> I've asked too many questions for this kind of forum, but if
> a seminal paper has come along in the last decade which
> resolves all this, I would greatly appreciate a reference.
>
> Thanks much, and I'm sorry if I've stepped on any toes. I
> tend to go stomping about without my glasses rather often.
>
> Joe Breeden
>
> ============================================================
> FRIAM Applied Complexity Group listserv
> Meets Fridays 9a-11:30 at cafe at St. John's College
> lectures, archives, unsubscribe, maps at http://www.friam.org
>
>

[Robert Holmes]

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
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[Russell Standish]

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

--
*PS: A number of people ask me about the attachment to my email, which
is of type "application/pgp-signature". Don't worry, it is not a
virus. It is an electronic signature, that may be used to verify this
email came from me if you have PGP or GPG installed. Otherwise, you
may safely ignore this attachment.

----------------------------------------------------------------------------
A/Prof Russell Standish                  Phone [phone] (mobile)
Mathematics                               0425 253119 (")
UNSW SYDNEY 2052                 R.Standish at unsw.edu.au            
Australia                                http://parallel.hpc.unsw.edu.au/rks
            International prefix  +612, Interstate prefix 02
----------------------------------------------------------------------------

[Robert Holmes]

>
>
> 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
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[Russell Standish]

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

--
*PS: A number of people ask me about the attachment to my email, which
is of type "application/pgp-signature". Don't worry, it is not a
virus. It is an electronic signature, that may be used to verify this
email came from me if you have PGP or GPG installed. Otherwise, you
may safely ignore this attachment.

----------------------------------------------------------------------------
A/Prof Russell Standish                  Phone [phone] (mobile)
Mathematics                               0425 253119 (")
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Australia                                http://parallel.hpc.unsw.edu.au/rks
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[Phil Henshaw-2]

perhaps because it's a sum?
 
 

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-----Original Message-----
From: [hidden email] [mailto:[hidden email]] On
Behalf Of Robert Holmes
Sent: Tuesday, July 25, 2006 8:46 PM
To: The Friday Morning Applied Complexity Coffee Group
Subject: Re: [FRIAM] Definition of Complexity




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



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[Robert Holmes]

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
>
> --
> *PS: A number of people ask me about the attachment to my email, which
> is of type "application/pgp-signature". Don't worry, it is not a
> virus. It is an electronic signature, that may be used to verify this
> email came from me if you have PGP or GPG installed. Otherwise, you
> may safely ignore this attachment.
>
>
> ----------------------------------------------------------------------------
> A/Prof Russell Standish                  Phone [phone] (mobile)
> Mathematics                                    0425 253119 (")
> UNSW SYDNEY 2052                         R.Standish at unsw.edu.au
> Australia
> http://parallel.hpc.unsw.edu.au/rks
>             International prefix  +612, Interstate prefix 02
>
> ----------------------------------------------------------------------------
>
>
> ============================================================
> FRIAM Applied Complexity Group listserv
> Meets Fridays 9a-11:30 at cafe at St. John's College
> lectures, archives, unsubscribe, maps at http://www.friam.org
>

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[Russell Standish]

Emergence is model dependent thing, and yes, as you point out
Newtonian gravity is emergent from General Relativity in the near flat
spacetime limit. As for string theory (or whatever), I think the grand
hope is for all of present day physics to be emergent from whatever
that theory says is the fundamental things, be they wiggly strings or
whatever.

But I don't think this is being overly broad. Did I
refer you to Mark Bedau's Principia paper where he discusses nominal,
weak and strong emergence? His nominal emergence is pretty much as I
define it. He proposes something called weak emergence, which relates
to simulability of a system.

Cheers


On Tue, Jul 25, 2006 at 10:30:07PM -0600, Robert Holmes wrote:

> 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
> >
> >--
> >*PS: A number of people ask me about the attachment to my email, which
> >is of type "application/pgp-signature". Don't worry, it is not a
> >virus. It is an electronic signature, that may be used to verify this
> >email came from me if you have PGP or GPG installed. Otherwise, you
> >may safely ignore this attachment.
> >
> >
> >----------------------------------------------------------------------------
> >A/Prof Russell Standish                  Phone [phone] (mobile)
> >Mathematics                                    0425 253119 (")
> >UNSW SYDNEY 2052                         R.Standish at unsw.edu.au
> >Australia
> >http://parallel.hpc.unsw.edu.au/rks
> >            International prefix  +612, Interstate prefix 02
> >
> >----------------------------------------------------------------------------
> >
> >
> >============================================================
> >FRIAM Applied Complexity Group listserv
> >Meets Fridays 9a-11:30 at cafe at St. John's College
> >lectures, archives, unsubscribe, maps at http://www.friam.org
> >

> ============================================================
> FRIAM Applied Complexity Group listserv
> Meets Fridays 9a-11:30 at cafe at St. John's College
> lectures, archives, unsubscribe, maps at http://www.friam.org

--
*PS: A number of people ask me about the attachment to my email, which
is of type "application/pgp-signature". Don't worry, it is not a
virus. It is an electronic signature, that may be used to verify this
email came from me if you have PGP or GPG installed. Otherwise, you
may safely ignore this attachment.

----------------------------------------------------------------------------
A/Prof Russell Standish                  Phone [phone] (mobile)
Mathematics                               0425 253119 (")
UNSW SYDNEY 2052                 R.Standish at unsw.edu.au            
Australia                                http://parallel.hpc.unsw.edu.au/rks
            International prefix  +612, Interstate prefix 02
----------------------------------------------------------------------------