So we've seen that phase transitoins
give rise to power laws
at the critical point, at the point where
the system goes from one phase to another
where the transition occurs, many quantities
of interest are distributed according to
a power law, and that's not the case
on either side of that transition.
So, power laws and phase transitions
are closely linked. The question then is
what fraction of the power laws that
we observe more generally in the
study of complex systems can be said to
be due to a phase-transition-like behavior
of some sort? So that's a question that I want
to address from a number of different
angles in this video. And I should mention
that, much more than some of the other
videos, I'll be giving some opinions,
and less an accounting of mathematical
or empirical facts. I think the
position that I'm going to carve out is
pretty much a standard one
within the study of complex systems,
but there is certainly some room for
disagreement. So, what fraction of
the power laws that we observe in
complex systems arise from phase transitions?
I think the answer is, "almost none".
That phase transitions, likely, are not
an explanation for the vast majority
of the power laws we see in complex systems
Why do I say that? Well, a phase transition is
a very unusual state of affairs. The
phase transition, it's a critical point...
a critical temperature, or transition probability
only one out of a whole number of
different things... so it's very unlikely
that we would sort of encounter that by chance.
Here in the physical world, things are
very rarely poised right at the critical point
Right between liquid and solid, or
magnet and non-magnet. Things tend to be
solid or liquid or gas. And not poised
right at that point in between the two.
So, um, because phase transitions,
by definition, occur at a very narrow, a very particular
point, a very particular set of parameters, it seems unlikely
that that could be a generic explanation
for power laws that we see quite commonly
throughout a whole range of social and biological and technological
phenomena. But phase transitions and
power laws have often been closely linked,
more so in the past, less so these days,
and I want to offer some thoughts on why
that may be. I think a lot of it has to do
with the culture and habits of mind of
physicists. So, let me explain what
I mean by this. So, within physics,
and I say this as a physicist, the theory
of phase transitions, also known as the
theory of critical phenomena, is a fantastic theory.
It has everything that a physicist would
get excited about. It explains a broad range of phenomena
in fairly simple terms... so this is the
idea of universality, that many different
transitions, even in systems that seem
very different, are characterized by
the same exponents, so it collects
a lot of phenomena together into a
similar quantitative framework.
There's some non-trivial mathematics behind it,
the renormalization group, explaining
why some of this is so. So it's the
sort of thing that most physicists just love.
It's a great theory. It deservedly got a
lot of attention in the 70s and the 80s
I would say... So the theory of
critical phenomena is a significant accomplishment
within the study of physics, both
theoretical and experimental. So a lot of
the claims that linked power
laws and other areas of complex systems
with phase transitions were originated
by physicists, who were accustomed to
associating power laws and phase transitions.
And again, phase transitions are seen
as a very interesting thing in physics,
It's an unusual state of affairs,
difficult to explain, but then can be
understood with some mathematics.
So there's a fascination, one is drawn
towards phase transitions. And phase
transitions give rise to power laws and so
in the minds of many physicists, the two
are closely linked. And of course that's
right, it is indeed the case that power laws
arise from phase transitions, but as we've
seen in this uinit, power laws also arise
from many many other types of
situations that have nothing to do with
a phase transition. This is something
that many physicists, I think, weren't
aware of. And so when they saw
power law behavior, they were quick to say
"Oh, this must indicate some sort
of a phase transition," that the system
is poised between two different states
A state of order and disorder.
So I think physicists, first, tended
to see, be drawn towards
power laws in the first place,
because power laws are associated with
phase transitions, which are interesting.
And then, seeing power laws,
immediately associated them with
phase transitions and critical phenomena.
There is one other reason having to do with
the culture of physicists that may
suggests why there was a little bit of
hype, or some alleged power laws
that turned out on reexamination
to not be that well-described by
a power law, and that is that
physicists tend to not learn much
statistics. It's rarely a degree requirement for physicists.
My background is in statistical physics but I was never required
to take a statistics class. And insofar
as we do learn things like statistics,
it's more about error propagation in
experiments, and less about testing hypotheses and model
verification. So those are skills that
are taught more often in the sciences,
I think, in biology and economics,
and less in physics. And, of course
it's taught in statistics all the time.
So all that is to say is that
physicists didn't necessarily know
about some of the more advanced or
modern data analysis techniques
that I described in unit 5, leading
some to claim that power laws were present
in light of, maybe, not such convincing evidence.
So... to sum up, phase transitons
most definitely do give rise to power laws. Of that there is
no dispute. But there are many other
mechanisms that give rise to power laws as well.
So I think that the physicists'
understandable fascination
with phase transitions and power laws
has sometimes maybe extended a little
too far into the field of complex systems.
That phase transitions are beautiful physics,
and a really impressive theory,
but maybe not always so useful
in the study of complex systems.