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.