In this class, I tried to explain why "power-law" behaviour are important.
In equibrium, we see power-law only at criticality. It is difficult to mantain
equilibrium criticality, because we have to tune  parameters  to critical
points.  Many non-equilibrium systems however self-organized to show
scale-free behaviour.  The self organized criticality  (SOC) is an important
concept in  non-equilibrium  physics.


The above figure  describes few things about   phase transition.  

Change is nature.  (Why things change ?)

Changes occur  from the  microscopic dynamics.
Although  dynamics act in the micro-scale,  sometimes macroscopic physical
changes are observed as  the system goes from  one phase to the other. In other
words phase transitions (for example  metal becoming a super-conductor or  a magnet)
are  always associated with macroscopic changes.  At the transition point, thus , the
system needs a diverging length scale (\xi) to connect the micro and the macro world.
Once you  have a diverging  length scale, all other  existing (length) scales become
un-important  (compared to \xi) and the system  becomes scale free. 

Mathematically  an arbitrary function f(x) is said to be scale free (scale-invariance)
when f(kx) =constant* f(x).
One can prove now  f(x) has the form
f(x) ~ x^a.

Thus, at criticalty,  all physical quantities  q_i diverge (when a<0) or vanish (when a>0) as
     q_i (T) ~ (T-T_c)^a_i .
a_i  are called the the critical  exponents or scaling exponents. 
What does a_i depend on ?
It turns out that a_i  are "universal", they do not dpend on details of the model,
depends only on the (space-) dimension of the system and the symmetry of the
order-parameter ?

What is a order parameter ?
Since during phase transition, the system  goes from the order to a dis-ordered phase (or the
other way) one can define a parameter which  is a quantitative  measure of order, name
it 'order-parameter' .  It could be a  scaler or a vector or  it might have a discrete or continuous
symmetry, which is all what matter in defining the universality class .  A set of universal values
of exponets a_i  defines the universality class.


I also did the following  :

Stationary stochastic process => probability  distribution P(C)
Stochasticity comes from the "loss of predictibility"
CM => chaos
QM => Uncertainty

Once we have P(C), we can measure any physical observable.


 


(There  are  many typo in  this note, I am just typing them  and would check it later).