1. The document discusses scaling hypothesis and statistical mechanics in the context of critical phenomena during phase transitions. It introduces concepts like critical exponents, universality, fractals, and ergodicity breaking. 2. It presents Rushbrooke's and Widom's identities which relate critical exponents to each other based on the assumption of homogeneity. This allows reducing the number of independent exponents needed to describe critical behavior. 3. The identities are consistent with experimental data and exact solutions, supporting the scaling hypothesis over saddle-point approximations for describing critical phenomena.

1. SCALING HYPOTHESIS
STATISTICAL MECHANICS
(SAP 2007)
REFERENCE – (MIT Video Lecture 6,7,8
PDFs- Kardar-Chap4
MIT8_334S14_Lec6 )
BY- ADITYA NARAYAN SINGH
I.M.Sc VIITH
SEMESTER
BIT MESRA ,RANCHI

2. As the name 'scaling' itself suggest that it is somehow related to measurements.
In thermodynamics we often can see the sytems violating the standard laws of thermodynamics .
Also in context to the statistical mechanics when we talk about the phase interaction and phase
transition then we face the problem of invalidation of formulations of these systems and these
invalid points is referred as critical points or singularity. In thermo these constants are there
as Tc , Pc , Vc , in the situation like given curve . Liquid liqiud critical points ,represents the
temperature -concentration extremum of the spinodal curves.
Critical phenomenon takes place in second order phase
transition and described by critical exponents,
universality,
fractal behavier, ergodicity breaking, etc.
Ex- power law divergences in ferromagnatism

3. In classical statistical mechanics critical singular points are studied using some constants {a,b,
g,d...} and they have separate meaning in order to read the dependence of these exponents we
need a concept that at least give a relation between these critical exponents as in normal
conditions these are unreliable due to fluctuations. we will try to find minimum number of
independent exponents .
Before we go further we need to understand some basic definitions involved in it .
1. Correlation length: A function that gives the statistical correlation between the random variable contingent on the
spatial or temporal distance between these variables.
2. Power law: A functional relationship between two quantity where a relative change in one quantity results in a
proportional change in other ,whatever be the initial values.
3. Ferromagnetism: A basic mechanism by which certain materials find permanent magnetic properties.
4. Critical exponents: Constants and exponents that are used to describe the behaviour of continuous phase transition ,
but still not proven.
5. Universality: A observation that there are properties for a large class of system that are independent of the dynamical
details of the system.
6. Fractals: Self similar subset of euclidean space whose fractal dimensions strictly exceeds its topological dimensions
and exhibits similar pattern.
7. Ergodicity: Property of dynamical system which expresses the form of irreducibility of system.
8. Homogeneity: Assumption that the statistical property of any part of an overall dataset are the same as any other part.
9. Singularity: Exists when there is a perfect correlation of explanatory variable,Presence of either effects the
interpretation of variables and response variables in regression it is the extreme form of multicollinearity.

4. HOMOGENEITY ASSUMPTIONS
The singular behaviour in the vicinity of a continuous transition was characterized by a set of critical
exponents (d,g,b,a,D..). The saddle-point estimates of these exponents were found to be unreliable due to
the importance of fluctuations. Since the various thermodynamic quantities are related, these exponents can
not be independent of each other. So we will try to find out the dependency.
Fig.2. The vicinity of the critical point in the (t,h) plane, with crossover
boundaries indicated by curved lines
The non-analytical structure is a coexistence line for t < 0 and h = 0 that
terminates at the critical point t = h = 0. The various exponents describe the
leading singular behaviour of a thermodynamic quantity Q(t,h), in the vicinity
of this point. A basic quantity in the canonical ensemble is the free energy,
which in the saddle point approximation is given by following function:
f(t,h) = min[tm2/2+um4 −h.m]m = (-1/16).t2/u for h=0,t<0
(-3/44/3). (h4/u)1/3 for
h≠0, t=0
As we found the experimental values of these exponents by the methods of light scattering
and satellite values, we made a theoretical attempt to find the relations by the method of
saddle point approximation but the deviation was very high, and we have to leave that.

5. These are the experimental results :
To overcome the situation of saddle point we introduced the new concept of scaling
hypothesis. In this we will go with homogeneity approximation and then we will find the
mathematical results establishing the relationship between these exponents, as we have seen
from table that they are not independent.
Before the mathematical results we will just note down the results here :
Rushbrooke’s
identity
Widom’s identity
We will prove it in next slides

11. Few more conclusions :
1. Because of the interconnections via thermodynamic derivatives, the same gap
exponent, D, occurs for all such quantities.
2. All (bulk) critical exponents can be obtained from only two independent ones,
(a ,D).
These identities can be checked against the previously mentioned table of critical exponents.
The first three rows are based on a number of theoretical estimates in d = 3; the last row comes
from an exact solution in d = 2. The exponent identities are completely consistent with these
values, as well as with all reliable experimental data.
IMPORTANT :
Divergence of the correlation length :-
The homogeneity assumption relates to the free energy and quantities derived from it. It says
nothing about the behavior of correlation functions. An important property of a critical point
is the divergence of the correlation length which is responsible for, and can be deduced from,
diverging response functions.