Ising model and Epsilon Expansion Monday, Jan 1 2007 

As one might know that topology plays an important role in the classical O(2) model in d=2 (i.e. x-y model in two dimension) becuase one can have non-trivial excitations, namely, vortices. These excitations, while having a huge energy, can contribute to the partition function at high temperatures becuase the entropy associated with them is also proportionately large. One can easily verify that perturbative epsilon-expansion (i.e. an expansion in no. of dimensions) fails here. The reason which is often cited is that such an expansion destroys all information related to topology since atleast till now there is no homotopy theory for fractional dimensions.

Considering an even simpler case, the ising transition in d=2 could be looked upon as condensation of domain walls which are again topological objects. My question is: why does an epsilon-expansion in this case works out quite nicely? (note: here one does an expansion near the lower critical dimension which is one). Is it that notion of domain walls can be extended to even fractional dimension in some way?

Btw, one might like to visit the website http://www.math.princeton.edu/~aizenman/OpenProblems.iamp/index.html  which contains a collection of open problems in mathematical physics. Coming to a problem related to this post (though somewhat remotely), one might be surprised to find through this link that the perturbative result of O(N > 2) model in d=2 being always in a single disordered phase at all temperatures has actually not been rigourosly proved. Infact, it is even disputed by some numerical results (though I don’t claim any understanding of these). See for example, http://prola.aps.org/abstract/PRB/v54/i10/p7177_1.

Counting of States in a Micro-canonical ensamble Sunday, Apr 23 2006 

We (batch mates) were discussing about time reversibility and Karan pointed out an example of system that is not in equilibrium but seems stable: all the particles in a box are moving back and forth in one direction perpendicular to a wall. Now this system is not in equilibrium but in a meta stable state. 

My question is that there are many-many such possible states and during the counting of the states of micro-canonical ensemble we add all such states. Even though such states can never be achieved by a system in equilibrium. Why even then we have correct results for observables ? 

A hint of renormalization Friday, Apr 21 2006 

Background : Some amount of Quantum Field theory would help. Though the article doesnot go very much deep into quantum field theory as such, it is a bit difficult to understand the context of renormalisation, if you’ ve not seen it either in field theory or statistical mechanics.

Link : http://arxiv.org/abs/hep-th/0212049

A hint of renormalization
Authors: B. Delamotte
Comments: 17 pages, pedagogical article
Subj-class: High Energy Physics – Theory; Statistical Mechanics
Journal-ref: Am.J.Phys. 72 (2004) 170-184

An elementary introduction to perturbative renormalization and renormalization group is presented. No prior knowledge of field theory is necessary because we do not refer to a particular physical theory. We are thus able to disentangle what is specific to field theory and what is intrinsic to renormalization. We link the general arguments and results to real phenomena encountered in particle physics and statistical mechanics.

The abstract says it all. And yeah, all the best for endsems everyone 🙂

Update : See also

Renormalization of a model quantum field theory(should be accessible inside IITK)

Per Kraus and David J. Griffiths
Physics Department, Reed College, Portland, Oregon 97202

(Received 9 December 1991; accepted 30 April 1992)

Renormalization is the technique used to eliminate infinities that arise in quantum field theory. This paper shows how to renormalize a particularly simple model, in which a single mass counterterm of second order in the coupling constant suffices to cancel all divergences. The model serves as an accessible introduction to Feynman diagrams, covariant perturbation theory, and dimensional regularization, as well as the renormalization procedure itself. ©1992 American Association of Physics Teachers

Posted by : Loganayagam.R.(Tom)

Time Reversal Invariance Saturday, Apr 8 2006 

There is something about the time reversal principle that I dont completely understand.

Consider an isolated gas of molecules that has undergone expansion in volume. Now, we know that Newtonian dynamics obeys time reversal symmetry, ie if I reverse the velocities of all particles in the system at some instant, the evolution from then on would be backward. In other words the system would retrace its path. Atleast , this definitely holds when there are no velocity dependent interactions, which we can assume here for the sake of simplicity.
However, we also know for sure (from both second law of thermodynamics and common experience) that isolated gas molecules would never contract (statistically speaking) and decrease in overall volume.

How then does one account for this apparent contradiction?

Posted By :Venkateshan

A Question in Real Space Renormalization Group Monday, Apr 3 2006 

Background: Scaling, Basic real space RG

The critical surface is formed of points in the phase space that have all relevant parameters = 0. Thus, any point on the critical surfact will move, under RG transformation towards the fixed point. It is because of this reason that we conclude that all points on the critical surface have the same critical exponents, although they may have different values of Tc. This also brings in the concept of universality, in that many different critical instances of the hamiltonian have the same exponents.

It seems to me that underlining this is an assumption that scaling does not change critical exponents. Can anyone provide a mathematically rigorous proof of this fact?

Posted By: Aakash Basu