Central Limit Theorem

An interactive illustration of the Central Limit Theorem. You will see a Gaussian distribution emerge by casting dice.

Fractal demo

An interactive illustration of how fractals are computed, including zooming in features. It includes the von Koch snowflake, see under fractals. (Denis Rapaort)

Also, a 3D trip through a Sierpinski gasket.

Faraday waves on the surface of water

These are not quite spontaneous fluctuations on the free surface of water, but rather acoustically excited by a speaker. But it will give you a nice image of different surface configurations as a function of time in this driven system, not unlike what you would observe (at a much smaller scale) in a spontaneously fluctuating water- vapor interface.

Ising model in two dimensions

An interactive applet that performs an equilibrium simulation of the 2D Ising model. It plots the configuration, as well as the value of the average magnetization and energy. Temperature and initial configuration can be adjusted.

Monte Carlo simulation of protein folding

A Monte Carlo simulation of a coarsed grained model describing the folding of the protein GB1. The search in the protein configuration space is carried out with the Metropolis algorithm and an energy function that is largely empirical. It treats the aminoacids in the protein as interaction centers. The forces between the centers are largely empirical with coefficients determined by fits to know structures in the protein data bases.

A history of the Renormalization Group

"The renormalization method is specifically aimed at connecting theories describing physical processes at different length scales and thereby connecting different theories in the physical sciences. The renormalization method used today is the outgrowth of one hundred and fifty years of scientific study of thermal physics and phase transitions. Different phases of matter show qualitatively different behavior separated by abrupt phase transitions. These qualitative differences seem to be present in experimentally observed condensed-matter systems. However, the "extended singularity theorem" in statistical mechanics shows that sharp changes can only occur in infinitely large systems. Abrupt changes from one phase to another are signaled by fluctuations that show correlation over infinitely long distances, and are measured by correlation functions that show algebraic decay as well as various kinds of singularities and infinities in thermodynamic derivatives and in measured system parameters.

Renormalization methods were first developed in field theory to get around difficulties caused by apparent divergences at both small and large scales.

The renormalization (semi-)group theory of phase transitions was put together by Kenneth G. Wilson in 1971 based upon ideas of scaling and universality developed earlier in the context of phase transitions and of couplings dependent upon spatial scale coming from field theory. Correlations among regions with fluctuations in their order underlie renormalization ideas. Wilson's theory is the first approach to phase transitions to agree with the extended singularity theorem.

Some of the history of the study of these correlations and singularities is recounted, along with the history of renormalization and related concepts of scaling and universality. Applications are summarized".

Article.

Critical exponent inequalities

This paper discusses simple but rigorous proofs of several of the critical exponent inequalities described in class.

Scaling, universality, and renormaliztion: Three pillars of modern critical phenomena

This paper prsents an overview of the concepts of scaling and renormalization in critical phenomena.

Examples of coarse grained kinetic equations

Ten component, non conserved scalar order parameter

This is a 10 component, purely relaxational model. Motion is driven by interfacial curvature reduction (excess free energy reduction). Vertices reach local equilibrium quickly and move adiabatically with the interfaces.

One dimensional Complex-Ginzburg Landau Equation

This is an example of a "non potential" model that leads to spatio temporal chaos (x axis is space, y axis is time).

Same equation but in two dimensions.

Coarsening of a modulated phase

In the symmetry that is spontaneously broken corresponds to a one dimensional solid. The model is purely relaxational and coarsening leads to an equilibrium state. Note the existence of topological defects (dislocations, disclinations, and grain boundaries).