statphysII

Bf. 27: Statistical Physics II, Fall 2002

Hans Fogedby

STATISTICAL PHYSICS

"Statistical physics enjoys a very special position among the subfields of physics. Its principles hold whenever there are many (more than three?) particles or subunits interacting with each other to make a "complex" system. Its subject vary from microsopic and mesoscopic scales to macroscopic and cosmic scales. Its working tools range from rigorous mathematics to numerical simulations. Its applications extend to almost all branches of natural sciences and, perhaps, economuy and sociology."
(see also About statistical physics)

List of notes handed out (dvi-format) (ps-format) (pdf-format)

Week 36 (Sep 2 - Sep 9)

In the this lecture we reviewed briefly the thermodynamics of phase transitions, the phase diagram for a fluid, and the van der Waal equation. We then turned to magnetic systems and introduced the Ising model as the simplest nontrivial interacting system with a phase transition. We also showed some links about the history of thermodynamics and statistical mechanics and about phase transitions.

Links
History of statistical mechanics and thermodynamics
Physics Encyclopędia
Statistical mechanics course (Tuckerman, NYU)
Statistical Thermodynamics of Materials (Veytsman and Kotelyanskii, Penn State)
Carbon dioxide at its critical state
Phases of matter
Phase transitions and phase equilibria (Penn State)
Ising model and its applications (Penn State)
Simulation of the Ising model (Young, Santa Cruz)

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Week 37 (Sep 9 - Sep 16)

In this lecture we returned to the Ising model and discussed mean field theory. We derived the equation of state and showed that MFT predicts a phase transition. We then developed the MFT approximation for the free energy, the Braggs-Williams approximation and determined the stable thermodynamic MFT state. Finally, we showed Landau's argument for the absence of a phase transition at a finite temperature in one dimension and Peirl's estimate for the critical temperature in a 2-D Ising model.

Links
Introduction to the Ising Model (Sethna, Cornell)
Three-dimensional proof for Ising Model impossible, Sandia researcher claims

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Week 38 (Sep 16 - Sep 23)

In this lecture we discussed the Metropolis importance sampling Monte Carlo method in the context of simulating statistical physics systems in thermal equilibrium. We showed how to adjust the transition probabilities in order to ensure detailed balance and thus characterize the interaction of the system with a heat bath. Using the link from Santa Cruz we next showed a Java applet for the Monte Carlo simulation of the Ising model. Finally, we discussed the properties of the mean field theory approximation as regards the critical point and introduced the critical exponents characterizing the phase transition.

Links
Monte Carlo Simulations for Statistical Physics (Coddington, Syracuse)
Markov Chain Monte Carlo (Minnesota)
Monte Carlo Simulation
Monte Carlo Methods and The Metropolis Algorithm (Caltech)
The Monte-Carlo method (Demidov, Nizhny Novgorod)
Monte Carlo Methods and Simulation (MacKinnon, Imperial College)
Monte Carlo methods and the Metropolis algorithm
Simulations of Ising models (de Araujo, Brandeis)
Monte Carlo Methods (Harada, Kyoto)
Quantum Monte Carlo: Mostly Path Integrals
Nicholas Constantine Metropolis

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Week 39 (Sep 23 - Sep 30)

In this lecture we reviewed the critical exponents and derived the correlation function for the 1D ising model in the absence of a magnetic field. We used a trick consisting of generalizing the Ising Hamiltonian to a model with a site dependent exchange coupling - a model which hsa been used in the context of disordered magnetic systems. We then solved the Isisng model i 1D using the transfer matrix method. In the thermodynamic limit the free energy is given in terms of the largest eigenvalue of the transfer matrix. We discussed the phase diagram and corroborated the result of the Landau argument, that there is no phase transition at a finite temperature. Finally, we started to discuss Landau theory for second order phase transitions.

Links
Ising Model (Wolfram Research)
Java Applets for Statistical and Thermal Physics
Critical Phenomena
Fluctuations and Critical Phenomena
One dimensional Ising model
Two-dimensional Ising Model
Nobel Prize 1982, Critical Phenomena, Wilson

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Week 40 (Sep 30 - Oct 7)

In this lecture we discussed Landau theory and showed that the Landau theory yields the mean field exponents. We then turned to the Ginzburg-Landau theory where the free energy is given in terms of a functional of the nonuniform magnetization. We derived the the magnetization induced by a localized field and derived the order parameter correlation function. Using the fluctuation-dissipation theorem we then showed that the long range behavior of the correlation function near the critical point is consistent with the divergent behavior of the susceptibility. Finally, we went through the Ginzburg criterion which shows that mean field theory or Landau theory is valid above four dimensions - foru being the so-called critical dimension.

Links
Review of Kadanoff's "Statistical Physics" by Paul Martin

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Week 41 (Oct 7 - Oct 14)

In this lecture we reviewed the scaling properties and the critical exponents. We then turned to the scaling ansatz for the free energy and derived two scaling laws. Finally, we turned to the conceptually important Kadanoff block construction and derived two new scaling laws. We commented on universality classes.

Links
Critical phenomena by light scattering
Press Release: The 1982 Nobel Prize in Physics

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Week 43 (Oct 21 - Oct 28)

Exercises moved to 520-731 on Monday October 21

In this lecture we gave a little overview of soft condensed matter. We then turned to a discussion of the the ideal chain model for a polymer and showed that the coil size, i.e., the radius of gyration scales with the number of monomers to the power 1/2 - a typical universal scaling behavior. Finally, we started using a random walk model to describe the conformational stratistics of polymers.

Links
Soft condensed matter network
Soft matter (Paris)
Principles of Soft Matter (21st CNLS)
Soft condensed matter (Penn)
Soft matter (Hahn-Meitner)
Soft matter (Juelich)
Soft condensed matter physics (Lubensky)
Polymers and liquid Crystals (lecture notes)
Building with snakes - the physics of long chain molecules
Theory of LC biomembrane-- a probe into soft condensed matter physics

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Week 44 (Oct 28 - Nov 4)

In this lecture we derived Flory theory and found expression for the scaling exponents. We also derived the Edwards-Doi field theoretical formulation.

Links
Soft Matter, Nobel lecture, de Gennes
Nobel Prize, 1991, de Gennes
de Gennes, Nobel 1991
Nobel Prize, de Gennes, 1991
Soft matter and other stuff, Sethna (Cornell)
Soft matter at Penn
Polymer fun
Fun Physics (Leeds)

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Week 45 (Nov 4 - Nov 11)

We continued the discussion of polymers and membranes. We discussed briefly solid and fluid membranes and derived the Edwards generalization. We also discussed the bending rigidity and the persistance length.

Links
Stochastic Processes in Physics (Birger Bergersen)
Einstein's Explanation of Brownian Motion
Particles Diffusion, Java Applet
Brownian Motor
Membranes
Statistical Mechanics of Membranes
Quantum Gravity Simulation
Random Surfaces and Quantum Gravity
Random Surfaces and Simplicial Quantum Gravity

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Week 46 (Nov 11 - Nov 18)

In this lecture we started our discussion of quantum liquids. We then discussed the ideal Bose gas and the concept of a Bose condensate. We introduced ODRL and discussed Penrose and Onsager's ideas.

Links
Chemical of the week, polymers
Mechanical Properties of Polymers
POLYMERS AND PEOPLE: An Informal History
What are polymers and why are they interesting?
Polymers are everywhere
Polymers
Liquid hydrogen turns superfluid
Soft Matter, deGennes Nobel Lecture
Alkali Quantum Gases, Ketterle's group at MIT
Liquid Helium
Bose Einstein Condensates Nobel Prize 1962, Landau
HyperPhysics
HyperMath
BOSE SYSTEMS
Jila Bec Homepage
BEC Homepage, Colorado
Reprint and Preprint Bibliography for BEC
Quantum Simulations of Condensed Matter Systems

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Week 47 (Nov 18 - Nov 25)

In this lecture we continued our discussion of Bose systems. We discussed the absence of a phase transition in an ideal Bose gas in dimensions below 2. We then turned to Landau's argument for superfluidity in a Bose fluid. Finally, we started on the two-fluid model and the Gross-Pitaevski equation for the condensate wave function.

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Week 49 (Dec 2 - Dec 9)

In this lecture we finished our discourse on Bose systems and in particular liquid Helium. We discussed in particular vortex quantization and the fountain effect. We then went on to a little discussion of quasiparticles in condensed matter physics.

Links
Stochastic Processes in Physics, Birger Bergersen
Superfluidity in Helium-3
He 3, The Original Paper
What Is Superfluidity
Superfluidity and Quantized Vortices, He-3

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Week 50 (Dec 9 - Dec 16)

Links
Advanced lecture notes in statistical physics

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Week 51 (Dec 16 - Dec 23)

In this lecture discussed response functions

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