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Course Title: COLLECTIVE PHENOMENA AND PHASE TRANSITIONSCourse Number: 360603Semester: Spring Term 2013Theory Professor: Giancarlo Franzese (Dep. Fsica Fonamental)Theory Practices Professor: M. Carmen Miguel (Dep. Fsica Fonamental)Computer Practices Professor: Giancarlo Franzese (Dep. Fsica Fonamental)Coordinators: J. M. Sancho & J. Casademunt (Dep. Estr. Const. Mat.)

Credit hours: Total 64./

Activities: ! ! Theory 42

Theory Practices 14 Computer Practices 8

Course requirements:It is recommended that students have previously studied the subjects of Thermodynamicsand Statistical Physics.

Skills developed:

Cross skills: Critical thinking and self-criticism.

Specific: Understanding of the physical phenomena: having good understanding of themost important physical theories, their mathematical and logical structure and its

experimental support.

Course goals:After completion of the core course in Statistical Physics, we assume that the student isfamiliar with the methods of statistical mechanics of equilibrium (ensembles and partitionfunctions) and has applied these concepts to ideal systems. The syllabus of this course isrestricted to equilibrium and focuses on applications to systems of interacting particle: realgases, liquids and ferromagnets, with emphasis on phase transitions and criticalphenomena.

Methods of evaluation:1. The rating takes into account home-works, home-works presentation, tests: 20%.2. The completion of all computer practices is mandatory to pass the course. Practices

reports: 10%.3. A final theory exam: 20%.4. A final problems exam: 50%.

For those choosing the Single Evaluation: To approve the tests 2, 3, 4 at the previouspoint: 100%

Re-evaluation: Same as Single Evaluation.

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Required and optional texts:

[H] HUANG, K. Statistical mechanics. 2nd Edition, Wiley, 1987.

[Pa] PATHRIA, R. K. Statistical mechanics, 2nd Edition, Elsevier, 1996.

[Pl] PLISCHKE, M.; BERGENSEN, B. Equilibrium statistical physics. 3rd Edition, World Scientific,2006.

[S] STANLEY, H. E. Introduction to phase transitions and critical phenomena, Oxford UniversityPress, 1987.

[G] GOODSTEIN, D. L. States of matter, Dover, 1975.

[Sa] SANCHO, J. M. Fsica Estadstica: Sistemas en Interaccin, Ed. Rey 2012.

[Mc] McQUARRY, D. A. Statistical Mechanics, HarperCollins, 1973

[Ma] MA, S-K. Statistical Mechanics, World Scientific, 1985

[R] REICHL, L.E., A Modern Course in Statistical Physics, 2nd Edition, John Wiley & Sons,1998

[F1] FRANZESE, G.; MALESCIO, G; SKIBINSKY, A; BULDYREV, SV; STANLEY, HE, PhysicalReview E, 66, 051206 (2002).

[F2] FRANZESE, G. "Monte Carlo dynamics for spin models", in "Frustrated Systems: ClusterDynamics and Precursor Phenomena", Ph.D. Thesis by G. Franzese, 1998.

[F3] FRANZESE, G. Scaling and Migdal-Kadanoff Renormalization: Introduction and application toPercolation and Potts Model, 1995.

[K] KADANOFF, L. P. Statistical Physics - Statics, Dynamics and Renomalization, WorldScientific, 2000

Course description

Syllabus with References (in parenthesis)

Part I. Classical Fluids: Real Gasses and LiquidsBasics Concepts

Summary of canonical and grancanonical ensembles and their relation to theThermodynamics. ([Sa] Appendix A, B)

Interaction potentials. ([Sa] Cap. 1); ([Pa] 11.1)

Thermodynamic Limit and 1st Yang-Lee Theorem. ([Sa] Cap. 1); ([Ma] Cap. 9)

Examples. ([Sa] Cap. 1); (List of problems)

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Real Gases

The Virial expansion. ([Pa] Cap. 9); ([Mc] Cap. 12)

The Second Virial coefficient. ([Pa] Cap. 9); ([Mc] Cap. 12)

Higher-Order Virial coefficients ([R] 9.C.3); ([Mc] Cap. 12)

Examples. ([Pa] Cap. 9); ([R] 9.C.2.1); ([Mc] Cap. 12); (List of problems)

Liquids

Radial distribution function and Thermodynamics ([Pa] 14.2); ([Mc] Cap. 13); ([R] 9.B);([S] 7.1-7.2)

Ornstein-Zernike Equation. ([F1] III, III.A, III.B, III.C); ([S] 7.4)

Molecular Dynamics. (Practice)

[PT] Examples. (List of problems, including the relation between the pair correlationfunction and the scattering of electromagnetic radiation) ([S] 7.3)

Part II. Phase Transitions and Critical Phenomena

Introduction

Phenomenology and Thermodynamics of phase transitions. ([S] 1.1)

Coexistence of phases: Gibbs phase rule. ([R] 3.B)

Classification of Phase transitions. ([R] 3.C)

[PT] Clausius-Clapyron Equation ([R] 3.D.2)

Modern era of Critical Phenomena: Critical exponents near a critical point. ([S] 1.2)

Phase transition in other systems. ([S] 1.3)

Critical point exponents

Definition of a critical exponents. ([S] 3.1)

The different exponents. ([S] 3.2)

Their numerical values. ([S] 3.3)

Useful relations among them. ([S] 3.5)

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Inequalities among them. ([S] 4.1-4.2)

Classical Theories of Cooperative Phenomena: Mean field

The van der Waals Theory ([S] 5.1-5.3)

The law of corresponding states for the van der Waals equation of state ([S] 5.4)

[PT] Critical-point exponents for the van der Waals Theory ([S] 5.5)

van der Waals equation of states as mean field theory ([S] 5.6)

Magnetic phase transitions: the mean field solution ([S] 6.1-6.3.4)

The law of corresponding states for the m.f. ferromagnetic equation of state ([S] 6.3.6)

Mean field as an approximation for the Heisenberg model ([S] 6.4)

Mean field theory as an infinite interaction range: the Kac model ([S] 6.5)

Lattice Models

Mapping to other equivalent systems: Lattice gas; Binary alloy ([H] 14.1-2)

[PT] Other lattice models.

Ising Model

Monte Carlo Dynamics. (Practice)

Spontaneous Symmetry Breaking. Lower Critical Dimension ([H] 14.3)

Bragg-Williams Approximation ([H] 14.4)

One-Dimensional Ising Model ([H] 14.6)

Temperature expansion series ([Sa] 5.4.3; 8.3)! ! ! ! ! ! ! (Suggested further reading [S] Cap 9) [PT] Bethe-Peierls Approximation ([H] 14.5)

Landau Theory

Expansion about the critical point and assumptions ([S] 10.1-2)

Critical point exponents calculation and criticisms ([S] 10.3-4)

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Field Theory formulation and the correlation length calculation ([H] 17.1; 17.4)

The Gaussian Model. Upper Critical Dimension. The Ginzburg Criterion. Anomalousdimensions ([H] 17.7-9)! ! ! ! (Suggested further reading: First-Order Transition [R]3G.2;

Tricritical point [H] 17.6)

Modern Theories of Phase Transitions

Scale Invariance and Renormalization. Formal approach to Renormalization and Blockspins. ([F3] 1.1-1.2.1)! ! ! ! ! ! (Suggested further reading: [K] 12.1-3; 12.6;

[H] 18.1-18.4;[S] 12.1)

Relations among the critical point exponents. ([F3] 1.2.2)

! ! ! ! ! ! ! (Suggested further reading: [K] 12.5;[S] 11.1-3)

Migdal-Kadanoff renormalization. ([F3] 1.3)(Suggested further reading: Application to the Percolation and Potts model [F3] 2.2-2.3; ! ! ! ! ! ! Application to the 1d Ising model [H] 18.2)

Part III. Computer Practices

Practice 1: Introduction to Molecular Dynamics of phase transitions and coexistence inArgon-like systems using a Lennard-Jones interaction potential.

Practice 2: Introduction to Monte Carlo Dynamics: Importance Sampling; DetailedBalance and Ergodicity; Metropolis and Heat Bath. Application to the Ising Model ([F2]).

Practice 3: Percolation. Basic concepts. Site and Bond Percolation. Lattice-dependentpercolation thresholds. Finite-size analysis of Percolation probability and Mean clustersize. Critical exponents.

Practice 4: Renormalization group applied to Percolation. Percolation and Potts model([F3] 2.1). Percolation applied to Cluster Monte Carlo Dynamics of Potts model.

GUIDELINES for Practice Reports

For all Reports: The Report should be organized as a short paper (in English) with:

TitleAuthors

AbstractMain textReferences

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The Main text must be divided in sections:

Introduction. (What is the aim of the practice?)

Method. (Which applet was used and how? How did you planned to analyze your results to

reach the aim of the practice? Define the quantities you will calculate.)

Results. (Elaborate the data collected and show here the plots with statistical errors.Include here the plot you will need to discuss the comparison of your results with theexpected results, for example the plot for the calculation of the critical exponents or of thecritical point. If you want to include tables of data, do it in an appendix, not in the maintext. )

Discussion. (Compare your results with the theoretical predictions. Do they compare well?Why? Do they not compare well? Why? At least make some reasonable guess).

Conclusions. (What did you find?)

SPECIFIC NOTES FOR EACH PRACTICE.

Practice 1: Molecular Dynamics of Lennard-Jones interaction potential.

a) Use the public freeware releases of the Virtual Molecular Dynamics Laboratory suite ofsoftware tools at http://argento.bu.edu/vmdl/Software/index.html

Download the software and run the application.

Go to Home>Additional Files>Sergey's Manual> Experiment 1Select the Heat Bathoption.Press Show Additional Parametersandselect No Walls (periodic).

In the Optionmenu, select Averaging for Simulations.Run the simulations withSelect Show Averages.

Run the simulations and use in Filemenu the Start Saving Datacommand to generate thedata files.

The code allows you to change the temperature and the number density. Change theseparameters and find the liquid-gas critical point and the coexistence of different phases.Discuss how you decide if your system is equilibrated by studying the Energies plot asfunction of time at each temperature and density.

Plot the isotherms in the Pressure - density plane and isobars in the Temperature - densityplane and discuss.

b) Use the public freeware athttp://www.physics.orst.edu/~rubin/CPUG/CPlab/MoleDynam/md.html

http://argento.bu.edu/vmdl/Software/index.htmlhttp://argento.bu.edu/vmdl/Software/index.html

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Here you can change temperature and density. Follow the instructions at the web page torun the code. Study the phase coexistence by varying temperature and density, plottingthe RDF and the other quantities. Discuss and compare with the result of part a).

Ask Prof. Franzese by email about a couple of useful references to compare with.

Practice 2: Monte Carlo Dynamics of the Ising Model in 2d.

a) Usethe applet athttp://ifisc.uib-csic.es/research/applet_complex/bidimensional/JIsing.phpfor a qualitative understanding of the 2d Ising Model phase diagram as function oftemperature T and external magnetic field H.

Check how the magnetization M changes by decreasing T from high to low at H=0.Estimate the Tc for this case.

Repeat the analysis changing T from low to high and estimate Tc in this case. Cmpare withthe previous and discuss.

At fixed T

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Practice 4: Cluster Monte Carlo Dynamics of Potts model.

Use the applet at http://spot.colorado.edu/~beale/PottsModel/MDFrameApplet.html

to study the Potts model for q=3, 4, 5, 6.

Select the size L of the system, by changing the size of the window.Select the WolfAlgorithm.Select the Temperature Tand the external field H.

For each qstudy how the energy E, the specific heat C, the magnetization Mand thesusceptibility(calculated from the fluctuation-dissipation relation) change with Tfor H=0for fixed large L.

Compare the cases q=3and 4and the case q=5and 6.Then compare the cases q=3and 6.Discuss the differences.

For each case estimate the critical temperature Tc(L)/qfor several sizes and extrapolate inthe thermodynamic limit by plotting Tc(L)/q vs 1/L. Compare with the theoretical valueTc/q=1/[ln(1+q1/2)].

Then for q=3and q=4estimate the critical exponents /and /by plotting ln vs ln L

and ln vs ln L , respectively, at Tc(L). Compare with the expected values that can becalculated knowing that the two independent critical exponents associated to the thermal

field and the magnetic field arefor q=3: yt=6/5, yh=28/15for q=4: yt=3/2, yh=15/8.

Discuss the results.

http://spot.colorado.edu/~beale/PottsModel/MDFrameApplet.htmlhttp://spot.colorado.edu/~beale/PottsModel/MDFrameApplet.html